Analysis and Design of Delayed Genetic Regulatory Networks [1st ed.] 978-3-030-17097-4;978-3-030-17098-1

Table of contents :
Front Matter ....Pages i-xxi
Backgrounds (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 1-18
Front Matter ....Pages 19-19
Stability Analysis for GRNs with Mixed Delays (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 21-56
Stability Analysis of Delayed GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 57-80
Stability Analysis for Delayed Switching GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 81-97
Stability Analysis for Delayed Stochastic GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 99-116
Stability Analysis for Delayed Reaction-Diffusion GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 117-154
Front Matter ....Pages 155-155
State Estimation for Delayed GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 157-181
Guaranteed Cost Control for Delayed GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 183-196
State Estimation for Delayed Reaction-Diffusion GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 197-220
\(H_{\infty }\) State Estimation for Delayed Stochastic GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 221-243
\(H_{\infty }\) State Estimation for Delayed Discrete-Time GRNs (Xian Zhang, Yantao Wang, Ligang Wu)....Pages 245-263

Citation preview

Studies in Systems, Decision and Control 207

Xian Zhang Yantao Wang Ligang Wu

Analysis and Design of Delayed Genetic Regulatory Networks

Studies in Systems, Decision and Control Volume 207

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control-quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

More information about this series at http://www.springer.com/series/13304

Xian Zhang Yantao Wang Ligang Wu •

•

Analysis and Design of Delayed Genetic Regulatory Networks

123

Xian Zhang School of Mathematical Science Heilongjiang University Harbin, China

Yantao Wang School of Mathematical Science Heilongjiang University Harbin, China

Ligang Wu School of Astronautics Harbin Institute of Technology Harbin, Heilongjiang, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-17097-4 ISBN 978-3-030-17098-1 (eBook) https://doi.org/10.1007/978-3-030-17098-1 Library of Congress Control Number: 2019936283 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Chunyan and Ruitong Xian Zhang To My Family Yantao Wang To My Family Ligang Wu

Preface

The research on genetic regulatory networks (GRNs) is multi-disciplinary, crossing biology, control science, computer science, electronic science mathematics, theoretical physics, etc. In the last two decades, mathematical models have become a powerful tool for studying of GRNs. In general, mathematical models of GRNs are divided into two classes: the discrete models and the continuous ones. In the continuous models, the variables describe the concentrations of mRNAs and proteins as continuous values, which can provide detailed understanding of the nonlinear dynamical behavior exhibited by GRNs. Recently, it has been shown that differential equation models including delayed states, named as delayed GRNs, can more accurately describe GRNs. As a result, much effort has been paid to the study of delayed GRNs, and many significant results have been reported in literature. From the point of control theory, the research on delayed GRNs includes mainly two aspects: analysis and design. In spite of the fact that there exist some Ph.D. theses related to delayed GRNs, there is no comprehensive book on this topic. The aim of this book is to provide an introduction for current advances of delayed GRNs and present the basic methods for analysis and design of delayed GRNs. The whole book is divided into 11 chapters and focuses on the analysis and design problems on continuous-time delayed GRNs except the last chapter. All the contexts are taken from the authors’ publications. This book is also intended to offer a collection of important references on analysis and design of delayed GRNs. The book is addressed to graduate students and research-level mathematicians. It is hoped that the book will be suitable for postgraduate use or as a reference. Many researchers in the world have made great contribution to analysis and design of delayed GRNs. Due to the length limitation and the structural arrangement, many of their published results are not included in the book. I would extend my apologies to these researchers. I would appreciate any comments and corrections from the readers. Please feel free to contact me by the e-mail: [email protected]. Harbin, China February 2019

Xian Zhang

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Acknowledgements

Two co-authors of this book and I have cooperated to study delayed GRNs for 5 years at least. My colleagues, Dr. Xiangyu Gao, Dr. Yu Xue, Dr. Jing ma, Dr. Yanjiang Li, and Dr. Guodong Zhang, and previous graduate students, Ms. Ahui Yu, Ms. Ying Zhou, Mr. Shaochun Cui, Ms. Tingting Yu, Ms. Jing Wang, Mr. Xingming Zhou, Ms. Jiahua Zou, Ms. Yuanyuan Han, Ms. Tingting Liu, and Ms. Xiaofei Fan, have put great efforts on research of delayed GRNs. My present graduate students, Mr. Ning Zhao, Ms. Xin Li, Ms. Haifang Li, Ms. Shasha Xiao, Ms. Xinxiao Liu, Ms. Lulu Sun, and Ms. Xinyue Zhang, all helped me to find the errors and typos in the manuscripts. Besides all the above, Ms. Xiaofei Fan has helped me with conducting a reasonable review of references, and Mr. Ning Zhao has helped me with the revision of mathematical symbols. Their help has greatly improved the quality of the manuscripts. Here, I would like to express my heartfelt appreciation of their contribution. My special thanks go to my wife Chunyan for her support during writing the book. I gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (No. 11371006), the Important Subjects Foundation of Heilongjiang University, and the Fund of Heilongjiang Education Committee (No. 12541603).

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Contents

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1 1 4 7 7 8 11 12 14

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Stability Analysis for GRNs with Mixed Delays . . . . . . . 2.1 Constant Distributed Delay Case . . . . . . . . . . . . . . . 2.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . 2.1.2 Existence of Nonnegative Equilibrium Points 2.1.3 Globally Asymptotic Stability Criteria . . . . . . 2.1.4 Numerical Examples . . . . . . . . . . . . . . . . . . 2.2 Unbounded Distributed Delay Case . . . . . . . . . . . . . 2.2.1 Model Description . . . . . . . . . . . . . . . . . . . . 2.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . 2.3 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stability Analysis of Delayed GRNs . 3.1 Problem Formulation . . . . . . . . . 3.2 An Improved Integral Inequality 3.3 Stability Criteria . . . . . . . . . . . .

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Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction to GRNs . . . . . . . . . . . . . . . . . . . . . 1.2 Functional Differential Equation Models of GRNs 1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Nonsingular M-Matrix . . . . . . . . . . . . . . . 1.3.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

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Analysis of Delayed GRNs

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3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stability Analysis for Delayed Switching GRNs 4.1 Model Description . . . . . . . . . . . . . . . . . . 4.2 Stability Criteria . . . . . . . . . . . . . . . . . . . . 4.2.1 Constant Time-Delay Case . . . . . . . 4.2.2 Time-Varying Delay Case . . . . . . . 4.3 Numerical Examples . . . . . . . . . . . . . . . . . 4.4 Remarks and Notes . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stability Analysis for Delayed Stochastic GRNs . 5.1 Model Description and Problem Formulation 5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Examples . . . . . . . . . . . . . . . . . . 5.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stability Analysis for Delayed Reaction-Diffusion GRNs . 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Infinite-Time Case . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Asymptotic Stability Criteria . . . . . . . . . . . . . 6.2.2 Theoretical Comparisons . . . . . . . . . . . . . . . 6.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . 6.3 Finite-Time Case . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Finite-Time Stability Criteria . . . . . . . . . . . . 6.3.2 A Numerical Example . . . . . . . . . . . . . . . . . 6.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 7

Design of Delayed GRNs

State Estimation for Delayed GRNs 7.1 Problem Formulation . . . . . . . . 7.2 Full-Order State Observer . . . . 7.2.1 Observer Design . . . . . 7.2.2 A Numerical Example . 7.3 Reduced-Order State Observer . 7.3.1 Observer Design . . . . . 7.3.2 A Numerical Example . 7.4 Remarks and Notes . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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Guaranteed Cost Control for Delayed GRNs . . . . . . 8.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 8.2 Design of Guaranteed Cost Controller . . . . . . . . 8.2.1 Existence of Guaranteed Cost Controllers 8.2.2 Design Method . . . . . . . . . . . . . . . . . . . 8.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . 8.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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State Estimation for Delayed Reaction-Diffusion GRNs 9.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 9.2 Infinite-Time State Estimation . . . . . . . . . . . . . . . . 9.2.1 Observer Design . . . . . . . . . . . . . . . . . . . . 9.2.2 Numerical Examples . . . . . . . . . . . . . . . . . 9.3 Finite-Time State Estimation . . . . . . . . . . . . . . . . . 9.3.1 Observer Design . . . . . . . . . . . . . . . . . . . . 9.3.2 Numerical Examples . . . . . . . . . . . . . . . . . 9.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 H 1 State Estimation for Delayed Discrete-Time GRNs 11.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 11.2 H1 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . 11.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 H 1 State Estimation for Delayed Stochastic GRNs . 10.1 Model Description . . . . . . . . . . . . . . . . . . . . . 10.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . 10.4 Remarks and Notes . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Notations and Acronyms

R Rnm Rn diagðx1 ; . . .; xn Þ or Dx colðx1 ; . . .; xn Þ kmax ðAÞ kmin ðAÞ vj ðAÞ vðAÞ jAj Q P In or I 0mn or 0 AT A detðAÞ trðAÞ qðAÞ symðAÞ A1 hmi  X>Y or Y6X X [ Y or Y\X L2 ½0; 1Þ J CðJ; Rn Þ k k2

Field of real numbers Set of all n  m matrices over R Set Rn1 Diagonal matrix Column vector Maximum eigenvalue of real symmetric matrix A Minimum eigenvalue of real symmetric matrix A Number of nonzero elements in the jth row of A diag ðv1ð
AÞ;
v2 ð AÞ; . . .; vn ð AÞÞ Matrix
aij
with A ¼ aij Product sign Sum sign n  n identity matrix m  n zero matrix Transpose of the matrix A Conjugate transpose of the matrix A Determinant of the square matrix A Trace of the square matrix A Spectral radius of the square matrix A Matrix A þ AT Inverse of the nonsingular matrix A Set f1; 2; . . .; mg Hadamard produce X  Y is real symmetric positive semi-definite X  Y is real symmetric positive definite Set of square integrable functions over ½0; 1Þ Connected subset of R Linear space of all continuous functions h : J ! Rn Euclidean norm on Rn , or its induced norm

xv

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Cðð1; 0; Rn Þ kwkC R @R C 1 ðR; Rn Þ k k k kd

Efg L A B or B A A  B or B A

Notations and Acronyms

Linear space of all bounded and uniformly continuous functions w : ð1; 0 ! Rn Norm on Cðð1; 0; Rn Þ defined by R0 kwkC ¼ sup1\s60 kwðsÞk2 þ 1 kwðsÞk2 ds Compact set in Rl Boundary of R Banach space of continuous differential functions mapping R into Rn Norm on C 1 ðR; Rn Þ defined by R 1=2 kyðxÞk ¼ R yT ðxÞyðxÞdx Norm on C 1 ð½d; 0  R; Rn Þ defined by k/ðt; xÞkd ¼ max sup k/ðt; xÞk; d6t60     @/ðt; xÞ @/ðt; xÞ     ; max sup sup  @t  1  k  l d6t60 @xk  d6t60 Mathematical expectation operator Weak infinitesimaloperator    Real matrices A ¼ aij and B ¼ bij satisfy aij 6bij for all i and j     Real matrices A ¼ aij and B ¼ bij satisfy aij \bij for all i and j

Next, we will briefly explain several phrases which are helpful to understand this book. • The LKF is a class of nonnegative functionals acting on a space of functions. • Jensen’s inequality relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities and results in a linear combination of positive functions weighted by the inverses of convex parameters. • The free-weighting matrix approach introduces parameter matrices, indicating the relationship between the terms in the Leibniz–Newton formula, into LMIs to be solved. • The convex technique simplifies LMIs, including a linear combination of finite items weighted by convex parameters, by using the property of convex functions. • The reciprocally convex technique estimates the lower boundary of a combination of positive functions weighted by the inverses of convex parameters. • The Wirtinger-type integral inequality is a class of inequalities that are a generalization of Jensen’s inequalities and that are more accurate than Jensen’s inequalities. • A delay-dependent(-independent) result indicates solvable conditions to a problem on time-delay systems are (not) related to the information of delay.

List of Figures

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15

Trajectories of mRNA and protein concentrations (Example 2.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.41) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.41) . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.41) . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 2.19 Fig. 2.20 Fig. 3.1 Fig. 3.2 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 5.1 Fig. 5.2 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 8.1 Fig. 8.2

List of Figures

Trajectories of mRNA and protein concentrations (Example 2.41) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 2.42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 3.11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 3.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching signal (Example 4.7) . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brownian motions (Example 5.5) . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 5.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations when rðtÞ 1:2 and ¿ðtÞ 1 (Example 6.20) . . . . . . . . . . . . Trajectories of mRNA and protein concentrations when rðtÞ 6 and ¿ðtÞ 1 (Example 6.20) . . . . . . . . . . . . . Trajectories of mRNA and protein concentrations (Example 6.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The mRNA concentrations and their estimations (Example 7.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The protein concentrations and their estimations (Example 7.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation errors (Example 7.4) . . . . . . . . . . . . . . . . . . . . . . . The mRNA and protein concentrations and their estimations (Example 7.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The mRNA and protein concentrations and their estimations (Example 7.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The mRNA and protein concentrations and their estimations (Example 7.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation errors (Example 7.8) . . . . . . . . . . . . . . . . . . . . . . . Trajectories of mRNA concentrations (Example 8.7) . . . . . . . Trajectories of protein concentrations (Example 8.7) . . . . . . .

..

52

..

52

..

52

..

53

..

53

..

75

.. ..

76 95

..

95

..

96

.. 96 . . 112 . . 112 . . 138 . . 139 . . 153 . . 163 . . 164 . . 164 . . 178 . . 179 . . . .

. . . .

179 179 194 194

List of Figures

Fig. Fig. Fig. Fig.

9.1 9.2 9.3 9.4

Fig. 9.5 Fig. 9.6 Fig. 10.1 Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 11.1 Fig. 11.2

The mRNA concentration and its estimation (Example 9.7) . . The protein concentration and its estimation (Example 9.7) . . Estimation errors (Example 9.7) . . . . . . . . . . . . . . . . . . . . . . . The mRNA concentration and its estimation (Example 9.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The protein concentration and its estimation (Example 9.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation errors (Example 9.12) . . . . . . . . . . . . . . . . . . . . . . Wiener processes (Example 10.12) . . . . . . . . . . . . . . . . . . . . . The mRNA concentrations and their estimations (Example 10.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The protein concentrations and their estimations (Example 10.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation errors (Example 10.12) . . . . . . . . . . . . . . . . . . . . . The mRNA concentrations and their estimations (Example 11.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The protein concentrations and their estimations (Example 11.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

. . 212 . . 212 . . 212 . . 219 . . 219 . . 220 . . 240 . . 241 . . 241 . . 241 . . 261 . . 261

List of Tables

Table Table Table Table

3.1 3.2 4.1 5.1

Table 6.1 Table 8.1 Table 9.1 Table 9.2 Table 9.3

Maximum values of ¿ 2 (Example 3.11) . . . . . . . . . . . . Maximum values of ¿ 2 (Example 3.12) . . . . . . . . . . . . Maximum allowable delays (Example 4.6) . . . . . . . . . Maximum values of r2 with different partitions (Example 5.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  with different ¿d ¼ r d ¼ l Maximum values of ¿ ¼ r (Example 6.19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The maximum values of ¿ 2 (Example 8.7) . . . . . . . . .  for different ¿d ¼ r d ¼ l Maximum values of ¿ ¼ r (Remark 9.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numbers of decision variables of different methods (Remark 9.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  with different ¿d ¼ r d ¼ l Maximum values of ¿ ¼ r (Remark 9.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....... ....... .......

75 76 94

. . . . . . . 113 . . . . . . . 136 . . . . . . . 194 . . . . . . . 209 . . . . . . . 209 . . . . . . . 217

xxi

Chapter 1

Backgrounds

In this chapter we will briefly introduce some background knowledge related to Genetic Regulatory Networks (GRNs).

1.1 Introduction to GRNs GRNs are collections of DNA segments in a cell which interact with each other indirectly through their mRNAs, protein expression products and other substances. The main processes of gene expression are gene transcription and translation of mRNAs. The research on GRNs is multi-disciplinary, crossing biology, control science, compute science, electronic science and theoretical physics, etc [30, 37, 42, 89, 91–93, 99]. Thus it is currently great interest to many students, scholars and experts (see [16, 17, 19, 20, 22, 82] and the references therein). With the appearance and development of DNA microarray technology [48], it has become possible to measure gene expression levels on a genomic scale, and further analyze GRNs. As a result, GRNs can help people understand the genome sequencing and the gene recognition. Mathematical models have been used for the study of GRNs in the last two decades. As we know, weighting matrix model is the first model used in the research of GRNs. It represents the mutual regulation impact among genes, however, it leads to a great amount of computation because of the large number of genes and the weighting matrices [76]. In order to avoid it, Boolean algebraic model is established, based on this model, Boolean networks depending on Boolean functions provided a framework for describing the complex interactions among genes [2]. After that, correlation matrices are used to reconstruct GRNs from gene expression. In addition, correlation coefficient method had been successfully applied to selecting of Drug NC160 [21]. Based on it, Butte and Kohane established a network of relationships between genes and drugs [10]. Followed by it, linear combination model and © Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_1

1

2

1 Backgrounds

functional differential equation model are used to simulate GRNs [15], which are more effective to describe the nonlinear dynamical behavior, however, it also leads to much more computation time than Boolean model. The mathematical models of GRNs have become a powerful tool for studying genetic regulatory processes in living organisms, and they can be roughly divided into two types: the discrete models [2, 6, 11, 16, 22, 39, 54, 59, 75, 76] and the continuous models [12, 15, 17, 19, 28, 63, 74]. In the continuous models, the variables describe the concentrations of mRNAs and proteins as continuous values, which can provide detailed understanding of the nonlinear dynamical behavior exhibited by GRNs. Usually, a continuous model is described by a differential equation. However, in GRNs, mRNAs and proteins may be synthesized at different locations; thus, the transcription or the diffusion of mRNAs and proteins among these locations results in sizable delays [12]. Therefore, theoretical models without consideration of delay may even provide wrong predictions. So, differential equation models including delayed states, named as delayed GRNs, can more accurately describe GRNs, and hence it can better show the nature of life. As a result, much effort has been paid to the study of delayed GRNs, and many significant results have been reported in literature on the stability analysis [4, 39, 41, 53, 73, 90–92, 96], passivity analysis [42], Hopf bifurcation analysis [79, 85, 86], controller synthesis [14, 27, 30, 40], estimator design [44, 74, 89, 99], identifying unknown parameters [13, 62], design [33], and so on. As we all known, stability is one of the most important properties for any dynamic system. So, it is important and necessary to analyze stability of delayed GRNs. There are mainly two methods to establish stability criteria for delayed GRNs: the linear matrix inequality (LMI) method [12, 36, 38, 52, 71, 73, 91] and the M-matrix method [64–66, 77, 78, 93]. The LMI method-based stability criteria are generally effective for reducing conservativeness, but they are computationally complex; while the Mmatrix method-based stability criteria are more computationally simple, because they just need to verify whether a constant matrix is a nonsingular M-matrix. Key points of the LMI method are the constructions of Lyapunov–Krasovskii functionals (LKFs) and the employments of analysis techniques, which determines the resultant stability criterion to be less or more conservative at a certain point. In order to reduce conservativeness of LMI method-based stability criteria, some useful approaches have been introduced, e.g., delay decomposition approach [87, 98], reciprocally convex combination approach [70], augmented LKF approach [69], free-weighting matrix approach [69, 80, 88, 97] and convex combination approach [88]. These approaches are generally available for reducing conservativeness, but they will also increase the number of LMIs or of variables in LMI(s), which results in the computational complexity. For this reason, a so-called M-matrix-based approach has been proposed in [66, 77, 78] to infer the stability for equilibrium points of delayed GRNs. Since modeling GRNs is an approximate process, it is necessary to introduce uncertainties into GRN models. As a result, much effort has been paid to establish robust stability criteria of uncertain delayed GRNs, and many significant results have been reported in literature [34, 38, 52, 68, 73].

1.1 Introduction to GRNs

3

For individual molecules, since movement of mRNA from a transcription site to translation sites is an active process with a significant range of transport times, so it is significant and necessary to model GRNs by using functional differential equations with mixed (i.e., discrete and distributed) delays [26, 92]. In addition, it is too simple to express the movement of macromolecule in actual networks only with distributed delay [87]. And stability criteria for GRNs only with discrete (distributed) delays are generally unavailable for GRNs with mixed delays. For this reason, the stability analysis for equilibrium points of GRNs with mixed delays has received more and more attention of scholars (see [51, 55, 69, 70, 87, 88, 98] and the references therein). All of these stability criteria in these papers are established by using the LMI method. It is worth noting that the Lyapunov asymptotic stability and finite-time stability are a pair of independent concepts: a finite-time stable system may not be Lyapunov asymptotically stable, and vice versa [29]. It is well known that Lyapunov asymptotic stability is concerned with the behavior of a system over an infinite interval of time, while finite-time stability is used to describe the behavior of a system over a fixed time interval [3]. Furthermore, a system which is Lyapunov asymptotically stable may have a bad transient performance [3], and an unpredictable transient nature can yield a bad effect in the engineering, and even cause a great loss. Therefore, it can be seen that finite-time stability plays an important role in the practical applications. In some mathematical modeling, it is assumed that GRNs are spatially homogeneous, namely, the concentrations of mRNAs and proteins are homogenous in space at all times. However, in some cases, it is imperative to introduce reaction-diffusion terms into models of GRNs, because it is necessary to consider the diffusion of mRNAs and proteins [8, 9, 17, 24, 94]. To the best of authors’ knowledge, the stability problem for delayed reaction-diffusion GRNs has been only studied in [24, 25, 43, 95]. Ma et al. [43] established delay-dependent asymptotic stability criteria. Ma et al.’s results have been gradually improved in [24, 25] by introducing novel LKF and utilizing Wirtinger-type integral inequality approach. The problem of finite-time robust stochastic stability analysis for uncertain stochastic delayed reaction-diffusion GRNs has been studied in [95]. With changes in environment, the feedback loops which can inherently regulate the concentrations of mRNAs and proteins of GRNs may be destroyed. This will make GRNs’ performance worse, and eventually lead to some fatal disease like cancer [1, 31, 84]. Therefore, it is necessary to adjust the feedback loops by artificial input control. For this end, the concentrations of mRNAs and proteins are needed. However, due to the complexity of GRNs, it is almost impossible to measure the exact concentrations. Hence, the state estimation for GRNs has been one of available methods to investigate dynamical behaviors. To the best of authors’ knowledge, the state estimation problem for delayed reaction-diffusion GRNs is only in [89], although the reaction-diffusion-free case has been researched (see [5, 67, 74, 75] and the references therein).

4

1 Backgrounds

1.2 Functional Differential Equation Models of GRNs The following functional differential equations have been used to model GRNs with time-varying feedback regulation delays and translational delays [12, 52]: m˙ i (t) = −ai m i (t) + i ( p(t − σi (t))), t  0, i ∈ n,

(1.1a)

p˙ i (t) = −ci pi (t) + di m i (t − τi (t)), t  0, i ∈ n,

(1.1b)

m i (t) = φi (t), pi (t) = ψi (t), t ∈ [−d, 0], i ∈ n,

(1.1c)

where m i (t) and pi (t) denote the concentrations of mRNA i and protein i at time t, respectively; ai and ci are positive real numbers that represent the rates of degradation of mRNA i and protein i, respectively; di is a positive real number that represents the translating rate from mRNA i to protein i; i is a nonlinear function of the variables pi (t − σi (t)) (i ∈ n) which denotes the regulation function of gene i, and ¯ with τ¯ = is monotonic with each variable; φi , ψi ∈ C([−d, 0], R), d = max{τ¯ , σ} maxi∈n τ¯i and σ¯ = maxi∈n σ¯ i ; 0  τi (t)  τ¯i and 0  σi (t)  σ¯ i are continuously differentiable functions which denote the time-varying translational delay for mRNA i and the time-varying feedback regulation delay for protein i, respectively. Note that when n = 1, GRN (1.1) degenerates into a single-gene network model, which has been proposed and investigated in [45]. Equation (1.1a) describes the transcriptional process, where i characterizes the relative promoter or repressor activity of all possible proteins to mRNA i as a function of the concentrations of all possible proteins. Usually, one mRNA molecule or gene is generally activated or repressed by multiple  proteins in the transcriptional process. This can be indicated by defining i (x) = nj=1 i j (x) for all x  0, which is called “SUM” logic [32]. Here, the regulation function i j is a function of the Hill form as follows: ⎧ ai j , ⎪ 1+(x/b j )h j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0, i j (x)= ⎪ ⎪ ⎪ ai j (x/b j )h j ⎪ ⎪ hj , ⎪ ⎪ 1+(x/b j) ⎩

if the transcription factor j represses gene i, if the transcription factor j does not regulate gene i, if the transcription factor j activates gene i,

where ai j is the dimensionless transcriptional rate of transcription factor j to gene i, which is a nonnegative and bounded constant, b j is a positive scalar, and h j is the Hill coefficient that represents the degree of cooperativity. Under the  SUM logic, the general form of i ( p(t − σi (t))) in (1.1a) is i ( p(t − σi (t))) = nj=1 wi j i j ( p j (t − σi j (t))). In this book, we consider only the special

1.2 Functional Differential Equation Models of GRNs

case i ( p(t − σi (t))) = be rewritten as:

n j=1

m˙ i (t) = −ai m i (t) +

5

wi j g j ( p j (t − σ j (t))). From which, GRN (1.1) can

n 

wi j g j ( p j (t − σ j (t))) + Ji , t  0,

(1.2a)

j=1

p˙ i (t) = −ci pi (t) + di m i (t − τi (t)), t  0,

(1.2b)

m i (t) = φi (t), pi (t) = ψi (t), t ∈ [−d, 0],

(1.2c)

where i ∈ n, ⎧ ⎨ −ai j , if transcription factor j represses gene i, if transcription factor j does not regulate gene i, wi j = 0, ⎩ ai j , if transcription factor j activates gene i,

Ji =



ai j , Si = { j : j ∈ n, wi j < 0}, g j (x) =

j∈Si

(x/b j )h j . 1 + (x/b j )h j

Clearly, g j is a monotonically increasing function with saturation, and satisfies that for all distinct x, y ∈ R, g j (0) = 0, 0 

g j (x) − g j (y)  k j , j ∈ n, x−y

(1.3)

where k j : = maxu0 g˙ j (u) =

(h j − 1)(h j −1)/ h j (h j + 1)(h j +1)/ h j > 0. 4b j h j

(1.4)

When all proteins have the same feedback regulation delay (i.e., σ j (t) ≡ σ(t)), and all mRNAs have the same translational delay (i.e., τi (t) ≡ τ (t)), the model (1.2) simplifies into: m˙ i (t) = −ai m i (t) +

n 

wi j g j ( p j (t − σ(t))) + Ji , t  0, i ∈ n,

(1.5a)

j=1

p˙ i (t) = −ci pi (t) + di m i (t − τ (t)), t  0, i ∈ n, (1.5b) m i (t) = φi (t), pi (t) = ψi (t), t ∈ [−d, 0], i ∈ n.

(1.5c)

6

1 Backgrounds

Rewriting GRN (1.5) into compact matrix form, we obtain m(t) ˙ = −Am(t) + W g( p(t − σ(t))) + J,

(1.6a)

p(t) ˙ = −C p(t) + Dm(t − τ (t)),

(1.6b)

m(s) = φ(s), p(s) = ψ(s), s ∈ [−d, 0],

(1.6c)

where A = diag(a1 , a2 , . . . , an ), W = [wi j ]n×n , C = diag(c1 , c2 , . . . , cn ), D = diag(d1 , d2 , . . . , dn ), m(t) = col(m 1 (t), m 2 (t), . . . , m n (t)), p(t) = col( p1 (t), p2 (t), . . . , pn (t)), φ(t) = col(φ1 (t), φ2 (t), . . . , φn (t)), ψ(t) = col(ψ1 (t), ψ2 (t), . . . , ψn (t)), g( p(t)) = col(g1 ( p1 (t)), g2 ( p2 (t)), . . . , gn ( pn (t))), J = col(J1 , J2 , . . . , Jn ). In Corollary 2.7 below, it will be shown that GRN (1.6) has at least one nonnegative equilibrium point. Let (m ∗ , p ∗ ) be an equilibrium point of (1.6), that is, it is a solution of the following equations: − Am ∗ + W g( p ∗ ) + J = 0, −C p ∗ + Dm ∗ = 0.

(1.7)

For convenience, we shift the equilibrium point (m ∗ , p ∗ ) to the origin by using the ˜ = p(t) − p ∗ , then we have transformation m(t) ˜ = m(t) − m ∗ and p(t) ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(1.8a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0,

(1.8b)

˜ = ψ(s) − p ∗ , s ∈ [−d, 0], m(s) ˜ = φ(s) − m ∗ , p(s)

(1.8c)

where f (·) = g(· + p ∗ ) − g( p ∗ ). From the relationship between g and f , one can easily find that f satisfies the following sector conditions: f j (0) = 0, 0 

f j (s)  k j , ∀0 = s ∈ R, j ∈ n, s

(1.9)

where k j is defined as in (1.4), and f j (s) is the jth entry of f (s). Set K = diag(k1 , k2 , . . . , kn ).

1.3 Preliminaries

7

1.3 Preliminaries This section introduces some existing results which will be used to present the main results in the later chapters.

1.3.1 Nonsingular M-Matrix In this subsection we will give the concept of nonsingular M-matrix and several related conclusions. Definition 1.1 [49] A matrix is said to be a Z-matrix if it is a real square matrix with non-positive off-diagonal elements. A matrix A is said to be nonnegative (positive), denoted by A 0 (A 0), if all of its elements are nonnegative (positive). Definition 1.2 [49] A Z-matrix A is said to be an M-matrix if A = s I − B for some nonnegative matrix B and scalar s  ρ(B). Definition 1.3 [49] A Z-matrix A is said to be a nonsingular M-matrix if A = s I − B for some nonnegative matrix B and scalar s > ρ(B). Lemma 1.4 [49] Let A be an n × n Z-matrix. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

A is a nonsingular M-matrix. All of its eigenvalues have positive real parts. A is nonsingular, and all elements of A−1 are non-negative. AT is a nonsingular M-matrix. There exists an n-dimensional vector γ 0 such that Aγ 0.

For a real matrix A = [ai j ] in Rm×n , let |A| represent the matrix [|ai j |] in Rm×n . Lemma 1.5 [77] If A1 , A3 and A4 are n × n positive diagonal matrices, A2 is an A1 −|A2 | is a nonsingular M-matrix if and n × n real matrix, then the matrix −A3 A4 only if A1 A4 − A3 |A2 | is a nonsingular M-matrix. Due to Lemmas 1.4 and 1.5, one can easily obtain the following result. Proposition 1.6 Let E ∈ Rn×n , and let A, C, D and K be n × n positive diagonal matrices. Then the following statements are equivalent: (i) AC − D K |E|T is a nonsingular M-matrix. (ii) AC − |E|K D is a nonsingular M-matrix. (iii) AC − D|E|K is a nonsingular M-matrix.

8

1 Backgrounds



A −|E|K is a nonsingular M-matrix. −D C  A −D is a nonsingular M-matrix. (v) −K |E|T C

(iv)

1.3.2 Inequalities First, we introduce the so-called Schur complementary Lemma as follows. Lemma 1.7 (Schur Complementary Lemma) [7] Let  S :=

S11 S12 T S12 S22



T T = S11 and S22 = S22 . Then the be a real matrix of appropriate sizes, where S11 following statements are equivalent:

(i) S < 0. T −1 (ii) S11 < 0 and S22 − S12 S11 S12 < 0. −1 T S12 < 0. (iii) S22 < 0 and S11 − S12 S22 Second, the following conclusion is important to design filters for systems with unknown states. Its proof is similar to one of [18, Theorem 1]. Lemma 1.8 For given matrices P T = P > 0 and Q T = Q > 0, the matrix inequality  −P −1 A 0 and F T F  I , then (A + D F E)T P(A + D F E)  AT (P −1 − ε−1 D D T )−1 A + εE T E for any scalar ε > 0 satisfying P −1 − ε−1 D D T > 0. m×n 1.11 [7] For given , W ∈ R p×q and X T = X ∈ Rm×m , set S =

U ∈R Lemman× p T : V V  I p . Then V ∈R

X + U V W + W T V T U T < 0, ∀V ∈ S if and only if there exists a scalar ε > 0 such that X + ε−1 UU T + εW T W < 0. Last, we introduce some integral inequalities as follows. Lemma 1.12 (Gronwall’s Inequality) [46] For given a scalar a  0, and two nonnegative and integrable functions u(t) and b(·, t) over [0, T ] such that ∂t∂ b(·, t)  0 and b(t, t) exist, if 

t

u(t)  a +

b(ξ, t)u(ξ)dξ, t ∈ [0, T ],

0

then u(t)  ae

t 0

b(ξ,t)dξ

, t ∈ [0, T ].

Lemma 1.13 (Wirtinger’s Inequality) [57] If a function f ∈ C 1 ([a, b], R) satisfies f (a) = f (b) = 0, then  a

b

f 2 (v)dv 

(b − a)2 π2



b

f˙2 (v)dv.

a

Lemma 1.14 (Jensen’s Inequality) For given an n × n matrix M T = M > 0, a pair of scalars a and b satisfying b  a, and an integral vector function w : [a, b] → Rn , the following inequalities hold: b (i) [23] (b − a) a w T (s)Mw(s)ds  1T M1 ;

10

1 Backgrounds

b0 T w (s)Mw(s)dsdθ  2T M2 ; ab θ0  0 T T a θ λ w (s)Mw(s)dsdλdθ  3 M3 . b b0 b00 Here 1 = a w(s)ds, 2 = a θ w(s)dsdθ and 3 = a θ λ w(s)dsdλdθ. (ii) [61] (iii) [50]

b2 −a 2 2 b3 −a 3 6

Lemma 1.15 [47] For given a pair of scalars a and b with a < b, an integral function w : [a, b] → Rn , and an n × n matrix M T = M > 0, the following inequalities hold:  b ˆ 0T Γ1T Γ2T ]T , w T (s)Mw(s)ds  (b − a)[Γ0T Γ1T Γ2T ] M[Γ a



b



b θ

a

where

˜ 3T Γ4T ]T , w T (s)Mw(s)dsdθ  (b − a)2 [Γ3T Γ4T ] M[Γ

Mˆ = diag(M, 3M, 5M), M˜ = diag(2M, 16M), Γ1 = Γ0 − 2Γ3 , Γ2 = Γ0 − 6Γ3 + 12Γ5 , Γ4 = Γ3 − 3Γ5 , Γ0 =

1 b−a



b

w(s)ds, Γ3 =

a

1 Γ5 = (b − a)3

 a

b

 θ

1 (b − a)2 b

 λ

b

 a

b

 θ

b

w(s)dsdθ,

w(s)dsdλdθ.

Lemma 1.16 [47] For given a pair of scalars a and b with a < b, a derivative function w : [a, b] → Rn , and an n × n matrix M T = M > 0, the following inequalities hold:  b 1 ˆ 0T Ω1T Ω2T ]T , [Ω T Ω1T Ω2T ] M[Ω w˙ T (s)M w(s)ds ˙  b−a 0 a 

b



b

a

θ

b

b

 a

θ

¯ 3T Ω4T ]T , w˙ T (s)M w(s)dsdθ ˙  [Ω3T Ω4T ] M[Ω



b λ

w˙ T (s)M w(s)dsdλdθ ˙ 

3(b − a) T Ω5 MΩ5 , 2

where M¯ = diag(2M, 4M), Ω0 = w(b) − w(a), Ω1 = w(b) + w(a) − 2Γ0 , Ω2 = Ω0 + 6Γ0 − 12Γ3 , Ω3 = w(b) − Γ0 , Ω4 = w(b) + 2Γ0 − 6Γ3 , Ω5 = w(b) − 2Γ3 , ˆ Γ0 and Γ3 are defined as in Lemma 1.15. and M,

1.3 Preliminaries

11

These inequalities in Lemmas 1.15 and 1.16 will be unitedly named as Wirtingertype integral inequality.

1.3.3 Miscellanea Lemma 1.17 (Brouwer’s Fixed Point Theorem) [58] Every continuous function from a convex compact subset of a Euclidean space to itself has a fixed point. Let R be a compact set in the vector space Rl with smooth boundary ∂R. Let C (R, Rn ) be the Banach space of functions which map R into Rn and have the continuous first derivatives. We define a pair of norms on C 1 (R, Rn ) and C 1 ([−d, 0] × R, Rn ) by · and · d as follows: 1

 y(x) =

R

1/2 y T (x)y(x)dx

and    ∂φ(t, x)   φ(t, x) d = max{ sup φ(t, x) , sup   ∂t  , −dt0 −dt0    ∂φ(t, x)   }, max sup  1kl −dt0  ∂x k respectively. Lemma 1.18 (Green’s Second Identity) [60] If R is a bounded C 1 –open set in Rn and μ, υ ∈ C 2 (∂R, R), then 

∂μ ∂υ −υ dS, μ μΔυdx = υΔμdx + ∂n ∂n R R ∂R







where ∂υ and ∂μ are the directional derivatives of υ and μ in the direction of the ∂n ∂n outward pointing normal n to the surface element dS, respectively. Lemma 1.19 [35] For a function Φ : [0, ∞) −→ R, if Φ˙ is bounded over the inter˙ val [0, ∞) (that is, there is a constant α > 0 satisfying |Φ(t)|  α for all t ∈ [0, ∞)), then Φ is uniformly continuous over the interval [0, ∞). Lemma  ∞ 1.20 [35] For a function Φ : [0, ∞) → R, if Φ is uniformly continuous and 0 Φ(s)ds < ∞, then limt→∞ Φ(t) = 0.

12

1 Backgrounds

1.4 Organization The rest of the book will be organized as follows: Chapter 2 will deal with the globally asymptotic stability issue for nonnegative equilibrium points of a kind of GRNs with discrete and distributed delays. Discrete delays are time-varying, and distributed delays are infinite or constant. The existence issues of nonnegative equilibrium points of the kind of GRNs under consideration here will be solved by employing the Brouwer’s fixed point theorem. Furthermore, by using the functional differential equation theory and the nonsingular M-matrix theory, we will present sufficient conditions to guarantee that the kind of GRNs under consideration here have a unique nonnegative equilibrium point which is globally asymptotically stable. The sufficient conditions are to verify whether a constant matrix is a nonsingular M-matrix. It is convenient to use, as it can be tested by the function eig in MATLAB. Some numerical examples and their simulations will be given to illustrate the effectiveness of the proposed approach. It is worth emphasizing that the globally asymptotic stability criteria established here are also available for GRNs only with discrete delays. So, this paper can be viewed as an extensive and supplementary version of the related literature. Chapter 3 will first establish an inequality concerning with double integrals, and then it will be mathematically proven that the proposed integral inequality is less conservative than the well-known Jensen’s inequality of double integers. Thereby, for a class of GRNs with time-varying delays, a pair of delay-range-dependent asymptotic stability criteria will be investigated by constructing an appropriate LKF and combining Jensen’s inequality approach and reciprocally convex combination technique whether the derivatives of time delays are known or not, respectively. The obtained stability criteria are in the form of LMIs, which can be easily solved by the Toolbox LMI or YALMIP of MATLAB. Then it will be mathematically proven that a stability criterion proposed here is less conservative than [71, Corollary 3.1]. Finally, numerical examples and their simulation results show that the proposed stability criteria are less conservative than ones in [52, 68, 71, 81, 83]. What is more, the approach proposed in this chapter can be easily applied to analyze robust asymptotic stability for GRNs with “SUM” logic, interval time-varying delays and structured uncertainties, such as the norm-bounded uncertainty, the linear fractional uncertainty, and so on. In Chap. 4, the global stability analysis problem will be investigated for switched GRNs with constant or time-varying delays. By employing a Wirtinger-type integral inequality and an improved reciprocally convex technique together with the average dwell time approach and the piecewise Lyapunov functional technique, we will present some sufficient conditions to guarantee the global exponential stability for the considered systems. Numerical examples will be provided to show the effectiveness of the proposed theories. Chapter 5 will investigate the global robust asymptotic mean square stability issue for a class of stochastic GRNs with both norm-bounded uncertainties and timevarying delays. By employing a novel “delay fractioning” approach, Itô’s differential

1.4 Organization

13

formula and the LMI method, sufficient conditions in terms of LMIs will be derived to ensure the global robust asymptotic mean square stability of GRNs under consideration here. Finally, one example and its simulation will be offered to show the advantages of the theoretical result. Chapter 6 will establish asymptotic stability criteria and finite-time stability criteria for the trivial solution of delayed reaction-diffusion GRNs under Dirichlet boundary conditions and Neumann boundary conditions, respectively. Asymptotic stability criteria and finite-time stability criteria will be derived by constructing novel LKFs and employing several existing results (e. g., integral inequalities, Green’s second identity, and reciprocally convex technique). Furthermore, it will be shown that the obtained asymptotic stability criteria are less conservative than the corresponding ones in [25, 43]. Moreover, our results remove the restriction conditions, μ1 < 1 and μ2 < 1, required in [25, 43], which extend the range of application of theoretical results. The results of numerical examples illustrate the correctness and effectiveness of theoretical results of this chapter. Chapter 7 will estimate the mRNA and protein concentrations by designing the full-order observers and reduced-order observers for delayed GRNs based on available network outputs. By employing the Wirtinger-type integral inequalities, the convex technique and the reciprocally convex technique, the delay-dependent sufficient conditions are obtained, which can guarantee the existence of full- and reduced-order state observers. Also the concrete expressions of desired full- and reduced-order state observers will be given, respectively. Finally, the results of numerical examples indicate that the obtained full- and reduced-order observers are available Chapter 8 will investigate the problem of guaranteed cost control via state feedback for GRNs with interval time-varying delays. Based on the construct of an appropriate LKF, a sufficient condition will be established for the existence of a state feedback guaranteed cost controller. Then, by employing the idea of cone complementarity linearization, a method to design the expected controller for the uncertain GRNs will be presented in terms of LMIs. Numerical results of an example are presented to show the effectiveness of the proposed approach. Chapter 9 will focus on the state estimation problem for a class of delayed reactiondiffusion GRNs. Infinite- and finite-time state observers will be designed to estimate the concentrations of mRNAs and proteins based on available network outputs, which guarantees that the error system is asymptotically stable and finite-time stable, respectively. By introducing new integral terms into LKFs and using the Green’s second identity, the Wirtinger-type integral inequality, the convex technique and the reciprocally convex technique, sufficient conditions guaranteeing the existence of state observers will be investigated. The concrete expression of the desired state observers will be given. Finally, we will present numerical examples to test our theoretical results. Chapter 10 will investigate the problem of robust H∞ filtering for uncertain stochastic GRNs with mixed time-varying delays. The systems under consideration involve Itô-type stochastic disturbance, norm-bounded uncertainties, time-varying discrete and distributed delays. By constructing a proper LKF and using reciprocal convex technique, we obtain sufficient conditions in terms of LMIs to guarantee that

14

1 Backgrounds

the filtering error systems are mean square robustly asymptotically stable with disturbance attenuation level γ. In the end, some simulation examples will be given to illustrate the effectiveness of the proposed main results. Chapter 11 will investigate the filtering problem for a class of discrete-time GRNs with random delays. The random delay is described as a Markovian chain, and hence the filtering error system is regarded as a Markovian switched system. By introducing an appropriate LKF, sufficient conditions for the solvability of concerned problems will be given in terms of LMIs. The designed filter can guarantee that the filtering error system is stochastically stable with a pre-specified H∞ disturbance attenuation level. Finally, the effectiveness of the obtained results will be demonstrated by numerical examples.

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Part I

Analysis of Delayed GRNs

Chapter 2

Stability Analysis for GRNs with Mixed Delays

The chapter will propose an M-matrix-based approach to establish globally asymptotic stability criteria for the nonnegative equilibrium point of GRNs with mixed (i.e., discrete and distributed) delays.

2.1 Constant Distributed Delay Case In this section we will develop an M-matrix-based approach to establish globally asymptotic stability criteria for the nonnegative equilibrium point of GRNs with discrete time-varying delays and constant distributed delays. By using the Brouwer’s fixed point theorem, the existence of nonnegative equilibrium points for the kind of GRNs under consideration here is investigated, i.e., sufficient conditions that the kind of GRNs has at least one nonnegative equilibrium point are given. Furthermore, based on the nonsingular M-matrix theory and the functional differential equation theory, M-matrix-based sufficient conditions are given to guarantee that the kind of considered GRNs has a unique nonnegative equilibrium point which is globally asymptotically stable. As presented in Theorem 2.22 and Remarks 2.23 and 2.24 below, the sufficient conditions proposed here are less conservative than ones in [17, 21–23]. Finally, the effectiveness of the theoretical results obtained in this section is illustrated by the simulation results of several numerical examples. It is worth emphasizing that the globally asymptotic stability criteria obtained in this section are also available for (i) GRNs with constant discrete and distributed delays by setting τ¯i,d = σ¯ i,d = 0 for all i ∈ n; and (ii) GRNs only with discrete delays by setting F = 0.

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_2

21

22

2 Stability Analysis for GRNs with Mixed Delays

Therefore, this section can be viewed as an extensive and supplementary version of the related literature (see Sects. 2.1.3 and 2.1.2 below for details).

2.1.1 Problem Formulation Consider the delayed GRN model (1.2) subject to (1.3), that is, m˙ i (t) = −ai m i (t) +

n 

wi j g j ( p j (t − σ j (t))) + Ji , t  0,

(2.1a)

j=1

p˙ i (t) = −ci pi (t) + di m i (t − τi (t)), t  0,

(2.1b)

m i (t) = φi (t), pi (t) = ψi (t), t ∈ [−d, 0],

(2.1c)

where i ∈ n, gi is a monotonically increasing function with saturation, and satisfies gi (0) = 0, 0 

gi (x) − gi (y)  ki , ∀x, y ∈ R, x = y, x−y

(2.2)

and ki is a positive scalar. Let K = diag(k1 , k2 , . . . , kn ). For individual molecules, since movement of mRNA from a transcription site to translation sites is an active process with a significant range of transport times, a GRN model with distributed delays is actually more practical [5]. In addition, it is too simple to express the movement of macromolecule in actual networks with only distributed delay [27]. Therefore, introducing the distributed delays into the GRN model (2.1) is necessary, which results in the following model of GRNs with discrete time-varying delays and constant distributed delays: m˙ i (t) = −ai m i (t) + +

n  j=1



n 

wi j g j ( p j (t − σ j (t)))

j=1 t

fi j

g j ( p j (s))ds + Ji , t  0, i ∈ n,

(2.3a)

t−δ j

p˙ i (t) = −ci pi (t) + di m i (t − τi (t)), t  0, i ∈ n,

(2.3b)

m i (t) = φi (t), pi (t) = ψi (t), t ∈ [−d, 0], i ∈ n,

(2.3c)

where δ j > 0 ( j ∈ n) are constants representing the distributed delays, f i j ∈ R, ¯ with δ¯ = maxi∈n δi , and the other param¯ δ} φi , ψi ∈ C([−d, 0], R), d = max{τ¯ , σ, eters are defined as previously.

2.1 Constant Distributed Delay Case

23

Throughout this section we will make the following assumptions: Assumption 2.1 The Hill coefficients h j  1 for j ∈ n. Assumption 2.2 The time-varying delays τi (t) and σi (t) (i ∈ n) satisfy 0  τi (t)  τ¯i , 0  σi (t)  σ¯ i , t  0, τ˙i (t)  τ¯i,d , σ˙ i (t)  σ¯ i,d , t  0, where τ¯i , σ¯ i , 0  τ¯i,d < 1 and 0  σ¯ i,d < 1 are known constants. Definition 2.1 The pair (m, ¯ p) ¯ ∈ Rn × Rn is called an equilibrium point of GRN (2.3) if it satisfies − ai m¯ i +

n  (wi j + δ j f i j )g j ( p¯ j ) + Ji = 0, i ∈ n,

(2.5a)

j=1

− ci p¯ i + di m¯ i = 0, i ∈ n,

(2.5b)

¯ respectively. where m¯ i and p¯ i are the i-th components of m¯ and p, For a given constant c > 0, set Ωc = {u ∈ C([−d, 0], Rn ) : sup u(s)2 < c}. s∈[−d,0]

Definition 2.2 [4] An equilibrium point (m, ¯ p) ¯ of GRN (2.3) is said to be globally asymptotically stable if: (i) for any  > 0 there is a δ = δ() such that φi , ψi ∈ Ωδ implies m i (t) − m¯ i 2  ε and  pi (t) − p¯ i 2  ε for any t  0 and i ∈ n; and (ii) limt→∞ m i (t) = m¯ i and limt→∞ pi (t) = p¯ i for any φi , ψi ∈ C([−d, 0], R) and all i ∈ n. The aim of this section is to give a sufficient condition under which GRN (2.3) has a unique nonnegative equilibrium point which is globally asymptotically stable.

2.1.2 Existence of Nonnegative Equilibrium Points In this subsection, by applying the Brouwer’s fixed point theorem on a convex compact subset of Rn , we will investigate a sufficient condition under which GRN (2.3) has at least one nonnegative equilibrium point.

24

2 Stability Analysis for GRNs with Mixed Delays

If (m, ¯ p) ¯ is an equilibrium point of GRN (2.3), then it is clear from Definition 2.1 that ¯ + J = 0, (2.6a) − Am¯ + (W + F Dδ )g( p) − C p¯ + D m¯ = 0,

(2.6b)

where g( p) ¯ = col(g1 ( p¯ 1 ), g2 ( p¯ 2 ), . . . , gn ( p¯ n )), J = col(J1 , J2 , . . . , Jn ), A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ), W = [wi j ]n×n , D = diag(d1 , d2 , . . . , dn ), Dδ = diag(δ1 , δ2 , . . . , δn ), F = [ f i j ]n×n . Now we define a map G : Rn → Rn by G(v) = (W + F Dδ )g(D A−1 C −1 v) + J, ∀v ∈ Rn .

(2.7)

¯ Then (m, ¯ p) ¯ is an equilibrium Proposition 2.3 For given m, ¯ p¯ ∈ Rn , let v¯ = Am. point of GRN (2.3) if and only if v¯ is a fixed point (that is, G(v) ¯ = v) ¯ of the map G defined in (2.7). Proof The “only if” part. If (m, ¯ p) ¯ is an equilibrium point of GRN (2.3), then (2.6) holds. Furthermore, ¯ + J = 0. −Am¯ + (W + F Dδ )g(C −1 D m) Due to v¯ = Am, ¯ it follows that v¯ = G(v), ¯ that is, v¯ is a fixed point of the map G defined in (2.7). The “if” part. If v¯ is a fixed point of the map G defined in (2.7), then v¯ = G(v), ¯ i.e., ¯ + J. v¯ = (W + F Dδ )g(D A−1 C −1 v) ¯ Then, by v¯ = Am, ¯ (2.6) is satisfied, i.e., (m, ¯ p) ¯ is an equilibSet p¯ = DC −1 A−1 v. rium point of GRN (2.3). By Proposition 2.3, in order to indicate that GRN (2.3) has at least one nonnegative equilibrium point, it suffices to show that the map G defined in (2.7) has at least one nonnegative fixed point. The following result presents a sufficient condition under which the map G defined in (2.7) has at least one nonnegative fixed point. Theorem 2.4 Set Ji = { j : j ∈ n, wi j + δ j f i j < 0}, i ∈ n. If Ji = ∅ or

 j∈Ji

|wi j + δ j f i j |  Ji , ∀i ∈ n,

(2.8)

2.1 Constant Distributed Delay Case

25

then the map G defined in (2.7) has at least one nonnegative fixed point in Φ (that is, GRN (2.3) has at least one nonnegative equilibrium point), where Φ = [0, Φ1 ] × [0, Φ2 ] × · · · × [0, Φn ] with Φi =

n j=1

|wi j + δ j f i j | + Ji for all i ∈ n.

Proof For every i ∈ n and v ∈ Rn , let G i (v) be the i-th component of G(v). Then      n  −1 −1  |G i (v)| =  (wi j + δ j f i j )g j (d j v j a j c j ) + Ji   j=1  

n 

−1 |wi j + δ j f i j |g j (d j v j a −1 j c j ) + Ji .

j=1

This, together with 0  g j (x)  1 for all x ∈ R and j ∈ n, implies that |G i (v)|  Φi .

(2.9)

On the other hand, it is clear that  −1 (wi j + δ j f i j )g j (d j a −1 G i (v) = j cj vj) j∈Ji

+



−1 (wi j + δ j f i j )g j (d j a −1 j c j v j ) + Ji .

j ∈J / i

Noting that 1  g j (x)  0 for all x ∈ Rn and j ∈ n, we derive that G i (v) 



(wi j + δ j f i j ) + Ji .

j∈Ji

By (2.8), one can conclude that G i (v)  0.

(2.10)

Let T be the restriction of the map G to Φ, i.e., T (v) = G(v) for all v ∈ Φ. By (2.9) and (2.10), T is a map from Φ to itself. Since Φ is a convex compact subset of Rn , one can conclude from Lemma 1.17 that T has at least one fixed point in Φ. So, the map G defined in (2.7) has at least one nonnegative fixed point. When W + F Dδ  0, it is easy to see Ji = ∅ for all i ∈ n, so the condition (2.8) is satisfied. Therefore, the following result can be derived from Theorem 2.4. Corollary 2.5 If W + F Dδ  0, then the map G defined in (2.7) has at least one nonnegative fixed point in Φ (that is, GRN (2.3) has at least one nonnegative equilibrium point).

26

2 Stability Analysis for GRNs with Mixed Delays

When F  0, one can conclude that Ji ⊆ Si from wi j =−ai j for any j ∈ Si , where Si is defined as previously. This, together with Ji = j∈Si ai j , implies that the condition (2.8) is naturally satisfied. So, we can obtain the following result from Theorem 2.4. Corollary 2.6 If F  0, then the map G defined in (2.7) has at least one nonnegative fixed point (that is, GRN (2.3) has at least one nonnegative equilibrium point). When F = 0 (that is, the distributed delays in GRN (2.3) vanish), the GRN model (2.3) is simplified into (2.1). In this case, Corollary 2.6 turns into: Corollary 2.7 GRN (2.1) has at least one nonnegative equilibrium point. Next, we further explain the relationship between our results in this subsection and ones in literature through the following remarks. Remark 2.8 In [15, 16, 21, 22], the uniqueness of equilibrium points of GRN (2.1) was presented by establishing globally asymptotic stability criteria. However, the existence and nonnegativity of equilibrium points of GRN (2.1) were not considered in [15, 16, 21, 22]. Remark 2.9 By employing the Brouwer’s fixed point theorem, Luo, Zhang and Liao [10] have investigated the existence of equilibrium points of GRN (2.1) without any restriction. Following the proof of Theorem 2.4, it is easy to see that the existence of equilibrium points can be obtained by the Brouwer’s fixed point theorem without the condition (2.8). Indeed, the condition (2.8) is employed to guarantee the nonnegativity of equilibrium points. Remark 2.10 Different from [10, 15, 16, 21, 22], Chen and Jiang [3] investigated the existence and uniqueness theorem of equilibrium points for a kind of GRNs with discrete delays and infinite distributed delays by constructing a homeomorphism of Rn onto itself. In order to judge the existence and uniqueness of equilibrium points by Chen and Jiang’s result, some nonlinear inequalities have to be solved. Compared with Chen and Jiang’s result, the condition (2.8) offered in Theorem 2.4 can be easily verified. In addition, the nonnegativity of equilibrium points was not considered in [3]. Remark 2.11 Note that equilibrium points of GRNs represent the concentrations of mRNAs and proteins in steady states, hence they should be nonnegative. However, all of these results in [3, 10, 15, 16, 21, 22] can not guarantee the nonnegativity of equilibrium points. Therefore, these results obtained in this subsection can be viewed as an extensive and supplementary version of the corresponding results in [3, 10, 15, 16, 21, 22]. Remark 2.12 Comparing [7, (4)] with (2.5), one can easily find that the definitions of equilibrium points for discrete-time and continuous-time GRNs are almost the same. So, by using the approach proposed in this subsection, one can easily give a sufficient condition under which the kind of delayed discrete-time GRNs considered in [7] has at least one nonnegative equilibrium point. However, the existence and nonnegativity of equilibrium points were not considered in [7].

2.1 Constant Distributed Delay Case

27

2.1.3 Globally Asymptotic Stability Criteria In this subsection, by applying the nonsingular M-matrix theory [13] we will give a sufficient condition under which GRN (2.3) has a unique nonnegative equilibrium point which is globally asymptotically stable. Set ˆ (2.11) L = A2 C 2 − 2Γ K 2 (Wˆ + Dδ2 F), where A, C, K and Dδ are defined as previously, Γ = diag(γ1 , γ2 , . . . , γn ), Wˆ = [wˆ i j ], Fˆ = [ fˆi j ], γi = (1 − τ¯i,d )−1 di2 , wˆ i j = (1 − σ¯ i,d )−1 χ j (W )w 2ji , fˆi j = χ j (F) f ji2 , and χ j (F) refers to the number of nonzero elements in the j-row of F. Theorem 2.13 If the condition (2.8) is satisfied, and the matrix L defined in (2.11) is a nonsingular M-matrix, then GRN (2.3) has a unique nonnegative equilibrium point which is globally asymptotically stable. Proof Since the condition (2.8) is satisfied, we obtain from Theorem 2.4 that GRN (2.3) has at least one nonnegative equilibrium point. Let (m ∗ , p ∗ ) be an arbitrary but fixed nonnegative equilibrium point of GRN (2.3). In order to conveniently show that the nonnegative equilibrium point (m ∗ , p ∗ ) is globally asymptotically stable, we shift it to the origin by defining xi (t) = m i (t) − m i∗ , yi (t) = pi (t) − pi∗ , i ∈ n, where m i∗ and pi∗ are the i-th components of m ∗ and p ∗ , respectively. Then it is easy to get from (2.12) that x˙i (t) = −ai xi (t) + +

n 



n 

wi j g˜ j (y j (t − σ j (t)))

j=1 t

g˜ j (y j (s))ds, t  0, i ∈ n,

(2.12a)

y˙i (t) = −ci yi (t) + di xi (t − τi (t)), t  0, i ∈ n,

(2.12b)

fi j

j=1

t−δ j

xi (t) = φi (t) − m i∗ , yi (t) = ψi (t) − pi∗ , t ∈ [−d, 0], i ∈ n,

(2.12c)

where g˜ j (·) = g j (· + p ∗j ) − g j ( p ∗j ). Based on the relationship between g j and g˜ j , we know from (2.2) that g˜ j satisfies the sector condition g˜ j (0) = 0, 0 

g˜ j (x)  k j , ∀0 = x ∈ R, j ∈ n. x

28

2 Stability Analysis for GRNs with Mixed Delays



Set Lˆ =

 A2 −Γ ˆ C2 . −2K 2 (Wˆ + Dδ2 F)

(2.13)

Since the matrix L defined in (2.11) is a nonsingular M-matrix, it follows from Proposition 1.6 that the matrix Lˆ defined in (2.13) is a nonsingular M-matrix, that ˆ  0 for some 2n-dimensional vector γ  0. Let is, Lγ  T γ = G 1 G 2 · · · G n T1 T2 · · · Tn . Then, from (2.13), αi := ai2 G i − γi Ti > 0, i ∈ n, ςi := −2ki2

(2.14a)

n  (wˆ i j + δi2 fˆi j )G j + ci2 Ti > 0, i ∈ n.

(2.14b)

j=1

Choose an LKF candidate V : [0, ∞) × C([−d, 0], Rn ) × C([−d, 0], Rn ) → [0, ∞) as follows: V (t, ϕ, ϑ) =

3 

Vi (t, ϕ, ϑ), ∀t  0, ϕ, ϑ ∈ C([−d, 0], Rn ),

(2.15)

i=1

where

n 

V1 (t, ϕ, ϑ) =

G i ai ϕi2 (0) +

i=1 n 

V2 (t, ϕ, ϑ) =

 Ti γi n 

n  n 

0

−τi (t)

Gi

i=1

V3 (t, ϕ, ϑ) = 2

Ti ci ϑi2 (0),

i=1

i=1

+2

n 

n 

ϕi2 (s)ds  wˆ ji k 2j

−σ j (t)

j=1

G j fˆi j δi

i=1 j=1



0 −δi

0



0 θ

ϑ2j (s)ds,

g˜i2 (ϑi (s))dsdθ.

Obviously, there exist scalars c1 > 0 and c2 > 0 such that V (t, ϕ, ϑ)  c1 (ϕ(0)22 + ϑ(0)22 ),

V (t, ϕ, ϑ)  c2 for all ϕ, ϑ ∈ C([−d, 0], Rn ).

(2.16a) 

sup

−ds0

ϕ(s)22

+ sup

−ds0

ϑ(s)22

(2.16b)

2.1 Constant Distributed Delay Case

29

By computing the derivatives of Vi (t, xt , yt ) (i ∈ 3) along the trajectory of (2.12), we can obtain that V˙1 (t, xt , yt ) n n   G i ai xi (t)x˙i (t) + 2 Ti ci yi (t) y˙i (t) =2 −

i=1 n 

i=1

G i ai2 xi2 (t) −

i=1

n 

Ti ci2 yi2 (t) +

i=1

n 

Ti di2 xi2 (t − τi (t))

i=1

⎛  n n n    + Gi ⎝ wi j g˜ j (y j (t − σ j (t))) + fi j i=1

−

n 

j=1

i=1

n 

Ti ci2 yi2 (t) +

i=1

n 

g˜ j (y j (s))ds ⎠

t−δ j

j=1

G i ai2 xi2 (t) −

⎞2 t

Ti di2 xi2 (t − τi (t))

i=1

⎞2 ⎛ n n   +2 Gi ⎝ wi j g˜ j (y j (t − σ j (t)))⎠ i=1

j=1

⎞2

⎛  n n   ⎝ +2 Gi fi j i=1

−

n 

+2 +2

j=1

G i χi (W )

n 

i=1

j=1

n 

n 

n 

G i χi (F)

+2

i=1 n  i=1

Ti ci2 yi2 (t) +

G i χi (W ) ⎛ ⎝

n 

Ti di2 xi2 (t − τi (t))

i=1

wi2j k 2j y 2j (t − σ j (t))

 f i2j

n 

t

2 g˜ j (y j (s))ds

t−δ j

j=1

G i ai2 xi2 (t) −

i=1 n 

+2

n  i=1

i=1

=−

g˜ j (y j (s))ds ⎠

t−δ j

G i ai2 xi2 (t) −

i=1 n 

t

Ti ci2 yi2 (t) +

i=1 n 

n 

Ti di2 xi2 (t − τi (t))

i=1

wi2j k 2j y 2j (t − σ j (t))

j=1 n  j=1

⎞

G j fˆi j ⎠



t

t−δi

2 g˜i (yi (s))ds

,

(2.17)

30

2 Stability Analysis for GRNs with Mixed Delays

V˙2 (t, xt , yt ) =

n 

 Ti γi xi2 (t) − (1 − τ˙i (t))xi2 (t − τi (t))

i=1

+2

n 

Gi

n 

i=1

−2

j=1

n 

Gi

n 

i=1



n 

wˆ ji k 2j y 2j (t) wˆ ji k 2j (1 − σ˙ j (t))y 2j (t − σ j (t))

j=1

Ti γi xi2 (t) −

i=1

+2

n 

Ti di2 xi2 (t − τi (t))

i=1 n n  

G j wˆ i j ki2 yi2 (t)

i=1 j=1

−2

n 

Gi

i=1

V˙3 (t, xt , yt ) = 2

n 

χi (W )wi2j k 2j y 2j (t − σ j (t)),

n n  

G j fˆi j δi2 g˜i2 (yi (t))

i=1 j=1

−2

n  n 

G j fˆi j δi



n n  

g˜i2 (yi (s))ds

G j fˆi j δi2 ki2 yi2 (t)

i=1 j=1

−2

t

t−δi

i=1 j=1

2

(2.18)

j=1

n n  

G j fˆi j



i=1 j=1

t

2 g˜i (yi (s))ds

.

(2.19)

t−δi

Now, it can be derived from (2.14) to (2.19) that V˙ (t, xt , yt ) n n n    G i ai2 xi2 (t) + Ti γi xi2 (t) − Ti ci2 yi2 (t) − i=1 n  n 

+2

i=1

i=1

ki2 (wˆ i j + δi2 fˆi j )G j yi2 (t)

i=1 j=1

=−

n 

αi xi2 −

i=1



−C3 (xt (0)22

n 

ςi yi2 (t)

i=1

+ yt (0)22 ),

(2.20)

2.1 Constant Distributed Delay Case

31

where  C3 = min

 min αi , min ςi .

1in

1in

Then by an approach used in the proofs of [21, Theorem 3] and [22, Theorem 3.4], one can indicate from (2.14) to (2.20) that GRN (2.3) has a unique nonnegative equilibrium point which is globally asymptotically stable. The proof is completed. Remark 2.14 The LKFs employed in [15–17, 21, 22, 31] are not available for GRNs with distributed delays. In the proof of Theorem 2.13, we introduce V3 (t, xt , yt) into our LKF. This guarantees that the global asymptotic stability criteria for GRNs with discrete and distributed delays can be derived. In addition, when the distributed delays vanish, we can theoretically investigate that the stability criterion given in Theorem 2.13 is less conservative than ones in [15, 21, 22] (see Theorem 2.22 and Remarks 2.23 and 2.24 below). When the discrete time delays in GRN (2.3) are constants, the matrix L defined in (2.11) becomes ˆ V := A2 C 2 − 2D 2 K 2 [(W ◦ W )T χ(W ) + Dδ2 F].

(2.21)

Thus, we can easily get the following result from Theorem 2.13. Corollary 2.15 If the condition (2.8) is satisfied, and the matrix V defined in (2.21) is a nonsingular M-matrix, then GRN (2.3) with constant delays has a unique nonnegative equilibrium point which is globally asymptotically stable. When W + F Dδ  0 or F  0, similar to Corollaries 2.5 and 2.6, the following result can be directly derived from Theorem 2.13. Corollary 2.16 If W + F Dδ  0 or F  0, and the matrix L defined in (2.11) is a nonsingular M-matrix, then GRN (2.3) has a unique nonnegative equilibrium point which is globally asymptotically stable. Remark 2.17 From the proof of Theorem 2.13, it can be found that the coefficient “2” ˆ in the matrix L originates in the coupling between of the item “Γ K 2 (Wˆ + Dδ2 F)” discrete delays and distributed delays. This is distinct between mixed delays and only discrete (distributed) delays. Therefore, in Corollary 2.18 below, there is no the coefficient “2”. When F = 0 (that is, the distributed delays in GRN (2.3) vanish), one can easily derive the following result from Corollary 2.16. Corollary 2.18 If the matrix L 0 := A2 C 2 − Γ K 2 Wˆ

(2.22)

32

2 Stability Analysis for GRNs with Mixed Delays

is a nonsingular M-matrix, then GRN (2.1) has a unique nonnegative equilibrium point which is globally asymptotically stable. It should be noted that Wu [22] has shown the following result. Proposition 2.19 [22, Theorem 3.4] If the matrix Lˆ 0 := AC − (I − Dτ )−1 D|W |K (I − Dσ )−1

(2.23)

is a nonsingular M-matrix, then GRN (2.1) has a unique equilibrium point which is globally asymptotically stable. Remark 2.20 Combining Proposition 2.19 and Corollary 2.7, one can indicate that if the matrix Lˆ 0 defined in (2.23) is a nonsingular M-matrix, then GRN (2.1) has a unique nonnegative equilibrium point which is globally asymptotically stable. The nonnegativity explains the real meaning of the unique equilibrium point very nicely. Tian et al. [15, Corollary 1] investigated that when (1 − σ¯ j,d )χi (W )  1, ∀i, j ∈ n,

(2.24)

the stability criterion in Corollary 2.18 is less conservative than one in Proposition 2.19. In order to weaken the condition (2.24), the following result is required. Proposition 2.21 Let u i , vi > 0, i ∈ n, E ∈ Rn×n , and let A, C and D be defined as previously. Set M = A2 C 2 − Du−1 D 2 χ(E)(E ◦ E)Dv−1 , Mˆ = AC − Du−1 D|E|Dv−1 . If χ(E)Du Dv  In and Mˆ is a nonsingular M-matrix, then so is M. Proof Since Mˆ is a nonsingular M-matrix, it follows from Lemma 1.4 that there ˆ  0, i.e., exists a positive vector β ∈ Rn such that Mβ ai ci βi > u i−1 di

n  (|ei j |v −1 j β j ), ∀i ∈ n, j=1

hence ⎛

⎞2 n  ⎠ ai2 ci2 βi2 > u i−2 di2 ⎝ (|ei j |v −1 j βj) j=1

 u i−2 di2

n 

2 (ei2j v −2 j β j ), ∀i ∈ n.

j=1

(2.25)

2.1 Constant Distributed Delay Case

33

2 Set η j = v −1 j β j for j ∈ n. Then η j > 0 for all j ∈ n. Now, one can indicate that

ai2 ci2 ηi > u i−1 di2 χi (E)

n  (ei2j v −1 j η j ), ∀i ∈ n.

(2.26)

j=1

Indeed, for a fixed i ∈ n, if χi (E) = 0, then (2.26) is trivial; if χi (E) = 0, then (2.25) can be written as ai2 ci2 ηi > εi−1 u i−1 di2 χi (E)

n  (ei2j v −1 j η j ), ∀i ∈ n, j=1

where εi = χi (E)vi u i . This, together with χ(E)Du Dv  In , implies that (2.26) holds. Set η = col(η1 , η2 , . . . , ηn ). Then η  0 and (A2 C 2 − Du−1 D 2 χ(E)(E ◦ E)Dv−1 )η  0, that is Mη  0. By Lemma 1.4, M is a nonsingular M-matrix. The following result generalizes [15, Corollary 1] by weakening the condition (2.24) to (2.27) (1 − τ¯i,d )(1 − σ¯ i,d )χi (W )  1, ∀i ∈ n. Theorem 2.22 If the condition (2.27) is satisfied, then the stability criterion in Corollary 2.18 is less conservative than one in Proposition 2.19. Proof In order to complete the proof, it suffices to show that if Lˆ 0 is a nonsingular M-matrix, then so is L 0 . Note that the matrices L 0 and Lˆ 0 defined as previously can be written as L 0 = A2 C 2 − D1−1 D 2 D2−1 (E ◦ E)T χ(E) and Lˆ 0 = AC − D1−1 D|E|D2−1 , respectively, where D1 = I − Dτ , E = W K and D2 = I − Dσ . It follows from Assumption 2.2 that 0  Di ≺ In , i = 1, 2. Since Lˆ 0 is a nonsingular M-matrix, we obtain from Proposition 2.21 that the matrix A2 C 2 − D1−1 D 2 χ(E)(E ◦ E)D2−1 is a nonsingular M-matrix. Noting that L 0 is similar to the matrix A2 C 2 − D2−1 (E ◦ E)T χ(E)D 2 D1−1 , one can conclude from Lemma 1.4 that L 0 is a nonsingular Mmatrix. Remark 2.23 When GRN (2.1) has ring structure (i.e., the matrix W can be written as the product of a permutation matrix and a diagonal matrix) [17, 23], we have χ(W ) = 1, and hence the condition (2.27) is naturally satisfied under

34

2 Stability Analysis for GRNs with Mixed Delays

Assumption 2.2. So, from Theorem 2.22, the stability criterion given in Corollary 2.18 is less conservative than one in Proposition 2.19 for GRN (2.1) with ring structure. Remark 2.24 When the discrete time delays are constants in GRN (2.1), the matrix Lˆ 0 defined in (2.23) becomes V0 := AC − D|W |K .

(2.28)

This implies that in this case Proposition 2.19 simplifies into [21, Theorem 5] and [22, Theorem 3.5]. From Theorem 2.22, in this case the globally asymptotic stability criterion given in Corollary 2.18 is less conservative than ones in [21, Theorem 5] and [22, Theorem 3.5] when χi (W )  1 for i ∈ n. Remark 2.25 Compared with the LMI-based stability criteria, an M-matrix-based one has lower computational complexity. Because there are more than 50 equivalent conditions to verify whether a matrix is an nonsingular M-matrix [1, 12]. One of the equivalent conditions is that a constant matrix is a nonsingular M-matrix if and only if all of its eigenvalues have a positive real part. While the eigenvalues of a matrix can be computed by using the function eig in MATLAB. So the M-matrix-based stability criteria can be easily verified.

2.1.4 Numerical Examples In this subsection, we will give four examples to show the correctness and effectiveness of our theoretical results. The first one illustrates the effectiveness of Corollaries 2.5 and 2.16 and Theorems 2.4 and 2.13, and the others are used to test Corollaries 2.15, 2.18 and Theorem 2.22, respectively. Comparisons with several existing results are made too. Example 2.26 Consider a three-node GRN in the form of (2.3), where ai = 1, ci = 1, di = 0.7, i ∈ 3, τi (t) = (sin t + 1)/4, σi (t) = (cos t + 1)/8, i ∈ 3, t  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. Case 1. δi = 0.1i for all i ∈ 3, and ⎡

⎤ 0 0.5 0 W = ⎣ 0 0 0.5⎦ , F = 0.3I3 , Ji = 0, i ∈ 3. 0.5 0 0

2.1 Constant Distributed Delay Case

35

It is clear that W + F Dδ  0. By Corollary 2.5, the GRN under consideration here has at least one nonnegative equilibrium point. √ Furthermore, noting that τ¯i,d = 1/4, σ¯ i,d = 1/8, χi (W ) = χi (F) = 1 and ki = 3 3/8 for all i ∈ 3, one can obtain the matrix L defined in (2.11) as follows: ⎡

⎤ 0.9995 0 −0.1575 ⎦. 0 L = ⎣−0.1575 0.9980 0 −0.1575 0.9955 By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix L are 0.8402 and 1.0764 ± 0.1364i. From Lemma 1.4, it is concluded that L is a nonsingular M-matrix. This, together with Corollary 2.16, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. In addition, we can indicate from Ji = 0 for all i ∈ 3 that the unique nonnegative equilibrium point is (0, 0). When φi (t) ≡ φi and ψi (t) ≡ ψi for all i ∈ 3 and t ∈ [−0.5, 0], we take 100 values of (φi , ψi ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (0, 0), which concludes that the GRN under consideration here is globally asymptotically stable. Partial simulation results are shown in Figs. 2.1, 2.2, 2.3 and 2.4. Case 2. δi = i and Ji = 0.5 for all i ∈ 3, and ⎡

⎤ ⎡ ⎤ −0.2 −0.3 0 0.3 0.3 0 W = ⎣ 0 −0.2 −0.3⎦ , F = ⎣ 0 0.15 0.1⎦ . −0.3 0 −0.2 0.3 0 0.1  Clearly, J1 = {1, 2}, J2 = {2, 3} and J3 = {1, 3} for all i ∈ 3. So, j∈Ji |wi j + δ j f i j |  Ji for all i ∈ 3, and hence the condition (2.8) is satisfied. By Theorem 2.4, the GRN under consideration here has at least one nonnegative equilibrium point. 3

2.5

2.5

2

2 1.5 1.5 1 1 0.5

0.5 0

0

1

2

3

4

5

6

7

8

9

10

0

0

1

2

3

4

Fig. 2.1 Trajectories of mRNA and protein concentrations (Example 2.26)

5

6

7

8

9

10

36

2 Stability Analysis for GRNs with Mixed Delays 2.5

2 1.8

2

1.6 1.4

1.5

1.2

1

0.8

1 0.6 0.5

0.4 0.2

0

0

1

2

3

4

5

6

7

8

9

10

0

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

5

6

7

8

9

10

Fig. 2.2 Trajectories of mRNA and protein concentrations (Example 2.26) 3

2.5

2.5

2

2 1.5 1.5 1 1 0.5

0.5 0

0 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

Fig. 2.3 Trajectories of mRNA and protein concentrations (Example 2.26) 1.6

2.5

1.4 2

1.2 1

1.5

0.8 1

0.6 0.4

0.5

0.2 0

0

1

2

3

4

5

6

7

8

9

10

0

0

1

2

3

4

Fig. 2.4 Trajectories of mRNA and protein concentrations (Example 2.26)

2.1 Constant Distributed Delay Case

37

√ Since τ¯i,d = 1/4, σ¯ i,d = 1/8, χi (W ) = χi (F) = 2 and ki = 3 3/8 for all i ∈ 3, the matrix L defined in (2.11) is ⎡

⎤ 0.8504 0 −0.2126 ⎦. 0 L = ⎣−0.5103 0.8504 0 −0.2126 0.8504 By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix L are 0.5657 and 0.9927 ± 0.2465i. From Lemma 1.4, L is a nonsingular Mmatrix. This, together with Theorem 2.13, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. In addition, we can indicate from Definition 2.1 that the unique nonnegative equilibrium point is (m ∗ , p ∗ ) with   m ∗ = 0.54 0.51 0.51 , p ∗ = 0.38 0.35 0.35 . Example 2.27 Consider a three-node GRN in the form of (2.3), where ai = 1, ci = 1, di = 0.3, Ji = 0, δi = 0.1i, i ∈ 3, ⎡

⎤ ⎡ ⎤ 0 2.7 0 100 W = ⎣ 0 0 2.7⎦ , F = ⎣0 1 0⎦ , 2.7 0 0 001 τi (t) ≡ 1/4, σi (t) ≡ 1/8, i ∈ 3, t  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. It follows from W + F Dδ  0 and Corollary 2.5 that the GRN under consideration here has √ at least one nonnegative equilibrium point. Since χi (W ) = χi (F) = 1 and ki = 3 3/8 for all i ∈ 3, the matrix V defined in (2.21) is ⎡

⎤ 0.9992 0 −0.5536 ⎦. 0 V = ⎣−0.5536 0.9970 0 −0.5536 0.9932 By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix V are 0.4429 and 1.2733 ± 0.4794i. From Lemma 1.4, it is concluded that V is a nonsingular M-matrix. This, together with Corollary 2.15, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. In addition, we can indicate from Definition 2.1 that the unique nonnegative equilibrium point is (0, 0). Example 2.28 Consider a three-node GRN in the form of (2.1), where ai = 1.2, ci = 1.2, di = 0.5, Ji = 3.1, i ∈ 3,

38

2 Stability Analysis for GRNs with Mixed Delays

⎡

⎤ 0 −3.1 0 0 −3.1⎦ , W =⎣ 0 −3.1 0 0 τi (t) = (sin t + 1)/4, σi (t) = (cos t + 1)/8, i ∈ 3, t  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. √ Since τ¯i,d = 1/4, σ¯ i,d = 1/8, χi (W ) = 1 and ki = 3 3/8 for all i ∈ 3, the matrix L 0 defined in (2.22) is ⎡

⎤ 2.0736 0 −1.5445 ⎦. 0 L 0 = ⎣−1.5445 2.0736 0 −1.5445 2.0736 By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix L 0 are 0.5291 and 2.8458 ± 1.3375i. From Lemma 1.4, L 0 is a nonsingular Mmatrix. This, together with Corollary 2.18, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. In addition, we can indicate from Definition 2.1 that the unique nonnegative equilibrium point is (m ∗ , p ∗ ) with m i∗ = 1.71 and pi∗ = 0.71 for i = 1, 2, 3. When φi (t) ≡ φi and ψi (t) ≡ ψi for all i ∈ 3 and t ∈ [−0.5, 0], we take 100 values of (φi , ψi ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (m ∗ , p ∗ ), which concludes that the GRN under consideration here is globally asymptotically stable. Partial simulation results are shown in Figs. 2.5, 2.6, 2.7 and 2.8. It is worth emphasizing that the stability conditions derived in [22, Theorem 3.4] and [31, Theorem 1] can not be satisfied for the GRN in Example 2.28. Therefore, Corollary 2.18 may be less conservative than [22, Theorem 3.4] and [31, Theorem 1].

2.2

2.2 2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0

1

2

3

4

5

6

7

8

9

10

0.2

0

1

2

3

4

Fig. 2.5 Trajectories of mRNA and protein concentrations (Example 2.28)

5

6

7

8

9

10

2.1 Constant Distributed Delay Case

39

3

3

2.5

2.5

2

2

1.5 1.5

1

1

0.5 0

0

1

2

3

4

5

6

7

8

9

10

0.5

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

7

8

9

10

Fig. 2.6 Trajectories of mRNA and protein concentrations (Example 2.28) 3

2 1.8

2.5

1.6 1.4

2

1.2

1.5

1 1

0.8 0.6

0

1

2

3

4

5

6

7

8

9

10

0.5

0

1

2

3

4

Fig. 2.7 Trajectories of mRNA and protein concentrations (Example 2.28) 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

6

7

8

9

10

0

0

1

2

3

4

Fig. 2.8 Trajectories of mRNA and protein concentrations (Example 2.28)

5

6

40

2 Stability Analysis for GRNs with Mixed Delays

Example 2.29 Consider a three-node GRN in the form of (2.1), where ai = 1, ci = 1, di = 0.3, Ji = 0, i ∈ 3, ⎡

⎤ 1 1.5 0 W = ⎣ 0 1 1.5⎦ , 1.5 1 0 τi (t) = (sin t + 1)/3, σi (t) = (cos t + 1)/4, i ∈ 3, t  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. √ Clearly, τ¯i,d = 1/3, σ¯ i,d = 1/4, χi (W ) = 2 and ki = 3 3/8 for all i ∈ 3. So, (1 − τ¯i,d )(1 − σ¯ i,d )χi (W ) < 1 for all i, and hence the condition (2.27) is satisfied. By Theorem 2.22, we can indicate that the stability criterion in Corollary 2.18 is less conservative than one in Proposition 2.19. However, this can not be indicated from [15, Corollary 1], since (1 − σ¯ 1,d )χ1 (W ) > 1 (that is, the condition (2.24) is not satisfied).

2.2 Unbounded Distributed Delay Case The section addresses the problem of stability analysis for nonnegative equilibrium points of GRNs with discrete time-varying delays and unbounded distributed delays. First, the existence issue of nonnegative equilibrium points for the kind of GRNs under consideration here was addressed. Then, by using the functional differential equation theory and the nonsingular M-matrix theory, we give sufficient conditions to guarantee that GRN has a unique nonnegative equilibrium point which is globally asymptotically stable. The sufficient conditions are to verify whether a constant matrix is a nonsingular M-matrix, which is very easy to be checked, as it can be judged by the function eig in MATLAB. Last, several numerical examples and their simulations are given to illustrate the effectiveness of the proposed approach. It is worth emphasizing that, for GRNs with discrete and unbounded distributed delays, the M-matrix method proposed in the previous section is available to present existence conditions of nonnegative equilibrium point; however, it does not apply to establish M-matrix-based globally asymptotic stability criteria, since the LKF employed in the previous section is invalid to the unbounded distributed delays.

2.2.1 Model Description In this section we will extend the results of the previous section to the following GRNs with discrete time-varying delays and unbounded distributed delays:

2.2 Unbounded Distributed Delay Case

m˙ i (t) = −ai m i (t) + +

n  j=1

n 

wi j g j ( p j (t − σ j (t)))

j=1

 vi j

41

t

−∞

Θ j (t − s)g j ( p j (s))ds + Ji , t  0,

(2.29a)

p˙ i (t) = −ci pi (t) + di m i (t − τi (t)), t  0,

(2.29b)

m i (t) = ϕi (t), t ∈ [−d, 0],

(2.29c)

pi (t) = ψi (t), t ∈ (−∞, 0],

(2.29d)

where i ∈ n, Θ j : [0, +∞) → [0, +∞) is the delay kernel which is a nonnegative continuous function satisfying 

+∞

Θ j (s)ds = 1, j = 1, 2, . . . , n;

(2.30)

0

vi j ∈ R, ψi ∈ C((−∞, 0], R), and the other notations are defined as previously. Assume that the discrete time delays satisfy 0  τi (t)  τ¯i , 0  σi (t)  σ¯ i , t  0,

(2.31a)

τ˙i (t)  τ¯i,d , τ˙i (t)  σ¯ i,d , t  0,

(2.31b)

where τ¯i , σ¯ i , 0  τ¯i,d < 1 and 0  σ¯ i,d < 1 are known constants. Definition 2.30 The pair (m, ¯ p) ¯ is called an equilibrium point of GRN (2.29) if it satisfies n  − ai m¯ i + (wi j + vi j )g j ( p¯ j ) + Ji = 0, (2.32a) j=1

− ci p¯ i + di m¯ i = 0,

(2.32b)

¯ respectively. where i ∈ n, m¯ i and p¯ i are the i-th components of m¯ and p, A connected subset of R is called an interval. For given an interval J and a positive integer n, the set of all continuous functions h : J → Rn is denoted by C(J, Rn ). It is a linear space with respect to the usual operations on functions, and is further a Banach space with respect to the norm  ·  defined by h = sup h(s)2 , ∀h ∈ C(J, Rn ). s∈J

42

2 Stability Analysis for GRNs with Mixed Delays

Let C((−∞, 0], Rn ) be the Banach space of functions ψ ∈ C((−∞, 0], Rn ) such that ψ is bounded and uniformly continuous, with norm  ψC :=

sup

−∞ 0, set ⊗c = {u ∈ C((−∞, 0], Rn ) : uC < c}. Definition 2.31 [4] An equilibrium point (m, ¯ p) ¯ of GRN (2.29) is said to be globally asymptotically stable if: (i) for any  > 0 there is a δ = δ() such that φi ∈ Ωδ and ψi ∈ ⊗δ implies m i (t) − m¯ i 2   and  pi (t) − p¯ i 2  ε for any t  0 and i ∈ n; and (ii) limt→∞ m i (t) = m¯ i and limt→∞ pi (t) = p¯ i for any ϕi ∈ C([−d, 0], R), ψi ∈ C((−∞, 0], R) and all i ∈ n.

2.2.2 Main Results In this section we will investigate M-matrix-based sufficient conditions such that GRN (2.29) has a unique nonnegative equilibrium point which is globally asymptotically stable. Firstly, we show the existence of nonnegative equilibrium points of GRN (2.29). Since Definitions 2.1 and 2.30 are similar, the following theorem and corollary follow directly from Sect. 2.1.2. Theorem 2.32 Set Ji = { j : j ∈ n, wi j + vi j < 0}, i ∈ n. If Ji = ∅ or



|wi j + vi j |  Ji , ∀i ∈ n,

(2.33)

j∈Ji

then GRN (2.29) has at least one nonnegative equilibrium point. Corollary 2.33 Set W = [wi j ] and V = [vi j ]. If W + V  0 or V  0, then GRN (2.29) has at least one nonnegative equilibrium point. Remark 2.34 When V = 0 (that is, the unbounded distributed delays in GRN (2.29) vanish), the GRN model (2.29) is simplified into (2.1). In this case, Corollary 2.33 turns into Corollary 2.7. Furthermore, the relationship among Theorem 2.32, Corollary 2.33 and the corresponding results in [3, 7, 15, 21, 22] has been clarified in the previous section.

2.2 Unbounded Distributed Delay Case

43

Theorem 2.32 shows that under the condition (2.33), GRN (2.29) has at least one nonnegative equilibrium point. Let (m ∗ , p ∗ ) is an arbitrary but fixed equilibrium point of GRN (2.29). For the convenience of showing that (m ∗ , p ∗ ) is globally asymptotically stable, we shift it to the origin by the transformation xi (t) = m i (t) − m i∗ , yi (t) = pi (t) − pi∗ , i ∈ n,

(2.34)

where m i∗ and pi∗ are the i-th components of m ∗ and p ∗ , respectively. Then GRN (2.29) can be transformed into the following form: x˙i (t) = −ai xi (t) + +

n 

 vi j

n 

t −∞

j=1

wi j g˜ j (y j (t − σ j (t)))

j=1

Θ j (t − s)g˜ j (y j (s))ds, t  0,

(2.35a)

y˙i (t) = −ci yi (t) + di xi (t − τi (t)), t  0,

(2.35b)

xi (t) = ϕi (t) − m i∗ , t ∈ [−d, 0],

(2.35c)

yi (t) = ψi (t) − pi∗ , t ∈ (−∞, 0],

(2.35d)

where i ∈ n, g˜ j (y j (t − σ j (t))) = g j (y j (t − σ j (t)) + p ∗j ) − g j ( p ∗j ). From the relationship between g j and g˜ j , we know from (1.3) that g˜ j satisfies the following sector condition g˜ j (0) = 0, 0  Set

g˜ j (x)  k j , ∀0 = x ∈ R, j ∈ n. x

L = A2 C 2 − 2Γ K 2 (Wˆ + Vˆ ),

where

(2.36)

Γ = diag(γ1 , γ2 , . . . , γn ), Wˆ = [wˆ i j ], Vˆ = [vˆi j ], γi = (1 − τ¯i,d )−1 di2 , wˆ i j = (1 − σ¯ i,d )−1 χ j (W )w 2ji , vˆi j = χ j (V )v 2ji .

Theorem 2.35 If the condition (2.33) is satisfied, and the matrix L defined in (2.36) is a nonsingular M-matrix, then GRN (2.29) has one unique nonnegative equilibrium point which is globally asymptotically stable. Proof Set Lˆ =

 −Γ A2 . −2K 2 (Wˆ + Vˆ ) C 2



(2.37)

44

2 Stability Analysis for GRNs with Mixed Delays

Since the matrix L defined in (2.36) is a nonsingular M-matrix, it follows from Proposition 1.6 that the matrix Lˆ defined in (2.37) is a nonsingular M-matrix. By ˆ  0. Let Lemma 1.4, there is a 2n-dimensional vector γ  0 such that Lγ T  γ = G 1 G 2 · · · G n T1 T2 · · · Tn . Then, from (2.37), ai2 G i − γi Ti > 0, i ∈ n, −

2ki2

(2.38a)

n  (wˆ i j + vˆi j )G j + ci2 Ti > 0, i ∈ n.

(2.38b)

j=1

Set xt (θ) = x(t + θ), θ ∈ [−d, 0], yt (θ) = y(t + θ), θ ∈ (−∞, 0]. Choose an LKF candidate to be V (t, xt , yt ) = V1 (t, xt , yt ) + V2 (t, xt , yt ) + V3 (t, xt , yt ), where V1 (t, xt , yt ) =

n 

G i ai xi2 (t) +

i=1

V2 (t, xt , yt ) =

n 

 Ti γi n 

t

t−τi (t)

Gi

n 

i=1

V3 (t, xt , yt ) = 2

n  i=1

Gi

n 

Ti ci yi2 (t),

i=1

i=1

+2

n 

(2.39)





wˆ ji k 2j

j=1 +∞

vˆ ji

xi2 (s)ds t t−σ j (t)

 Θ j (σ)

0

j=1

t

t−σ

y 2j (s)ds,

g˜ 2j (y j (r ))dr dσ.

By simple computations, one can find that there exist scalars λ1 > 0 and λ2 > 0 satisfying (2.40a) V (t, xt , yt )  λ1 (x(t)22 + y(t)22 ),

V (t, xt , yt )  λ2

 sup

s∈[−d,0]

xt (s)22

This implies that the LKF V (t, xt , yt ) is a legitimate.

+

yt 2C

.

(2.40b)

2.2 Unbounded Distributed Delay Case

45

By computing the derivatives of Vi (t, xt , yt ), i ∈ 3 along the trajectory of (2.35), we can obtain that V˙1 (t, xt , yt ) n n   G i ai xi (t)x˙i (t) + 2 Ti ci yi (t) y˙i (t) =2 −

i=1 n 

i=1

G i ai2 xi2 (t) −

i=1

n 

Ti ci2 yi2 (t) +

i=1

n 

Ti di2 xi2 (t − τi (t))

i=1

⎛  n n n    + Gi ⎝ wi j g˜ j (y j (t − σ j (t)))+ vi j i=1

−

j=1

n 

j=1

G i ai2 xi2 (t) −

i=1

n  i=1

⎛  n n   +2 Gi ⎝ vi j i=1

+2

n 

j=1

−

+2

n 

−∞

n  i=1

G i χi (V )

n 

i=1

+2

Ti di2 xi2 (t − τi (t))

i=1

t

Θ j (t − s)g˜ j (y j (s))ds ⎠

⎞2

Θ j (t − s)g˜ j (y j (s))ds ⎠

j=1

G i ai2 xi2 (t) −

i=1 n 

n 

−∞

⎞2 ⎛ n  Gi ⎝ wi j g˜ j (y j (t − σ j (t)))⎠

i=1 n 

Ti ci2 yi2 (t) +

⎞2 t

Ti ci2 yi2 (t) + 

vi2j

j=1

G i χi (W )

i=1

n 

n 

Ti di2 xi2 (t − τi (t))

i=1 t −∞

2

Θ j (t − s)g˜ j (y j (s))ds

wi2j k 2j y 2j (t − σ j (t)),

(2.41)

j=1

V˙2 (t, xt , yt ) n    Ti γi xi2 (t) − (1 − τ˙i (t))xi2 (t − τi (t)) = i=1

+2

n 

Gi

i=1



n 

n  j=1

Ti γi xi2 (t) −

i=1

−2

  wˆ ji k 2j y 2j (t) − (1 − σ˙ j (t))y 2j (t − σ j (t)) n 

Ti di2 xi2 (t − τi (t)) + 2

i=1 n  i=1

Gi

n  j=1

χi (W )wi2j k 2j y 2j (t − σ j (t)),

n  n 

G j wˆ i j ki2 yi2 (t)

i=1 j=1

(2.42)

46

2 Stability Analysis for GRNs with Mixed Delays

V˙3 (t, xt , yt )  n n   Gi vˆ ji =2 i=1

−2

0

j=1

n 

G i χi (xt )

n 

i=1

2

n 

n 

i=1

−2

 vi2j

j=1

G i χi (xt )

n 

−2

Θ j (σ)g˜ 2j (y j (t − σ))dσ

+∞

 Θ j (σ)dσ

0

0

+∞

Θ j (σ)g˜ 2j (y j (t − σ))dσ

G j vˆi j ki2 yi2 (t)

i=1 j=1 n 

0

 vi2j

j=1

n n  

+∞

vˆ ji k 2j y 2j (t)

i=1

2

Θ j (σ)g˜ 2j (y j (σ))dσ

j=1

Gi

n 

+∞

G i χi (xt )

n 

i=1

 vi2j

j=1

+∞

2 Θ j (σ)g˜ j (y j (t − σ))dσ

.

(2.43)

0

Now, it can be derived from (2.41) to (2.43) that V˙ (t, xt , yt )  −

n 

G i ai2 xi2 (t) +

i=1 n n  

+2

n 

Ti γi xi2 (t) −

i=1

n 

Ti ci2 yi2 (t)

i=1

ki2 (wˆ i j + vˆi j )G j yi2 (t)

i=1 j=1

=−

n   i=1

−

n  i=1

G i ai2 − Ti γi xi2 (t)

⎡

⎤ n  ⎣Ti ci2 − 2ki2 (wˆ i j + vˆi j )G j ⎦ yi2 (t). j=1

From (2.38), there exists a scalar λ3 > 0 such that V˙ (t, xt , yt ) < −λ3 (x(t)22 + y(t)22 ).

(2.44)

By [6, Theorem 3.1.5], the nonnegative equilibrium point (m ∗ , p ∗ ) of GRN (2.29) is asymptotically stable. This, together with (2.40), implies that the nonnegative equilibrium point (m ∗ , p ∗ ) of GRN (2.29) is globally asymptotically stable [4], and hence it is the unique nonnegative equilibrium point of GRN (2.29). Remark 2.36 The LKFs employed in [15, 21, 22] and the previous section are not available for GRNs with unbounded distributed delays. In the proof of Theorem 2.35,

2.2 Unbounded Distributed Delay Case

47

the item V3 (t, xt , yt ) are employed to guarantee that the globally asymptotic stability criteria for GRNs with discrete time-varying delays and unbounded distributed delays can be established. In addition, when the distributed delays vanish, it has been previously investigated that the stability criteria given in Theorem 2.35 is less conservative than ones in [15, 21, 22]. When the discrete delays in GRN (2.29) are constants, the matrix L defined in (2.36) becomes U := A2 C 2 − 2D 2 K 2 [(W ◦ W )T χ(W ) + (V ◦ V )T χ(V )].

(2.45)

Thus, we can easily get the following result from Theorem 2.35. Corollary 2.37 If the condition (2.33) is satisfied, and the matrix U defined in (2.45) is a nonsingular M-matrix, then GRN (2.29) with constant discrete delays and unbounded distributed delays has a unique nonnegative equilibrium point which is globally asymptotically stable. When W + V  0 or V  0, similar to Corollary 2.33, the following result can be directly derived from Theorem 2.35. Corollary 2.38 If W + V  0 or V  0, and the matrix L defined in (2.36) is a nonsingular M-matrix, then GRN (2.29) has a unique nonnegative equilibrium point which is globally asymptotically stable. Remark 2.39 Compared with the LMI-based stability criteria, M-matrix-based ones have lower computational complexity, because there are more than 50 equivalent conditions to verify whether a matrix is a nonsingular M-matrix [13]. One of them is that a Z-matrix is a nonsingular M-matrix if and only if all of its eigenvalues have a positive real part. While the eigenvalues of a matrix can be computed by using the function eig in MATLAB. So the M-matrix-based stability criteria can be easily verified.

2.2.3 Numerical Examples In this section, we will give three numerical examples to illustrate the correctness of our theoretical results. Example 2.40 Consider a three-node GRN in the form of (2.35), where ai = 5, ci = 1.5, di = 0.5, i ∈ 3, τi (t) = (sin t + 1)/4, σi (t) = (cos t + 1)/8, i ∈ 3, t  0,

48

2 Stability Analysis for GRNs with Mixed Delays

gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0, Θi (x) = e−x , i ∈ 3, x  0, ⎤ ⎤ ⎡ −5 0 −3 4 0 2.5 W = ⎣ 0 5 −2⎦ , V = ⎣0 −4.5 1 ⎦ . 0 3 −2 0 −2 4 ⎡

 Clearly, J1 = {1, 3}, J2 = {3}, J3 = ∅, J1 = 8 and J2 = J3 = 2. So, j∈Ji |wi j + vi j |  Ji for all i ∈ 2, and hence the condition (2.33) is satisfied. By Theorem 2.32, the GRN under consideration here has at least one nonnegative √ equilibrium point. Since τ¯i,d = 1/4, σ¯ i,d = 1/8, χi (W ) = χi (V ) = 2 and ki = 3 3/8 for all i ∈ 3, the matrix L defined in (2.36) is ⎤ 31.1786 0 0 0 28.7879 −8.0357⎦ . L=⎣ −9.3013 −3.1339 44.6786 ⎡

By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix L are 46.1307, 27.3358 and 31.1786. From Lemma 1.4, L is a nonsingular Mmatrix. This, together with Theorem 2.35, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. Partial simulation results are shown in Figs. 2.9, 2.10, 2.11 and 2.12. Example 2.41 Consider a three-node GRN in the form of (2.35), where ai = 1.2, ci = 1.2, di = 0.3, i ∈ 3, ⎡

⎤ ⎡ ⎤ 0 2.5 0 2.8 0 0 W = ⎣ 0 0 2.5⎦ , V = ⎣ 0 2.8 0 ⎦ , 2.5 0 0 0 0 2.8 τi (t) = (sin t + 1)/4, σi (t) = (cos t + 1)/8, i ∈ 3, t  0, Θi (x) = e−x , i ∈ 3, x  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. It follows from W + V  0 and Corollary 2.33 that the GRN under consideration here has √ at least one nonnegative equilibrium point. Since χi (W ) = χi (V ) = 1 and ki = 3 3/8 for all i ∈ 3, the matrix L defined in (2.36) is ⎡

⎤ 1.2798 0 −0.7232 ⎦. 0 L = ⎣−0.7232 1.2798 0 −0.7232 1.2798

2.2 Unbounded Distributed Delay Case

49

30

4

20

3

10

2

0 1 -10 0

-20

-1

-30 -40

0

5

10

15

20

25

30

-2

0

5

10

15

20

25

30

15

20

25

30

20

25

30

Fig. 2.9 Trajectories of mRNA and protein concentrations (Example 2.40) 60

4.5 4

50

3.5

40

3 2.5

30

2

20

1.5 1

10 0

0.5 0

5

10

15

20

25

30

0

0

5

10

Fig. 2.10 Trajectories of mRNA and protein concentrations (Example 2.40) 40

2.5

30

2

20

1.5

10

1

0

0.5

-10

0

-20

-0.5

-30

-1

-40

-1.5

0

5

10

15

20

25

30

0

5

10

15

Fig. 2.11 Trajectories of mRNA and protein concentrations (Example 2.40)

50

2 Stability Analysis for GRNs with Mixed Delays 40

3.5 3

30

2.5

20

2 1.5

10

1

0

0.5 0

-10 -20

-0.5 0

5

10

15

20

25

30

-1

0

5

10

15

20

25

30

Fig. 2.12 Trajectories of mRNA and protein concentrations (Example 2.40)

By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix L are 0.5566 and 1.6414 ± 0.6263i. From Lemma 1.4, it is concluded that L is a nonsingular M-matrix. This, together with Corollary 2.38, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. Partial simulation results are shown in Figs. 2.13, 2.14, 2.15 and 2.16. Example 2.42 Consider a three-node GRN in the form of (2.35), where ai = ci = 1, di = 0.1, i ∈ 3, ⎡

⎤ ⎡ ⎤ 0 2.7 0 100 W = ⎣ 0 0 2.7⎦ , V = ⎣0 1 0⎦ , 2.7 0 0 001 τi (t) ≡ 1/4, σi (t) ≡ 1/8, i ∈ 3, t  0, Θi (x) = e−x , i ∈ 3, x  0, gi (x) = x 2 /(1 + x 2 ), i ∈ 3, x  0. It follows from W + V  0 and Corollary 2.33 that the GRN under consideration here has √ at least one nonnegative equilibrium point. Since χi (W ) = χi (V ) = 1 and ki = 3 3/8 for all i ∈ 3, the matrix U defined in (2.45) is ⎡

⎤ 0.9916 0 −0.0615 ⎦. 0 U = ⎣−0.0615 0.9916 0 −0.0615 0.9916 By using the function eig in MATLAB, we can easily get that the eigenvalues of the matrix U are 0.9301 and 1.0223 ± 0.0533i. From Lemma 1.4, it is concluded

2.2 Unbounded Distributed Delay Case

51

120

30

100

25

80

20

60

15

40

10

20

5

0

0

10

20

30

40

50

60

0

0

10

20

30

40

50

60

40

50

60

40

50

60

Fig. 2.13 Trajectories of mRNA and protein concentrations (Example 2.41) 120

30

100

25

80

20

60

15

40

10

20

5

0

0

10

20

30

40

50

60

0

0

10

20

30

Fig. 2.14 Trajectories of mRNA and protein concentrations (Example 2.41) 120

30

100

25

80

20

60

15

40

10

20

5

0

0

10

20

30

40

50

60

0

0

10

20

30

Fig. 2.15 Trajectories of mRNA and protein concentrations (Example 2.41)

52

2 Stability Analysis for GRNs with Mixed Delays 120

30

100

25

80

20

60

15

40

10

20

5

0

0

10

20

30

40

50

60

0

0

10

20

30

40

50

60

40

50

60

40

50

60

Fig. 2.16 Trajectories of mRNA and protein concentrations (Example 2.41) 50

5

45

4.5

40

4

35

3.5

30

3

25

2.5

20

2

15

1.5

10

1

5

0.5

0

0

10

20

30

40

50

60

0

0

10

20

30

Fig. 2.17 Trajectories of mRNA and protein concentrations (Example 2.42) 45

4.5

40

4

35

3.5

30

3

25

2.5

20

2

15

1.5

10

1

5

0.5

0

0

10

20

30

40

50

60

0

0

10

20

30

Fig. 2.18 Trajectories of mRNA and protein concentrations (Example 2.42)

that U is a nonsingular M-matrix. This, together with Corollary 2.37, implies that the GRN under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. Partial simulation results are shown in Figs. 2.17, 2.18, 2.19 and 2.20.

2.3 Remarks and Notes

53

50

5

45

4.5

40

4

35

3.5

30

3

25

2.5

20

2

15

1.5

10

1

5

0.5

0

0

10

20

30

40

50

60

0

0

10

20

30

40

50

60

40

50

60

Fig. 2.19 Trajectories of mRNA and protein concentrations (Example 2.42) 5

60

4.5

50

4 3.5

40

3 2.5

30

2

20

1.5 1

10 0

0.5 0

10

20

30

40

50

60

0

0

10

20

30

Fig. 2.20 Trajectories of mRNA and protein concentrations (Example 2.42)

2.3 Remarks and Notes This chapter addressed the problem of establishing globally asymptotic stability criteria for nonnegative equilibrium points of a kind of GRNs with mixed delays. The results about the case of constant and unbounded distributed delays are taken from [30] and [29], respectively. The stability criteria obtained in this chapter are delay-range-independent and delay-rate-dependent. It is worth emphasizing that the globally asymptotic stability criteria established in this chapter are also available for GRNs only with discrete delays by setting F = 0 or V = 0. So, this chapter can be viewed as an extensive and supplementary version of the corresponding results in [15–17, 21, 22]. The difference among these Mmatrix-based stability criteria is that the stability criteria proposed in [21] are delayrange-independent and delay-rate-independent, while the stability criteria proposed in [15–17, 22] are delay-range-independent and delay-rate-dependent. We end this chapter by introducing the following items, which are related to this chapter:

54

2 Stability Analysis for GRNs with Mixed Delays

1. Zhang et al. [32] derive robust asymptotic stability criteria for uncertain GRNs with random discrete time delays and distributed time delays by utilizing an LKF method combined with delay decomposing technique and reciprocally convex technique; 2. Tian et al. [14] investigated several criteria for the absolute ultimate boundedness of GRNs with mixed time-delays by using comparing theorem and Dini derivation method. Thereby, it is obtained that GRNs with mixed time-delays are absolute ultimate bounded in sense of Lagrange without any additional conditions; 3. Ling et al. [9] analyze the stability and bifurcations of cyclic GRNs with discrete and Gamma-type distributed delays. The Gamma-type distributed delay kernels can present more realistic biological background among genes. The existence of positive equilibria is verified, and exact conditions of biochemical parameters for stability and bifurcations in cyclic GRNs with both positive and negative gains are deduced, respectively. While the stability and bifurcations of GRNs with multiple delays are achieved in [23, 24]; 4. Yin and Liu [25, 26] investigate the delay-dependent global exponential stability criteria for GRNs with time-varying delays and continuous distributed delays by employing the LKF method and convex technique; 5. Chen and Jiang [3] investigated the existence and uniqueness theorem of equilibrium points for a kind of GRNs with discrete delays and infinite distributed delays by constructing a homeomorphism of Rn onto itself; 6. The problem of stability analysis of stochastic (respectively, Markov jump, switched) GRNs with mixed delayed has been addressed in [11, 18–20, 28, 33]. A brief introduce on these results will be given in Sect. 5.4 below. 7. For a new form of coupled repressilators of GRNs, Ling et al. [8] investigate the existence of a unique equilibrium and analyze the stability of the equilibrium by studying the root distribution of the characteristic equation. 8. A model of genetic regulatory system with delay is considered in [2], and it is proved that under certain conditions the model has a unique constant equilibrium which is globally attractive.

References 1. Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979) 2. Chen, S., Wei, J.: Global attractivity in a model of genetic regulatory system with time delay. Appl. Math. Comput. 232, 411–415 (2014) 3. Chen, Z., Jiang, H.: Advances in neural network. In: Stability Analysis of Genetic Regulatory Networks with Mixed Time-delays. Lecture Notes in Computer Science, vol. 6677, pp. 280– 289. Springer, Berlin, Heidelberg (2011) 4. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer-Verlag, New York (1993) 5. He, W., Cao, J.: Robust stability of genetic regulatory networks with distributed delay. Cogn. Neurodyn. 2(4), 355–361 (2008)

References

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6. Kolmanovskii, V.B., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Boston (1992) 7. Li, Y., Zhang, X., Tan, C.: Global exponential stability analysis of discrete-time genetic regulatory networks with time delays. Asian J. Control 15(5), 1448–1457 (2013) 8. Ling, G., Guan, Z.H., He, D.X., Liao, R.Q., Zhang, X.H.: Stability and bifurcation analysis of new coupled repressilators in genetic regulatory networks with delays. Neural Netw. 60, 222–231 (2014) 9. Ling, G., Guan, Z.H., Liao, R.Q., Cheng, X.M.: Stability and bifurcation analysis of cyclic genetic regulatory networks with mixed time delays. SIAM J. Appl. Dyn. Syst. 14(1), 202–220 (2015) 10. Luo, Q., Zhang, R., Liao, X.: Unconditional global exponential stability in Lagrange sense of genetic regulatory networks with SUM regulatory logic. Cogn. Neurodyn. 4(3), 251–261 (2010) 11. Meng, Q., Jiang, H.J.: Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays. Neurocomputing 74(1), 362–368 (2010) 12. Minc, H.: Nonnegative Matrices. Wiley, New York (1988) 13. Plemmons, R.J.: M-matrix characterizations. I–nonsingular M-matrices. Linear Algebra Appl. 18(2), 175–188 (1977) 14. Tian, F.X., Zhou, G.P., Liao, X.X.: Absolute ultimate boundedness of genetic regulatory networks with mixed time-delays. In: Proceedings of the 35th Chinese Control Conference (CCC), pp. 3503–3508. IEEE, Chengdu (2016) 15. Tian, L.P., Shi, Z.K., Liu, L.Z., Wu, F.X.: M-matrix-based stability conditions for genetic regulatory networks with time-varying delays and noise perturbations. IET Syst. Biol. 7(5), 214–222 (2013) 16. Tian, L.P., Shi, Z.K., Wu, F.X.: New global stability conditions for genetic regulatory networks with time-varying delays. In: Proceedings of 2012 IEEE 6th International Conference on Systems Biology (ISB), pp. 185–191. IEEE (2012) 17. Tian, L.P., Wu, F.X.: Globally delay-independent stability of ring-structured genetic regulatory networks. In: Proceedings of the 24th Canadian Conference on Electrical and Computer Engineering (CCECE), pp. 000308–000311 (2011) 18. Wang, L., Cao, J.: Global stability of switched genetic regulatory networks with noises and mixed time-delays. In: Proceedings of the 32nd Chinese Control Conference (CCC), pp. 1498– 1502. IEEE (2013) 19. Wang, W., Zhong, S., Liu, F., Cheng, J.: Robust delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays. Int. J. Robust Nonlinear Control 24(16), 2574–2596 (2014) 20. Wang, Z., Liao, X., Mao, J., Liu, G.: Robust stability of stochastic genetic regulatory networks with discrete and distributed delays. Symp. (Int.) Combust. 13(12), 1199–1208 (2009) 21. Wu, F.X.: Delay-independent stability of genetic regulatory networks. IEEE Trans. Neural Netw. 22(11), 1685–1693 (2011) 22. Wu, F.X.: Global and robust stability analysis of genetic regulatory networks with time-varying delays and parameter uncertainties. IEEE Trans. Biomed. Circuits Syst. 5(4), 391–398 (2011) 23. Wu, F.X.: Stability and bifurcation of ring-structured genetic regulatory networks with time delays. IEEE Trans. Circuits Syst. I: Regul. Pap. 59(6), 1312–1320 (2012) 24. Xiao, M., Zheng, W.X., Cao, J.D.: Stability and bifurcation of genetic regulatory networks with small RNAs and multiple delays. Int. J. Comput. Math. 91(5), 907–927 (2014) 25. Yin, L., Liu, Y.: New global exponential stability criteria for genetic regulatory networks with mixed delays. In: Proceedings of the 32nd Chinese Control Conference (CCC), pp. 1503–1508. IEEE (2013) 26. Yin, L.Z., Liu, Y.G.: Exponential stability analysis for genetic regulatory networks with both time-varying and continuous distributed delays. Abstr. Appl. Anal. 2014 (Article ID 897280, 10 pages, 2014)

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27. Zhang, W., Fang, J., Tang, Y.: Stochastic stability of Markovian jumping genetic regulatory networks with mixed time delays. Appl. Math. Comput. 217(17), 7210–7225 (2011) 28. Zhang, W.B., Fang, J.A., Tang, Y.: New robust stability analysis for genetic regulatory networks with random discrete delays and distributed delays. Neurocomputing 74(14–15), 2344–2360 (2011) 29. Zhang, X., Han, Y.Y., Wu, L., Zou, J.H.: M-matrix-based globally asymptotic stability criteria for genetic regulatory networks with time-varying discrete and unbounded distributed delays. Neurocomputing 174, 1060–1069 (2016) 30. Zhang, X., Wu, L., Zou, J.H.: Globally asymptotic stability analysis for genetic regulatory networks with mixed delays: an M-matrix-based approach. IEEE/ACM Trans. Comput. Biol. Bioinf. 13(1), 135–147 (2016) 31. Zhang, X., Yu, A.H., Zhang, G.D.: M-matrix-based delay-range-dependent global asymptotical stability criterion for genetic regulatory networks with time-varying delays. Neurocomputing 113, 8–15 (2013) 32. Zhang, X.W., Li, R.X., Han, C., Yao, R.: Robust stability analysis of uncertain genetic regulatory networks with mixed time delays. Int. J. Mach. Learn. Cybern. 7(6), 1005–1022 (2016) 33. Zhu, Y., Zhang, Q., Wei, Z., Zhang, L.: Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing 110, 44–50 (2013)

Chapter 3

Stability Analysis of Delayed GRNs

The chapter first proves an inequality concerning with double integrals by partitioning the integral domain into two parts and exchanging the order of double integrals over a sub-domain. Then it is mathematically proven that the proposed integral inequality is less conservative than Lemma 1.14(ii). Thereby, for a class of GRNs with time-varying delays, a pair of delay-range-dependent and delay-rate-dependent asymptotic stability criteria are investigated by constructing an appropriate LKF and applying reciprocally convex techniques. The obtained stability criteria are given in the form of LMIs, which can be easily checked by the Toolbox YALMIP of MATLAB. Furthermore, it is theoretically proven that a stability criterion proposed in this chapter is less conservative than [25, Corollary 3.1]. Finally, numerical examples and their simulation results show that the stability criteria proposed in this chapter may be less conservative than ones in [13, 21, 25, 28, 30].

3.1 Problem Formulation Consider the delayed GRN model (1.8) subject to (1.9), that is, ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(3.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0,

(3.1b)

m(s) ˜ = φ(s), p(s) ˜ = ϕ(s), s ∈ [−d, 0],

(3.1c)

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_3

57

58

3 Stability Analysis of Delayed GRNs

where d = max{σ2 , τ2 }, and f satisfies the following sector condition: f j (0) = 0, 0 

f j (s)  k j , ∀0 = s ∈ R, j ∈ n, s

(3.2)

k j is a nonnegative scalar, and f j (·) is the jth entry of f (·). Let K = diag(k1 , k2 , . . . , kn ). In this chapter, we will always assume that the delays τ (t) and σ(t) are differentiable functions satisfying 0  τ1  τ (t)  τ2 , 0  σ1  σ(t)  σ2 ,

(3.3a)

˙  σd < ∞, τ˙ (t)  τd < ∞, σ(t)

(3.3b)

where τ1 , τ2 , σ1 , σ2 , τd and σd are known constants. Definition 3.1 The trivial solution of delayed GRN 3.1 is asymptotically stable, if the following (i) and (ii) are satisfied: (i) for any ε > 0, there exists a δ(ε) > 0 such that sup

−ds0

  φ(s)22 + ϕ(s)22 < δ

2 2 ˜ ⇒ m(t) ˜ 2 +  p(t)) 2 < ε, ∀t  0;

(ii) there exists a δ˜ > 0 such that sup

−ds0

  φ(s)22 + ϕ(s)22 < δ˜

⇒ m(t) ˜ → 0, p(t) ˜ → 0 as t → +∞. Wang et al. [25] investigated asymptotic stability criteria for GRN (3.1) by using an LKF containing triple–integral items. A key step of [25] is to estimate the double– integral items in the derivative of LKF by using Lemma 1.14(ii), which is an integral inequality concerning with double-integrals. This motivates us to research the following three questions: (i) Improve the inequality in Lemma 1.14(ii) to a less conservative form (see (3.5) below); (ii) Establish asymptotic stability criteria for GRN (3.1); (iii) Give theoretical and/or numerical comparisons of the stability criteria proposed in the chapter and ones in literature. The answer to the question (i) will offer a mathematical tool to solve the question (ii), and the answer to the question (iii) will ensure that the approach to solve the question (ii) is more effective.

3.2 An Improved Integral Inequality

59

3.2 An Improved Integral Inequality In this section, we will improve the integral inequality in Lemma 1.14(ii) to a less conservative form (see (3.5) below), which plays an important role in stability analysis of systems with interval time-varying delays. For this end, we first investigate the following lemma. Lemma 3.2 For given a symmetric positive definite matrix M ∈ Rn×n , scalars a and b satisfying b > a > 0, and vectors p, q ∈ Rn , the following inequality holds: 2 2 1 ( p + q)T M( p + q)  p T M p + q T Mq. b+a b−a a In addition, the sign of equality in (3.4) holds if and only if p =

(3.4)

b−a q. 2a

Proof The conclusions can be obtained from 2 2 1 ( p + q)T M( p + q) − p T M p − q T Mq b+a b−a a   2a T b−a T 2 T T =− p M p+ q Mq − p Mq −q M p b+a b−a 2a T      2a b−a 2a b−a 2 p− q M p− q =− b+a b−a 2a b−a 2a  0. The proof is completed. In order to reduce the conservativeness of Lemma 1.14(ii), we investigate the following proposition. Proposition 3.3 For given a symmetric positive definite matrix M ∈ Rn×n , scalars a and b satisfying b  a  0, and an integral vector function w : [−b, −a] → Rn , the following inequality holds: −a 0 1 2 a(b − a) w T (s)Mw(s)dsdθ 2 −b θ −a −a −a −a w T (s)dsdθM w(s)dsdθ a −b

θ

1 + (b − a)3 2



−b

0 −a



w T (s)ds M

θ

0 −a

w(s)ds.

(3.5)

Proof When a = 0 or b = a, it is clear that the inequality (3.5) is true. If, on the other hand, a = 0 and b = a, then

60

3 Stability Analysis of Delayed GRNs





−a

−b −a

=

0

θ



−b

w T (s)Mw(s)dsdθ

−a

θ

w (s)Mw(s)dsdθ + T

−a



−b

0

−a

w T (s)Mw(s)dsdθ.

Based on the method of variable substitute, let T = θ + a, then



−a

−b



−a

w (s)Mw(s)dsdθ =

0

T

θ

−(b−a)



−a −a+T

w T (s)Mw(s)dsdT.

Then, replacing variable symbol T with θ, we have

=



−a

−b 0

0

θ

w T (s)Mw(s)dsdθ



−(b−a)



−a

−a+θ

w T (s)Mw(s)dsdθ + (b − a)

0 −a

w T (s)Mw(s)ds.

This, together with (i) and (ii) of Lemma 1.14, implies that

−a

−b





0

θ

w T (s)Mw(s)dsdθ

0 0 −a −a 2 T w (s)dsdθM w(s)dsdθ (b − a)2 −(b−a) −a+θ −(b−a) −a+θ 0 b−a 0 T + w (s)ds M w(s)ds. a −a −a

Furthermore,

−a



0

w T (s)Mw(s)dsdθ −b θ −a −a −a −a 2 T w (s)dsdθM w(s)dsdθ  (b − a)2 −b θ −b θ 0 0 b−a w T (s)ds M w(s)ds. + a −a −a The proof is completed. The following proposition presents that the inequality (3.5) is less conservative than Lemma 1.14(ii) when b > a > 0. Proposition 3.4 For given a symmetric positive definite matrix M ∈ Rn×n , scalars a and b satisfying b > a > 0, and an integral vector function w : [−b, −a] → Rn , the following inequality holds:

3.2 An Improved Integral Inequality

61

−a 0 −a 0 2 T w (s)dsdθM w(s)dsdθ b2 − a 2 −b θ −b θ −a −a −a −a 2 w T (s)dsdθM w(s)dsdθ  (b − a)2 −b θ −b θ 0 b−a 0 T w (s)ds M w(s)ds. + a −a −a Proof The proof is completed by applying Lemma 3.2 to p=

−a −b



−a θ

w(s)dsdθ, q = (b − a)

0 −a

w(s)ds.

In next section, we will analyze the stability of GRN (3.1) by using the LKF approach and LMI technique. Due to introducing triple–integral terms into our LKF, double–integral items will appear in the derivative of LKF. Proposition 3.4 can guarantee that a less conservative stability criterion is obtained if we use (3.5) instead of Lemma 1.14(ii) to deal with the double-integral items.

3.3 Stability Criteria In this section we will analyze stability of a class of GRNs with time-varying delays. The following theorem provides a delay-range-dependent and delay-rate-dependent asymptotic stability criterion for GRN (3.1). Theorem 3.5 For given constants τ1 , τ2 , σ1 , σ2 , τd and σd with 0 < τ1 < τ2 and 0 < σ1 < σ2 , GRN (3.1) subject to (3.3) and (3.2) is asymptotically stable if there exist matrices P T = P > 0, Q iT = Q i > 0, RiT = Ri > 0 (i ∈ 7), Λ j := diag(λ1 j , λ2 j , . . . , λn j ) > 0, T j := diag(t1 j , t2 j , . . . , tn j ) > 0 ( j = 1, 2), Q˜ 5 and R˜ 5 of appropriate sizes, such that



 Q 5 Q˜ 5 R5 R˜ 5  0, ˜ T 0 Q˜ T5 Q 5 R5 R5

and Ψ := sym(Ψ0 + Ψ1 ) +

6

Ψi < 0,

i=2

where



e9 Ψ1 = Γ PΓd + K e2 − e9 T

T

 Λ1 (−Ce2 + De4 ), Λ2

Ψ2 = e1T (Q 1 + Q 3 )e1 + e3T (Q 2 − Q 1 )e3 − e5T Q 2 e5 − (1 − τd )e4T Q 3 e4 ,

(3.6)

(3.7)

62

3 Stability Analysis of Delayed GRNs

Ψ3 = e2T R1 e2 + e6T (R2 − R1 )e6 − e8T R2 e8

T 

T  e2 e e2 e − (1 − σd ) 7 + R3 R3 7 , e9 e9 e10 e10 T 

e1 e1 2 2 Ψ4 = (τ1 Q 4 +τ12 Q 5 ) −Ae1 + W e10 −Ae1 + W e10 T



e11 e11 − Q4 e1 − e3 e1 − e3 ⎡ ⎤T ⎤ ⎡ e13 e13

 ⎢e3 − e4 ⎥ Q 5 Q˜ 5 ⎢e3 − e4 ⎥ ⎢ ⎥ ⎥ −⎢ ⎣ e12 ⎦ Q˜ T Q 5 ⎣ e12 ⎦ , 5 e4 − e5 e4 − e5





T  e2 e2 2 2 Ψ5 = (σ1 R4 + σ12 R5 ) −Ce2 + De4 −Ce2 + De4 T



e14 e14 − R4 e2 − e6 e2 − e6 ⎡ ⎤T ⎤ ⎡ e16 e16

 ⎢e6 − e7 ⎥ R5 R˜ 5 ⎢e6 − e7 ⎥ ⎢ ⎥ ⎥ −⎢ ⎣ e15 ⎦ R˜ T R5 ⎣ e15 ⎦ , 5 e7 − e8 e7 − e8 Ψ6 = (−Ae1 + W e10 )T (

τ14 Q 6 + τs2 Q 7 )(−Ae1 + W e10 ) 4

+(−Ce2 + De4 )T (

σ14 R6 + σs2 R7 )(−Ce2 + De4 ) 4

−(τ1 e1 − e11 )T Q 6 (τ1 e1 − e11 ) 2τs (τ12 e3 − e12 − e13 )T Q 7 (τ12 e3 − e12 − e13 ) 2 τ12 τs τ12 (e1 − e3 )T Q 7 (e1 − e3 ) − (σ1 e2 − e14 )T R6 (σ1 e2 − e14 ) − τ1

−

2σs (σ12 e6 − e15 − e16 )T R7 (σ12 e6 − e15 − e16 ) 2 σ12 σs σ12 (e2 − e6 )T R7 (e2 − e6 ), − σ1 −

Ψ0 = −(e9 − K e2 )T T1 e9 − (e10 − K e7 )T T2 e10 , Γ = col(e1 , e2 , e11 , e12 + e13 , e14 , e15 + e16 ),

3.3 Stability Criteria

63

Γd = col(−Ae1 + W e10 , −Ce2 + De4 , e1 − e3 , e3 − e5 , e2 − e6 , e6 − e8 ), τ12 = τ2 − τ1 , τs =

1 2 1 (τ − τ12 ), σ12 = σ2 − σ1 , σs = (σ22 − σ12 ), 2 2 2

ei = [0n×(i−1)n In 0n×(16−i)n ], i ∈ 16. Proof Set  η(t) = col m(t), ˜ p(t), ˜ m(t ˜ − τ1 ), m(t ˜ − τ (t)), m(t ˜ − τ2 ), p(t ˜ − σ1 ), p(t ˜ − σ(t)), p(t ˜ − σ2 ), f ( p(t)), ˜ f ( p(t ˜ − σ(t))),



t−τ1

t−τ (t)

m(s)ds, ˜



t

p(s)ds, ˜

t−σ1

m(s)ds, ˜

t−τ1 t−σ(t)



t−σ1

m(s)ds, ˜

 p(s)ds ˜ ,

t−σ1

t−σ(t)



t

m(s)ds, ˜ t−τ1  t−σ1 p(s)ds, ˜ p(s)ds ˜ .

t

t−τ (t)

t−τ2

p(s)ds, ˜

t−σ2

 η1 (t) = col m(t), ˜ p(t), ˜



t

t−τ1

m(s)ds, ˜

t−τ2

t−σ2

˜˙ = (−Ce2 + De4 )η(t), η1 (t) = Γ η(t) and Then m(t) ˜˙ = (−Ae1 + W e10 )η(t), p(t) η˙1 (t) = Γd η(t). Choose an LKF candidate as V (t, m˜ t , p˜ t ) =

6

Vi (t, m˜ t , p˜ t ),

(3.8)

i=1

where V1 (t, m˜ t , p˜ t ) = η1T (t)Pη1 (t) + 2 +2

n i=1



λi2

λi1

p˜i (t)



+

(ki s − f i (s))ds, t−τ1

m˜ (s)Q 1 m(s)ds ˜ +

t−τ2 t

t−τ (t)

f i (s)ds

0

T

t−τ1

p˜i (t)

0

i=1



t

V2 (t, m˜ t , p˜ t ) =



n

m˜ T (s)Q 3 m(s)ds, ˜

m˜ T (s)Q 2 m(s)ds ˜

64

3 Stability Analysis of Delayed GRNs

V3 (t, m˜ t , p˜ t ) =



t

t−σ1

p˜ T (s)R1 p(s)ds ˜ +

t−σ1





t

+

t−σ(t)

p(s) ˜ f ( p(s)) ˜

p˜ T (s)R2 p(s)ds ˜

t−σ2

T

R3

 p(s) ˜ ds, f ( p(s)) ˜



T  m(s) ˜ m(s) ˜ V4 (t, m˜ t , p˜ t ) = τ1 dsdθ Q4 ˙ ˙˜ m(s) ˜ −τ1 t+θ m(s)

T  −τ1 t

m(s) ˜ m(s) ˜ +τ12 dsdθ, Q 5 ˙ ˙˜ m(s) ˜ −τ2 t+θ m(s)



0

t

T

 p(s) ˜ p(s) ˜ dsdθ R 4 ˙˜ ˙˜ p(s) −σ1 t+θ p(s) T

 −σ1 t

p(s) ˜ p(s) ˜ +σ12 dsdθ, R5 ˙ ˙˜ p(s) ˜ t+θ p(s) −σ2

V6 (t, m˜ t , p˜ t ) =

τ12 2



+τs

0





t



0

t

˙˜ m˙˜ T (s)Q 6 m(s)dsdλdθ

−τ1 θ t+λ −τ1 0 t

σ2 + 1 2 +σs



0

V5 (t, m˜ t , p˜ t ) = σ1

−τ2 0



θ



0

˙˜ m˜˙ T (s)Q 7 m(s)dsdλdθ

t+λ t



−σ1 θ t+λ −σ1 0 t

−σ2

θ

˙˜ p˙˜ T (s)R6 p(s)dsdλdθ ˙˜ p˜˙ T (s)R7 p(s)dsdλdθ,

t+λ

and the matrices P, Q i , Ri , (i ∈ 7), Λ j := diag(λ1 j , λ2 j , . . . , λn j ) > 0 ( j = 1, 2) are taken from a feasible solution to (3.6) and (3.7). The time derivative of V (t, m˜ t , p˜ t ) along the trajectories of GRN (3.1) can be easily obtained as follows: V˙ (t, m˜ t , p˜ t ) =

6

V˙i (t, m˜ t , p˜ t ),

(3.9)

i=1

where T ˙˜ + 2[K p(t) ˙˜ ˜ Λ2 p(t) ˜ − f ( p(t))] ˜ V˙1 (t, m˜ t , p˜ t ) = 2η1T (t)P η˙1 (t) + 2 f T ( p(t))Λ 1 p(t)

= η T (t)sym(Ψ1 )η(t),

(3.10)

˜ − m˜ T (t − τ2 )Q 2 m(t ˜ − τ2 ) V˙2 (t, m˜ t , p˜ t ) = m˜ T (t)(Q 1 + Q 3 )m(t) +m˜ T (t − τ1 )(Q 2 − Q 1 )m(t ˜ − τ1 ) T −(1 − τ˙ (t))m˜ (t − τ (t))Q 3 m(t ˜ − τ (t))  η T (t)Ψ2 η(t),

(3.11)

3.3 Stability Criteria

65

˜ − p˜ T (t − σ2 )R2 p(t ˜ − σ2 ) V˙3 (t, m˜ t , p˜ t ) = p˜ T (t)R1 p(t) + p˜ T (t − σ1 )(R2 − R1 ) p(t ˜ − σ1 )

 T

p(t) ˜ p(t) ˜ R3 + f ( p(t)) ˜ f ( p(t)) ˜ 

T

p(t ˜ − σ(t)) p(t ˜ − σ(t)) R3 −(1 − σ(t)) ˙ f ( p(t ˜ − σ(t))) f ( p(t ˜ − σ(t)))  η T (t)Ψ3 η(t),

(3.12)



T  m(t) ˜ m(t) ˜ 2 V˙4 (t, m˜ t , p˜ t ) = ˙ (τ12 Q 4 + τ12 Q5) ˙ m(t) ˜ m(t) ˜

T  t

m(s) ˜ m(s) ˜ ds Q −τ1 4 ˙ ˙˜ m(s) ˜ t−τ1 m(s)

T  t−τ1

m(s) ˜ m(s) ˜ −τ12 Q5 ˙ ds, m(s) ˜˙ m(s) ˜ t−τ2



T  p(t) ˜ p(t) ˜ 2 2 ˙ (σ1 R4 + σ12 R5 ) ˙ V5 (t, m˜ t , p˜ t ) = ˙ p(t) ˜ p(t) ˜ T

 t

p(s) ˜ p(s) ˜ ds R −σ1 4 ˙ ˙˜ p(s) ˜ t−σ1 p(s) T

 t−σ1

p(s) ˜ p(s) ˜ −σ12 ds, R 5 ˙ ˙˜ p(s) p(s) ˜ t−σ2

(3.13)

(3.14)

τ σ ˙˜ ˙˜ V˙6 (t, m˜ t , p˜ t ) = m˙˜ T (t)( 1 Q 6 + τs2 Q 7 )m(t) + p˙˜ T (t)( 1 R6 + σs2 R7 ) p(t) 4 4 t τ2 0 ˙˜ m˙˜ T (s)Q 6 m(s)dsdθ − 1 2 −τ1 t+θ −τ1 t ˙˜ −τs m˙˜ T (s)Q 7 m(s)dsdθ 4

σ2 − 1 2 −σs

−τ2 0



4

t+θ t



−σ1 t+θ −σ1 t −σ2

From Lemma 1.14, we have

t+θ

˙˜ p˙˜ T (s)R6 p(s)dsdθ ˙˜ p˙˜ T (s)R7 p(s)dsdθ.

(3.15)

66

3 Stability Analysis of Delayed GRNs

T  m(s) ˜ m(s) ˜ ds −τ1 Q4 ˙ ˙˜ m(s) ˜ t−τ1 m(s) T



e11 e11 η(t),  −η T (t) Q4 e1 − e3 e1 − e3



t

T

 p(s) ˜ p(s) ˜ ds R 4 ˙ ˙˜ p(s) ˜ t−σ1 p(s) T



e14 e14 T η(t),  −η (t) R4 e2 − e6 e2 − e6

−σ1

(3.16)



t

(3.17)

and from Lemmas 1.14 and 1.9, we can obtain



T  m(s) ˜ m(s) ˜ ds −τ12 Q5 ˙ ˙˜ m(s) m(s) ˜ t−τ2 T

 t−τ1

m(s) ˜ m(s) ˜ = −τ12 ds Q 5 ˙ ˙˜ m(s) ˜ t−τ (t) m(s)

T  t−τ (t)

m(s) ˜ m(s) ˜ −τ12 ds Q 5 ˙ ˙˜ m(s) m(s) ˜ t−τ2 ⎡ ⎤T ⎤ ⎡ e13 e13

 ⎢e3 − e4 ⎥ Q 5 Q˜ 5 ⎢e3 − e4 ⎥ ⎢ ⎥ ⎥  −η T (t) ⎢ ⎣ e12 ⎦ Q˜ T Q 5 ⎣ e12 ⎦ η(t). 5 e4 − e5 e4 − e5

t−τ1

(3.18)

Similarly, we have T

 p(s) ˜ p(s) ˜ ds R 5 ˙˜ ˙˜ p(s) p(s) t−σ2 ⎡ ⎤T ⎤ ⎡ e16 e16

 ⎢e6 − e7 ⎥ R5 R˜ 5 ⎢e6 − e7 ⎥ ⎢ ⎥ ⎥  −η T (t) ⎢ ⎣ e15 ⎦ R˜ T R5 ⎣ e15 ⎦ η(t). 5 e7 − e8 e7 − e8

−σ12



t−σ1

(3.19)

When w : D → Rn is a derivative vector function, by exchanging integral order, one can easily obtain

−a

−b

θ

−a

w(s)dsdθ ˙ =

−a

−b



s

−b

w(s)dθds ˙ =

−a −b

(s + b)w(s)ds. ˙

3.3 Stability Criteria

67

By using integration by parts, we have

−a



−b

−a θ

w(s)dsdθ ˙ = (b − a)w(−a) −

−a −b

w(s)ds.

(3.20)

Then it follows from Lemma 1.14 and (3.20) that t τ12 0 ˙˜ − m˙˜ T (s)Q 6 m(s)dsdθ 2 −τ1 t+θ 0 t 0 t T ˙ ˙˜ − m˜ (s)dsdθ Q 6 m(s)dsdθ −τ1



t+θ

−τ1



t

˜ − = − τ1 m(t)

T m(s)ds ˜

t+θ

Q 6 τ1 m(t) ˜ −

t−τ1

t

 m(s)ds ˜

t−τ1

= −η T (t)(τ1 e1 − e11 )T Q 6 (τ1 e1 − e11 )η(t),

(3.21)

t σ12 0 ˙˜ − p˙˜ T (s)R6 p(s)dsdθ 2 −σ1 t+θ 0 t 0 t ˙˜ − p˙˜ T (s)dsdθ R6 p(s)dsdθ −σ1



t+θ

˜ − = − σ1 p(t)



−σ1

t

T p(s)ds ˜

t+θ



R6 σ1 p(t) ˜ −

t−σ1

= −η T (t)(σ1 e2 − e14 )T R6 (σ1 e2 − e14 )η(t).



t

 p(s)ds ˜

t−σ1

(3.22)

Due to Lemma 1.14, Proposition 3.3 and (3.20), we have −τs

−τ1 −τ2



t

t+θ

˙˜ m˙˜ T (s)Q 7 m(s)dsdθ

−τ1 t−τ1 2τs −τ1 t−τ1 ˙ T ˙˜ (s)dsdθ Q m ˜ m(s)dsdθ 7 2 τ12 −τ2 t+θ −τ2 t+θ t τs τ12 t ˙ T ˙˜ − m˜ (s)ds Q 7 m(s)ds τ1 t−τ1 t−τ1 T



t−τ1 t−τ1 2τs ˜ − τ1 ) − m(s)ds ˜ Q 7 τ12 m(t ˜ − τ1 ) − m(s)ds ˜ = − 2 τ12 m(t τ12 t−τ2 t−τ2 τs τ12 T − [m(t) ˜ − m(t ˜ − τ1 )] Q 7 [m(t) ˜ − m(t ˜ − τ1 )] τ1 2τs = − 2 η T (t)[τ12 e3 − e12 − e13 ]T Q 7 [τ12 e3 − e12 − e13 ]η(t) τ12 τs τ12 T − η (t)(e1 − e3 )T Q 7 (e1 − e3 )η(t). (3.23) τ1 −

68

3 Stability Analysis of Delayed GRNs

Similarly, −σs

−σ1



−σ2

t

˙˜ p˙˜ T (s)R7 p(s)dsdθ

t+θ

2σs T η (t)[σ12 e6 − e15 − e16 ]T R7 [σ12 e6 − e15 − e16 ]η(t) 2 σ12 σs σ12 T − η (t)(e2 − e6 )T R7 (e2 − e6 )η(t). σ1

−

(3.24)

Next, from the sector condition (3.2) one can obtain T 0  −2[ f ( p(t)) ˜ − K p(t)] ˜ T1 f ( p(t)) ˜ −2[ f ( p(t ˜ − σ(t))) − K p(t ˜ − σ(t))]T T2 f ( p(t ˜ − σ(t))) T = η (t)sym(Ψ0 )η(t).

(3.25)

Then, the combination of (3.8)–(3.19) and (3.21)–(3.25) yields V˙ (t, m˜ t , p˜ t )  η T (t)Ψ η(t). It is easy to see that V˙ (t, m˜ t , p˜ t ) < 0 from (3.7). Thus, GRN (3.1) is asymptotically stable. The proof is completed. Remark 3.6 In the case that P = diag(P1 , P2 , 0), the item η1T (t)Pη1 (t) in V1 (t, ˜ + p˜ T (t)P2 p(t), ˜ which has been used in m˜ t , p˜ t ) above simplifies to m˜ T (t)P1 m(t) [28]. Therefore, Theorem 3.5 may be less conservative than [28, Corollary 1]. This will be checked by a numerical example in Sect. 3.4. Remark 3.7 In the proof of Theorem 3.5, we introduce a new term of

0

−τ1





T  m(s) ˜ m(s) ˜ dsdθ Q4 ˙ ˙˜ m(s) ˜ t+θ m(s) t

into LKF. Set  Q 41 Q 42 . Q4 = Q T42 Q 43

Then the new term contains not only

0

−τ1



t

t+θ

˙˜ (m˜ T (s)Q 41 m(s) ˜ + m˙˜ T (s)Q 43 m(s))dsdθ

3.3 Stability Criteria

69

used in [25] but also a cross product term

0 −τ1



t

˙˜ m˜ T (s)Q 42 m(s)dsdθ.

t+θ

Since the matrix Q 42 brings a higher degree of freedoms, which leads to a less conservative stability criterion (see Theorem 3.10 below). Similarly, the other items in V4 (t, m˜ t , p˜ t ) and V5 (t, m˜ t , p˜ t ) can also offer more freedoms. Remark 3.8 Although the same V6 (t, m˜ t , p˜ t ) has been chosen in the LKF of this chapter and [25], the derivative of V6 (t, m˜ t , p˜ t ) is estimated by using the improved integral inequality (3.5) instead of Lemma 1.14(ii) used in [25]. It should be emphasized that Proposition 3.4 ensures our approach to be less conservative than one in [25]. Theorem 3.5 is applicable to known τd and σd . However, it will be invalid when the information of the derivatives of delays is unknown. By setting Q 3 = R3 = 0 in our LKF V (t, m˜ t , p˜ t ), a delay-range-dependent stability criterion for GRN (3.1) with time delays only satisfying (3.3a) can be obtained as follows. Corollary 3.9 For given constants τ1 , τ2 , σ1 and σ2 with 0 < τ1 < τ2 and 0 < σ1 < σ2 , GRN (3.1) subject to (3.3a) and (3.2) is asymptotically stable if there exist matrices P T = P > 0, Q iT = Q i > 0, RiT = Ri > 0 (i = 1, 2, 4, 5, 6, 7), Λ j := diag(λ1 j , λ2 j , . . . , λn j ) > 0, T j := diag(t1 j , t2 j , . . . , tn j ) > 0 ( j = 1, 2), Q˜ 5 and R˜ 5 of appropriate sizes, such that (3.6) and the following LMI holds: Ψˆ := sym(Ψ0 + Ψ1 ) + Ψˆ 2 + Ψˆ 3 +

6

Ψi < 0,

i=4

where

Ψˆ 2 = e1T Q 1 e1 + e3T (Q 2 − Q 1 )e3 − e5T Q 2 e5 , Ψˆ 3 = e2T R1 e2 + e6T (R2 − R1 )e6 − e8T R2 e8 ,

and Ψi (i = 0, 1, 4, 5, 6) are defined as in Theorem 3.5. We end this section by offering a theoretical comparison between Theorem 3.5 and [25, Corollary 3.1]. Theorem 3.10 For given constants τ1 , τ2 , σ1 , σ2 , τd and σd with 0 < τ1 < τ2 and 0 < σ1 < σ2 , two delays τ (t) and σ(t) satisfying (3.3), the inequalities (3.6) and (3.7) are feasible if the inequalities (L − = 0, L + = K ) in [25, Corollary 3.1] are feasible.

70

3 Stability Analysis of Delayed GRNs

Proof Set εi = [0n×(i−1)n In 0n×(22−i)n ], i ∈ 22,  ˜ p(t), ˜ ξ1 (t) = col m(t),

0

−τ1 t

m(s)ds, ˜

t−τ1



t



˙˜ m(s)dsdθ,

t+θ



p(s)ds, ˜

t−σ1 0 t −σ1



t

−τ1

t−σ1

t−σ2

t

˙˜ m(s)dsdθ,

t+θ

p(s)ds, ˜

˙˜ p(s)dsdθ,

m(s)ds, ˜

t−τ2



−τ2

t−τ1

−σ1



−σ2

t+θ

 ˙p(s)dsdθ ˜ ,

t

t+θ

 ξ(t) = col m(t), ˜ p(t), ˜ m(t ˜ − τ1 ), m(t ˜ − τ (t)), m(t ˜ − τ2 ), p(t ˜ − σ1 ), p(t ˜ − σ(t)), p(t ˜ − σ2 ), ˙ ˙ f ( p(t)), ˜ f ( p(t ˜ − σ(t))), m(t), ˜ p(t), ˜ t t−τ (t) t−τ1 m(s)ds, ˜ m(s)ds, ˜ m(s)ds, ˜ t−τ1 0 t



−τ1 t

˙˜ m(s)dsdθ,

t+θ



p(s)ds, ˜

t−σ1 0 t −σ1

t−τ2

t+θ



−τ2

t−σ(t)

t−σ2

˙˜ p(s)dsdθ,

−τ1

t−τ (t)



t

t+θ

˙˜ m(s)dsdθ,

p(s)ds, ˜



−σ1

−σ2



t−σ1

p(s)ds, ˜

t−σ(t)

t

 ˙˜ p(s)dsdθ .

t+θ

Then ξ1 (t) = Γˆ ξ(t) and ξ˙1 (t) = Γˆd ξ(t), where Γˆ = col(ε1 , ε2 , ε13 , ε14 + ε15 , ε16 , ε17 , ε18 , ε19 + ε20 , ε21 , ε22 ), Γˆd = col(ε11 , ε12 , ε1 − ε3 , ε3 − ε5 , τ1 ε11 − ε1 + ε3 , τ12 ε11 − ε3 + ε5 , ε2 − ε6 , ε6 − ε8 , σ1 ε12 − ε2 + ε6 , σ12 ε12 − ε6 + ε8 ). Since the inequalities in [25, Corollary 3.1] are feasible, there exist matrices Pˆ T = Pˆ > 0, Qˆ iT = Qˆ i > 0 (i ∈ 9), Rˆ iT = Rˆ i > 0 (i = 1, 2, 4, . . . , 9), Λˆ j := diag(λˆ 1 j , λˆ 2 j , . . . , λˆ n j ) > 0, Tˆ j := diag(tˆ1 j , tˆ2 j , . . . , tˆn j ) > 0 ( j = 1, 2), Rˆ 3 , Rˆ 10 , E, Qˇ 5 , Qˇ 7 , Rˇ 5 , Rˇ 7 , M1 , M2 , N1 and N2 of appropriate sizes, such that the

3.3 Stability Criteria

71

inequalities in [25, Corollary 3.1] hold. By simple computation, the LMI (30) in [25, Corollary 3.1] can be rewritten as follows: Φ := sym(Φ0 + Φ1 + Φ7 ) +

6

Φi < 0,

(3.26)

i=2

where

Φ1 = Γˆ T Pˆ Γˆd + εT9 Λˆ 1 ε12 + (K ε2 − ε9 )T Λˆ 2 ε12 ,

Φ2 = εT1 ( Qˆ 1 + Qˆ 3 )ε1 + εT3 ( Qˆ 2 − Qˆ 1 )ε3 − εT5 Qˆ 2 ε5 − (1 − τd )εT4 Qˆ 3 ε4 , Φ3 = εT2 Rˆ 1 ε2 + εT6 ( Rˆ 2 − Rˆ 1 )ε6 − εT8 Rˆ 2 ε8

T

T

    ε ε7 Rˆ 3 E ε2 Rˆ 3 E ε7 + 2 ) − (1 − σ , d ε9 ε10 E T Rˆ 10 ε9 E T Rˆ 10 ε10 2 ˆ Q 5 )ε1 − εT13 Qˆ 4 ε13 Φ4 = εT1 (τ12 Qˆ 4 + τ12  

T

Qˆ 5 Qˇ 5 ε14 ε14 − ε15 Qˇ T5 Qˆ 5 ε15 2 ˆ +εT11 (τ12 Qˆ 6 + τ12 Q 7 )ε11 − (ε1 − ε3 )T Qˆ 6 (ε1 − ε3 ) T





Qˆ 7 Qˇ 7 ε4 − ε5 ε − ε5 , − 4 ε 3 − ε4 Qˇ T7 Qˆ 7 ε3 − ε4 2 ˆ Φ5 = εT2 (σ12 Rˆ 4 + σ12 R5 )ε2 − εT18 Rˆ 4 ε18  

T

Rˆ 5 Rˇ 5 ε19 ε − 19 ε20 Rˇ 5T Rˆ 5 ε20 2 ˆ +εT12 (σ12 Rˆ 6 + σ12 R7 )ε12 − (ε2 − ε6 )T Rˆ 6 (ε2 − ε6 ) T





Rˆ 7 Rˇ 7 ε7 − ε8 ε − ε8 , − 7 ε 6 − ε7 Rˇ 7T Rˆ 7 ε6 − ε7

τ14 ˆ σ4 Q 8 + τs2 Qˆ 9 )ε11 + εT12 ( 1 Rˆ 8 + σs2 Rˆ 9 )ε12 4 4 T ˆ T ˆ T ˆ −ε16 Q 8 ε16 − ε17 Q 9 ε17 − ε21 R8 ε21 − εT22 Rˆ 9 ε22 ,

Φ6 = εT11 (

Φ7 = −(ε10 − K ε7 )T Tˆ2 ε10 − (ε9 − K ε2 )T Tˆ1 ε9 , Φ0 = (εT1 M1 + εT11 M2 )(−Aε1 + W ε10 − ε11 ) +(εT2 N1 + εT12 N2 )(−Cε2 + Dε4 − ε12 ).

72

3 Stability Analysis of Delayed GRNs

Set X = col(X 1 , X 2 , . . . , X 22 ) with X i = ei (i ∈ 10), X 11 = −Ae1 + W e10 , X 12 = −Ce2 + De4 , X j = e j−2 ( j = 13, 14, 15), X 16 = τ1 e1 − e11 , X 17 = τ12 e1 − e12 − e13 , X k = ek−4 , (k = 18, 19, 20), X 21 = σ1 e2 − e14 , X 22 = σ12 e2 − e15 − e16 , and ei (i ∈ 16) are defined as in Theorem 3.5. Pre- and post-multiplying (3.26) by X T and X , respectively, we have that X T Φ X = sym(Φ˜ 1 + Φ˜ 7 ) +

6

Φ˜ i < 0,

(3.27)

i=2

where Φ˜ 1 = Γˇ T Pˆ Γˇd + e9T Λˆ 1 (−Ce2 + De4 ) + (K e2 − e9 )T Λˆ 2 (−Ce2 + De4 ), Γˇ = col(e1 , e2 , e11 , e12 + e13 , τ e1 − e11 , τ12 e1 − e12 − e13 , e14 , e15 + e16 , σ1 e2 − e14 , σ12 e2 − e15 − e16 ), Γˇd = col(−Ae1 + W e10 , −Ce2 + De4 , e1 − e3 , e3 − e5 , τ1 (−Ae1 + W e10 ) − e1 + e3 , τ12 (−Ae1 + W e10 ) − e3 + e5 , e2 − e6 , e6 − e8 , σ1 (−Ce2 + De4 ) − e2 + e6 , σ12 (−Ce2 + De4 ) − e6 + e8 ), Φ˜ 2 = e1T ( Qˆ 1 + Qˆ 3 )e1 + e3T ( Qˆ 2 − Qˆ 1 )e3 − e5T Qˆ 2 e5 − (1 − τd )e4T Qˆ 3 e4 , Φ˜ 3 = e2T Rˆ 1 e2 + e6T ( Rˆ 2 − Rˆ 1 )e6 − e8T Rˆ 2 e8  

T

Rˆ 3 E e2 e2 + e9 E T Rˆ 10 e9  

T

Rˆ 3 E e7 e , −(1 − σd ) 7 e10 E T Rˆ 10 e10

3.3 Stability Criteria

73

2 ˆ T ˆ Φ˜ 4 = e1T (τ12 Qˆ 4 + τ12 Q 5 )e1 − e11 Q 4 e11  

T

Qˆ 5 Qˇ 5 e12 e − 12 e13 Qˇ T5 Qˆ 5 e13 2 ˆ +(−Ae1 +W e10 )T (τ12 Qˆ 6 +τ12 Q 7 )(−Ae1 +W e10 ) T





Qˆ 7 Qˇ 7 e4 − e5 e4 − e5 T ˆ , −(e1 − e3 ) Q 6 (e1 − e3 ) − e3 − e4 Qˇ T7 Qˆ 7 e3 − e4 2 ˆ T ˆ Φ˜ 5 = e2T (σ12 Rˆ 4 + σ12 R5 )e2 − e14 R4 e14  

T

Rˆ 5 Rˇ 5 e15 e − 15 e16 Rˇ 5T Rˆ 5 e16 2 ˆ +(−Ce2 + De4 )T (σ12 Rˆ 6 + σ12 R7 )(−Ce2 + De4 ) T





Rˆ 7 Rˇ 7 e7 − e8 e7 − e8 T ˆ −(e2 − e6 ) R6 (e2 − e6 ) − , e6 − e7 Rˇ 7T Rˆ 7 e6 − e7

τ14 ˆ Q 8 + τs2 Qˆ 9 )(−Ae1 + W e10 ) 4 σ4 +(−Ce2 + De4 )T ( 1 Rˆ 8 + σs2 Rˆ 9 )(−Ce2 + De4 ) 4 T ˆ −(τ1 e1 − e11 ) Q 8 (τ1 e1 − e11 ) −(τ12 e1 − e12 − e13 )T Qˆ 9 (τ12 e1 − e12 − e13 ) −(σ1 e2 − e14 )T Rˆ 8 (σ1 e2 − e14 ) −(σ12 e2 − e15 − e16 )T Rˆ 9 (σ12 e2 − e15 − e16 ),

Φ˜ 6 = (−Ae1 + W e10 )T (

Φ˜ 7 = −(e10 − K e7 )T Tˆ2 e10 − (e9 − K e2 )T Tˆ1 e9 . Applying Lemma 3.2 to p = τ12 e3 − e12 − e13 , q = τ12 (e1 − e3 ), M = Qˆ 9 , b = τ2 and a = τ1 , we obtain that 2τs (τ12 e3 − e12 − e13 )T Qˆ 9 (τ12 e3 − e12 − e13 ) 2 τ12 τs τ12 (e1 − e3 )T Qˆ 9 (e1 − e3 ) − τ1

−

 −(τ12 e1 − e12 − e13 )T Qˆ 9 (τ12 e1 − e12 − e13 ). Similarly,

(3.28)

74

3 Stability Analysis of Delayed GRNs

2σs (σ12 e6 − e15 − e16 )T Rˆ 9 (σ12 e6 − e15 − e16 ) 2 σ12 σs σ12 − (e2 − e6 )T Rˆ 9 (e2 − e6 ) σ1  −(σ12 e2 − e15 − e16 )T Rˆ 9 (σ12 e2 − e15 − e16 ). −

(3.29)

Let ⎡

I 0 ⎢ 0 I ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ τ1 I 0 L=⎢ ⎢τ12 I 0 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎣ 0 σ1 I 0 σ12 I Then

0 0 I 0 −I 0 0 0 0 0

0 0 0 I 0 −I 0 0 0 0

0 0 0 0 0 0 I 0 −I 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎥ ⎥ I ⎥ ⎥ 0 ⎦ −I

Γˇ = LΓ, Γˇd = LΓd ,

(3.30)

where Γ and Γd are defined as in Theorem 3.5. Now, choose P = L T Pˆ L , Q j = Qˆ j , ( j ∈ 3), Ri = Rˆ i , Λi = Λˆ i , Ti = Tˆi (i = 1, 2), 

Rˆ 3 E , Q 6 = Qˆ 8 , Q 7 = Qˆ 9 , R6 = Rˆ 8 , R7 = Rˆ 9 , R3 = E T Rˆ 10 Q 4 = diag( Qˆ 4 , Qˆ 6 ), Q 5 = diag( Qˆ 5 , Qˆ 7 ), R4 = diag( Rˆ 4 , Rˆ 6 ), R5 = diag( Rˆ 5 , Rˆ 7 ), Q˜ 5 = diag( Qˇ 5 , Qˇ 7 ), R˜ 5 = diag( Rˇ 5 , Rˇ 7 ). Then, by (3.27)–(3.30), it is easy to see that the inequalities (3.6) and (3.7) are feasible. The proof is completed.

3.4 Numerical Examples In this section we will present a pair of examples to illustrate the effectiveness of our theoretical results. Example 3.11 Consider GRN (3.1) with

3.4 Numerical Examples

75

A = diag(3, 3), C = diag(2, 2), D = diag(1, 1), W =

 −1 0 . 1 2

Assume that the Hill coefficient is 2. By simple computation, we can find that k j = max{ f j (s)/s} = 0.65 ( j = 1, 2), and hence K = diag(0.65, 0.65). For τ1 ∈ {0.2, 0.4, 0.6, 0.8, 1, 1.2}, σ1 = 0.1, σ2 = 0.3, τd = 1.5 and σd = 0.7, the corresponding admissible upper bounds of τ2 obtained by Theorem 3.5, [28, Theorem 1], [21, Corollary 3.3] and [25, Corollary 3.1] are shown in Table 3.1, which shows that the stability criterion presented in Theorem 3.5 is less conservative than the corresponding ones in [21, 25, 28]. Furthermore, when the time-varying delays τ (t) and σ(t) are assumed to be τ (t) = 3sin2 (1.5t) + 0.4, σ(t) = 0.2sin2 (3.5t) + 0.1, the trajectories of mRNAs and proteins are given in Fig. 3.1 with the initial conditions φ(t) ≡ [0.6, 1.5]T , ϕ(t) ≡ [1, 0.8]T , t ∈ [−3.4, 0]. Example 3.12 Consider GRN (3.1) with

A = C = diag(2, 2), D = diag(1, 1), W =

Table 3.1 Maximum values of τ2 (Example 3.11) τ1 0.2 0.4 0.6 [28, Theorem 1] [25, Corollary 3.1] [21, Corollary 3.3] Theorem 3.5

2.3870 4.5208 4.5587 4.7784

2.5871 4.6483 4.7571 4.9676

2.7871 4.8236 4.9571 5.1666

0.8

1

1.2

2.9871 5.0190 5.1571 5.3665

3.1871 5.2170 5.3571 5.5666

3.3871 5.4158 5.5571 5.7665

1

1.6 1.4

0.8

1.2

0.6

1 0.8

0.4

0.6 0.4

0.2

0.2

0

0 -0.2

 1 −2 , 0.8 0

0

1

2

3

4

5

6

7

8

9

10

-0.2

0

1

2

3

4

Fig. 3.1 Trajectories of mRNA and protein concentrations (Example 3.11)

5

6

7

8

9

10

76

3 Stability Analysis of Delayed GRNs 1

1.6 1.4

0.8

1.2 1

0.6

0.8

0.4

0.6 0.4

0.2

0.2 0

0

-0.2 -0.4

0

2

4

6

8

10 12 14 16 18 20

-0.2

0

2

4

6

8

10 12 14 16 18 20

Fig. 3.2 Trajectories of mRNA and protein concentrations (Example 3.12) Table 3.2 Maximum values of τ2 (Example 3.12) τ1 0.1 0.4 [28, Corollary 1] [30, Theorem 3.1] Corollary 3.9

4.3664 3.7283 6.6803

4.6664 4.0283 6.8939

0.7

1

4.9664 4.3283 7.1656

5.2664 4.6283 7.4636

Assume that K = diag(0.65, 0.65). For τ1 ∈ {0.1, 0.4, 0.7, 1}, σ1 = 0.4 and σ2 = 0.6, the corresponding admissible upper bounds of τ2 obtained by Corollary 3.9, [30, Theorem 3.1] and [28, Corollary 1] are shown in Table 3.2. In addition, the LMIs in [13, Theorem 1] are infeasible for the example. Compared with [13, 28, 30], it is clear that our method is less conservative. Furthermore, when the time-varying delays τ (t) and σ(t) are assumed to be τ (t) = 4.2 sin(t) + 3.2, σ(t) = 0.5 cos(t) + 0.1, the trajectories of mRNAs and proteins are given in Fig. 3.2 with the initial conditions φ(t) ≡ [0.6, 1.5]T , ϕ(t) ≡ [1 0.8]T , t ∈ [−7.4, 0].

3.5 Remarks and Notes This chapter solves these problems: • Improvement of the inequality in Lemma 1.14(ii); • Establishing stability criteria for GRNs with interval time-varying delays; • Theoretical comparison of stability criteria proposed in the chapter and one in [25];

3.5 Remarks and Notes

77

• Numerical comparisons of stability criteria proposed in the chapter and [13, 21, 25, 28, 30]. This chapter is taken from [33]. We end this chapter by introducing the following items, which are related to this chapter: 1. Koo et al. [4] proposed a delay-range-dependent robust stability criterion for GRNs with delays which vary in an interval by employing an LKF method and convex combination technique. An LKF similar to one in [4] is adopted by Liu and Yue [6] to give a stability criterion for a class of delayed GRNs described by the Takagi–Sugeno fuzzy model; 2. By constructing an LKF containing delay factors and using convex combination technique, piecewise analysis technique and free-weighted matrix technique, Yan and Liu [30] developed robust asymptotic stability criteria for GRNs with time-varying delays; 3. Wang and Zhong [21] proposed robust stability criteria for GRNs with timevarying delays and nonlinear disturbance by introducing an LKF containing regulatory functions, and further generalized the result to the case of stochastic GRNs. 4. To investigate a delay-range-dependent and delay-rate-dependent stability criterion for GRNs with time-varying delays and linear fractional uncertainties, Wang et al. [25] constructed an augmented LKF which contains some tripleintegral terms, and then dealt with its derivative by employing reciprocal convex combination technique, integral inequality technique and free-weighted matrix technique. 5. By constructing an LKF based on the partitioning approach, Senthilraj et al. [14] researched the asymptotic stability of delayed GRNs with impulse control when time delays varies within a given interval. 6. Li et al. [5] investigated the problem of global stability for a class of delayed GRNs. An LKF including triple-integral terms is constructed, and its derivative is dealt with by employing the Jensen’s inequality, the free-weighting matrix technique and the convex combination technique. 7. Liu et al. [8] obtained a sufficient condition of the global exponential stability for a class of continuous-time delayed GRNs by applying Dini derivative method. Moreover, by using the semi-discretization technique and the IMEX-θ method, they derived two discrete-time delayed GRN models which preserve the dynamical characteristics of the continuous-time ones. 8. Based on the Lyapunov stability theory, Wang et al. [24] and Zhou et al. [34] investigated the robust stability criteria for uncertain delayed GRNs. The former partitions the interval time-varying delays into non-uniformly subintervals, and the latter uses the delta operator approach. 9. By using the LKF method, LMI technique and convex combination technique, Liu and Wu [9] presented a global asymptotical stability criterion for a class of delayed GRNs.

78

3 Stability Analysis of Delayed GRNs

10. Rakkiyappan et al. [12] investigated a delay-dependent sufficient condition for the global asymptotic stability of GRNs with mixed delays, which include constant delays in the leakage term (i.e., ‘leakage delay’), interval time-varying delays and continuously distributed delays. 11. Based on the LKF method, the lower bound lemma and the Jensen’s inequality, Wang et al. [23] investigated a sufficient condition for the exponential convergence of uncertain delayed GRNs in the case of the unknown equilibrium. 12. Takada et al. [15] give the necessary and sufficient condition for the local stability of GRNs with constant time delay. 13. For uncertain delayed GRNs, Ren and Cao [13], Wang et al. [27], He and Cao [2], Tian, Wang and Wu [16] and Liu [7] established robust asymptotic stability criteria by using the LKF method. To reduce the conservativeness of robust asymptotic stability criteria, Wang et al. [22] employed the Jensen’s inequality combined with series compensation technique, which is proposed in [31], while Wu et al. [28] adopt the free-weighting matrix technique. 14. Zhang and Wei [32] and Hu et al. [3] presented sufficient conditions for ensuring the global exponential stability of delayed GRNs. The former transforms the kinetics of networks into a single delay differential equation and utilizes the method of delay differential inequalities, and the laster employs the LKF method and some inequality techniques. 15. Based on the Lyapunov stability theory, Wan and Zhou [19] and Wang et al. [20] investigated the stability switches when the delay varies, and showed that Hopf bifurcations may occur within certain range of the model parameters. By combining the normal form method with the center manifold theorem, they determined the direction of the bifurcation and the stability of the bifurcated periodic solutions. Xiao and Cao [29] assessed Hopf bifurcation and derived sufficient conditions for the oscillation of the delayed GRNs. The relative works can be found in [10]. 16. Wang et al. [26] estimated the exponential convergence region of uncertain delayed GRNs. 17. Tu and Lu [17] introduced a model for delayed GRNs, and provided sufficient conditions to ensure the local asymptotical stability and global exponential stability. 18. Chen and Aihara [1] proposed a functional differential equations model for delayed GRNs, and analyzed the nonlinear properties of the model in terms of local stability and bifurcation. 19. The problem of robust stability analysis of delayed Takagi-Sugeno fuzzy GRNs has been addressed in [18]. 20. Pan et al. [11] derived a sufficient condition for the multistability of GRNs with multiple time delays and multivariable regulation functions.

References

79

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23. Wang, W.Q., Nguang, S.K., Zhong, S.M., Liu, F.: Exponential convergence analysis of uncertain genetic regulatory networks with time-varying delays. ISA Trans. 53(5), 1544–1553 (2014) 24. Wang, W.Q., Wang, Y.Z., Nguang, S.K., Zhong, S.M., Liu, F.: Delay partition method for the robust stability of uncertain genetic regulatory networks with time-varying delays. Neurocomputing 173, 899–911 (2016) 25. Wang, W.Q., Zhong, S.M., Liu, F.: New delay-dependent stability criteria for uncertain genetic regulatory networks with time-varying delays. Neurocomputing 93, 19–26 (2012) 26. Wang, Z., Liao, X., Guo, S., Liu, G.: Stability analysis of genetic regulatory network with time delays and parameter uncertainties. IET Control Theory Appl. 4(10), 2018–2028 (2010) 27. Wang, Z.D., Gao, H.J., Cao, J.D., Liu, X.H.: On delayed genetic regulatory networks with polytopic uncertainties: robust stability analysis. IEEE Trans. Nanobiosci. 7(2), 154–163 (2008) 28. Wu, H., Liao, X.F., Wei, F., Guo, S.T., Zhang, W.: Robust stability for uncertain genetic regulatory networks with interval time-varying delays. Inf. Sci. 180(18), 3532–3545 (2010) 29. Xiao, M., Cao, J.D.: Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays. Math. Biosci. 215(1), 55–63 (2008) 30. Yan, R., Liu, J.: New results on asymptotic and robust stability of genetic regulatory networks with time-varying delays. Int. J. Innov. Comput. Inf. Control 8(4), 2889–2900 (2012) 31. Zhang, H., Liu, Z., Huang, G.B.: Global stability analysis for a class of neutral networks with varying delays and control input. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40(6), 1480–1491 (2010) 32. Zhang, Q., Wei, X.: Analysis on exponential stability of uncertain genetic regulatory networks with variable delays. J. Comput. Theor. Nanosci. 10(11), 2604–2608 (2013) 33. Zhang, X., Wu, L., Cui, S.C.: An improved integral inequality to stability analysis of genetic regulatory networks with interval time-varying delays. IEEE/ACM Trans. Comput. Biol. Bioinf. 12(2), 398–409 (2015) 34. Zhou, Q., Shao, X.Y., Karimi, H.R., Zhu, J.: Stability of genetic regulatory networks with time-varying delay: Delta operator method. Neurocomputing 149, 490–495 (2015)

Chapter 4

Stability Analysis for Delayed Switching GRNs

This chapter is concerned with analyzing the global exponential stability of delayed switching GRNs. By using the average dwell time approach together with a novel LKF, we derived sufficient conditions to ensure the switched GRNs with time delays are globally exponentially stable. Two numerical examples are presented to clarify that our theoretical results are effective.

4.1 Model Description Consider the delayed GRN model (1.8), that is, ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0, ˙p(t) ˜ = −C p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(4.1a) (4.1b)

As we all known, most of gene networks contain some kinds of switching mechanisms. For instance, by increasing stimulation or by changing some regulatory mechanisms, a bistable system can switch from one steady state to the other [1]. When we take a switching signal into model (4.1), a class of switched GRN model with time-varying delays can be described as follows: ˙˜ m(t) = −A(α)m(t) ˜ + W (α) f ( p(t ˜ − σ(t))), t  0, ˙p(t) ˜ = −C(α) p(t) ˜ + D(α)m(t ˜ − τ (t)), t  0,

(4.2b)

m(t) ˜ = φ(t), p(t) ˜ = ϕ(t), −d  t  0,

(4.2c)

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_4

(4.2a)

81

82

4 Stability Analysis for Delayed Switching GRNs

where φ, ϕ ∈ C 1 ([−d, 0], Rn ), A(α), C(α), D(α) and W (α) are matrices parameterized by a piecewise constant function α : [0, +∞) → N , here α is called as a switching signal, and the other ones are defined as previously. Assume as previously that f i (0) = 0, f i (y)[ f i (y) − ki y]  0, ∀i ∈ n, y = 0.

(4.3)

Set K = diag(k1 , k2 , . . . , kn ). For convenience, for each α(t) = k ∈ N , we denote A(k) = A(α), W (k) = W (α), C(k) = C(α) and D(k) = D(α). Definition 4.1 [2] For any T2 > T1  0, let Nα (T1 , T2 ) denote the number of switchings of α(t) over [T1 , T2 ). If there exists Ta > 0 such that Nα (T1 , T2 )  (T2 − T1 )/Ta , then Ta is called the average dwell time. Definition 4.2 [7] GRN (4.2) is said to be globally exponentially stable if there exist constants μ > 0 and λ > 0 such that z(t) 2  μ z 0 e−λt

(4.4)

for any t  0, z 0 ∈ C 1 ([−d, 0], R2n ) and switching signal α, where  z 0 = max

 sup z 0 (θ) 2 , sup ˙z 0 (θ) 2 .

−dθ0

−dθ0

4.2 Stability Criteria In this section we will analyze the global exponential stability for switched GRN (4.2) in two cases: constant time delays and interval time-varying delays.

4.2.1 Constant Time-Delay Case In this subsection, we will derive a global exponential stability criterion for switched GRN (4.2) with constant time delays. Assume that 0 < τ (t) ≡ τ , 0 < σ(t) ≡ σ, where τ and σ are known positive constants. In this case, d = max{τ , σ}.

(4.5)

4.2 Stability Criteria

83

For a fixed positive integer r , set A1 (k) = −A(k)e1 + W (k)e2r +6 , A2 (k) = −C(k)er +3 + D(k)er +1 , ei = [0n×(i−1)n In 0n×(2r +6−i)n ], i ∈ 2r + 6 and

  ψ(t) = col ξ1 (t), ξm (t), ξ2 (t), ξ p (t)

with     2τ (r − 1)τ τ , m˜ t − , . . . , m˜ t − , ˜ m˜ t − ξ1 (t) = col m(t), r r r     2σ (r − 1)σ σ ξ2 (t) = col p(t), , p˜ t − , . . . , p˜ t − , ˜ p˜ t − r r r

 r t ξm (t) = col m(t ˜ − τ ), m(s)ds ˜ , τ t− τr

 r t ξ p (t) = col p(t ˜ − σ), p(s)ds, ˜ f ( p(t)), ˜ f ( p(t ˜ − σ)) . σ t− σr Then

˙˜ ˙˜ = A2 (k)ψ(t), m(t) = A1 (k)ψ(t), p(t) ξ1 (t) = E1 ψ(t), ξ2 (t) = E2 ψ(t), τ σ ξ1 (t − ) = E1r ψ(t), ξ2 (t − ) = E2r ψ(t), r r

(4.6a) (4.6b) (4.6c)

where E1 = col(e1 , e2 , . . . , er ), E2 = col(er +3 , er +4 , . . . , e2r +2 ), E1r = col(e2 , e3 , . . . , er +1 ), E2r = col(er +4 , er +5 , . . . , e2r +3 ). Theorem 4.3 For given an integer r  1 and constants μ  1 and β > 0, then the switched GRN (4.2) subject to (4.3) and (4.5) is globally exponentially stable for any switching signal with the average dwell time satisfying Ta > Ta∗ := lnβμ , if there exist symmetric positive definite matrices Pi (k) ∈ Rn×n , Ri (k) ∈ Rn×n , Q i (k) ∈ Rr n×r n (i = 1, 2) and Λ := diag (λ1 , λ2 , . . . , λn ) such that for any k, m ∈ N , Pi (m)  μPi (k), Q i (m)  μQ i (k), Ri (m)  μRi (k), i = 1, 2, Γ (k) := Γ0 + Γ1 (k) + Γ2 (k) + Γ3 (k) < 0,

(4.7) (4.8)

84

4 Stability Analysis for Delayed Switching GRNs

where T T Γ0 = sym(e2r +5 ΛK er +3 + e2r +6 ΛK e2r +3 ) T T −2e2r +5 Λe2r +5 − 2e2r +6 Λe2r +6 ,

Γ1 (k) = sym(e1T P1 (k)A1 (k) + erT+3 P2 (k)A2 (k)) +βe1T P1 (k)e1 + βerT+3 P2 (k)er +3 , βτ

βσ

T T Γ2 (k) = E1T Q 1 (k)E1+E2T Q 2 (k)E2−e− r E1r Q 1 (k)E1r −e− r E2r Q 2 (k)E2r ,

τ T σ A (k)R1 (k)A1 (k) + AT2 (k)R2 (k)A2 (k) r 1 r T ˜ T ˜ −H1 R1 (k)H1 − H2 R2 (k)H2 ,

r − βτ H1 = e r col(e1 − e2 , e1 + e2 − 2er +2 ), τ

Γ3 (k) =

H2 =

r − βσ e r col(er +3 − er +4 , er +3 + er +4 − 2e2r +4 ), σ

R˜ i (k) = diag (Ri (k), 3Ri (k)) , i = 1, 2. Moreover,

z(t) 2  ηe−λt z 0

(4.9)

with   1 ln μ λ2 , η= λ= β−  1, 2 Ta λ1 λ1 = min {λmin (P1 (k)), λmin (P2 (k))}, k∈N 

λ2 = max {λmax (P1 (k)) + λmax (P2 (k)) + τ λmax (Q 1 (k)) k∈N 

+σλmax (Q 2 (k)) +

τ2 σ2 λmax (R1 (k)) + λmax (R2 (k))}. 2 2

(4.10)

Proof For an arbitrary piecewise constant switching signal α, let 0 = t0 < t1 < · · · < tl < · · · denote the switching points of α. For any but fixed t > 0, it is clear that t ∈ [tl , tl+1 ) for some nonnegative integer l. Choose the LKF of the following form: V (t, k) = V1 (t, k) + V2 (t, k) + V3 (t, k), t ∈ [tl , tl+1 ) with ˜ + p˜ T (t)P2 (k) p(t), ˜ V1 (t, k) = m˜ T (t)P1 (k)m(t)

(4.11)

4.2 Stability Criteria

85

 V2 (t, k) =

t



+  V3 (t, k) =

eβ(s−t) ξ1T (s)Q 1 (k)ξ1 (s)ds

t− τr



0

− τr

t− σr t

eβ(s−t) ξ2T (s)Q 2 (k)ξ2 (s)ds,

˙˜ eβ(s−t) m˙˜ T (s)R1 (k)m(s)dsdθ

t+θ



+

t



0 − σr

t

˙˜ eβ(s−t) p˙˜ T (s)R2 (k) p(s)dsdθ,

t+θ

where Pi (k) > 0, Q i (k) > 0 and Ri (k) > 0 (i = 1, 2) are real matrices satisfying (4.7) and (4.8). Then, along the solution of system (4.2), we have from (4.6) that ˙˜ ˙˜ + 2 p˜ T (t)P2 (k) p(t) V˙1 (t, k) = 2m˜ T (t)P1 (k)m(t) = −βV1 (t, k) + ψ T (t)Γ1 (k)ψ(t), t ∈ [tl , tl+1 ),

(4.12)

V˙2 (t, k) = −βV2 (t, k) + ξ1T (t)Q 1 (k)ξ1 (t) + ξ2T (t)Q 2 (k)ξ2 (t)   βτ τ τ Q 1 (k)ξ1 t − −e− r ξ1T t − r r    βσ σ σ Q 2 (k)ξ2 t − −e− r ξ2T t − r r = −βV2 (t, k) + ψ T (t)Γ2 (k)ψ(t), t ∈ [tl , tl+1 ),

(4.13)

τ σ ˙˜ ˙˜ V˙3 (t, k)  −βV3 (t, k) + m˙˜ T (t)R1 (k)m(t) + p˙˜ T (t)R2 (k) p(t) r r  t βτ ˙˜ −e− r m˙˜ T (s)R1 (k)m(s)ds t− τr

βσ

−e− r



t

t− σr

˙˜ t ∈ [tl , tl+1 ). p˙˜ T (s)R2 (k) p(s)ds,

Applying Lemma 1.16, we obtain βτ

−e− r and −e

− βσ r



t t− τr



t

t− σr

˙˜  −ψ T (t)H1T R˜ 1 (k)H1 ψ(t) m˙˜ T (s)R1 (k)m(s)ds

˙˜  −ψ T (t)H2T R˜ 2 (k)H2 ψ(t). p˙˜ T (s)R2 (k) p(s)ds

(4.14)

86

4 Stability Analysis for Delayed Switching GRNs

This, together with (4.14), implies that V˙3 (t, k)  −βV3 (t, k) + ψ T (t)Γ3 (k)ψ(t), t ∈ [tl , tl+1 ).

(4.15)

Moreover, one can obtain from (4.3) that ˜ p(t) ˜ − 2 f T ( p(t))Λ ˜ f ( p(t)) ˜  0, 2 f T ( p(t))ΛK ˜ − σ))ΛK p(t ˜ − σ) − 2 f T ( p(t ˜ − σ))Λ f ( p(t ˜ − σ))  0 2 f T ( p(t

(4.16a) (4.16b)

for any t ∈ [tl , tl+1 ). The combination of (4.11)–(4.13), (4.15) and (4.16) gives V˙ (t, k) + βV (t, k)  ψ T (t)Γ (k)ψ(t), t ∈ [tl , tl+1 ). Using (4.8), we can obtain V˙ (t, k) + βV (t, k) < 0, t ∈ [tl , tl+1 ), that is

d dt

(4.17)

  βt e V (t, k) < 0. Integrating the both sides from tl to t, we have V (t, α(t))  e−β(t−tl ) V (tl , α(tl )), t ∈ [tl , tl+1 ).

(4.18)

This, together with υ := Nα (0, t)  (t − 0)/Ta , (4.7) and (4.17), implies that V (t, α(t))  e−β(t−tl ) μV (tl− , α(tl− ))  ···  e−βt μ Nα (0,t) V (0, α(0))  e−(β−ln μ/Ta )t V (0, α(0)), t  0.

(4.19)

Notice from (4.11) that V (t, α)  λ1 z(t) 22 , t  0, V (0, α(0))  λ2 z 0 2 ,

(4.20)

where λ1 and λ2 are defined as in (4.10). Combining (4.19) and (4.20), we can easily obtain that 1 λ2 −(β−ln μ/Ta )t V (t, α)  e z 0 2 , t  0. z(t) 22  λ1 λ1 The arbitrariness of φ and ϕ yields that (4.9) holds. From Definition 4.2, one can easily get that the switched GRN (4.2) is globally exponentially stable. Remark 4.4 The LKF used in the proof of Theorem 4.3 is similar to one in [6]. Indeed, by referring to the references [7, 8], we can construct a more complex LKF, which will decrease the conservativeness of the criterion.

4.2 Stability Criteria

87

4.2.2 Time-Varying Delay Case In this subsection, we will present a global exponential stability criterion for switched GRN (4.2) with time-varying delays. Assume that ˙  σd , 0 < τ1  τ (t)  τ2 , 0 < σ1  σ(t)  σ2 , τ˙ (t)  τd , σ(t)

(4.21)

where τ1 , τ2 , σ1 , σ2 , τd and σd are known real constants. In this case, d = max{τ2 , σ2 }. Set eˆi = [0n×(i−1)n In 0n×(2r +16−i)n ], i ∈ 2r + 16 and

  ζ(t) = col ξ1 (t), ζm (t), ξ2 (t), ζ p (t) ,

where

 t r ˜ − τ1 ), m(t ˜ − τ (t)), m(t ˜ − τ2 ), m(s)ds, ˜ ζm (t) = col m(t τ1 t− τr1  t−τ1  t−τ (t) 1 1 m(s)ds, ˜ m(s)ds ˜ , τ (t) − τ1 t−τ (t) τ2 − τ (t) t−τ2

˜ − σ1 ), p(t ˜ − σ(t)), p(t ˜ − σ2 ), ζ p (t) = col p(t 1 σ(t) − σ1



t−σ1

t−σ(t)

1 p(s)ds, ˜ σ2 − σ(t)



r σ1

t−σ(t)



t

t−

σ1 r

p(s)ds, ˜

p(s)ds, ˜

t−σ2

f ( p(t)), ˜ f ( p(t ˜ − σ1 )), f ( p(t ˜ − σ(t))), f ( p(t ˜ − σ2 ))), and ξ1 (t) and ξ2 (t) are defined as in Theorem 4.3. Then ˙˜ = Aˆ 2 (α)ζ(t), ˙˜ m(t) = Aˆ 1 (α)ζ(t), p(t) ξ1 (t) = Eˆ1 ζ(t), ξ2 (t) = Eˆ2 ζ(t), ζm (t) = Eˆm ζ(t), ζ p (t) = Eˆp ζ(t), τ1 σ1 ξ1 (t − ) = Eˆ1r ζ(t), ξ2 (t − ) = Eˆ2r ζ(t), r r

(4.22) (4.23) (4.24)

where Aˆ 1 (α) = −A(α)eˆ1 + W (α)eˆ2r +15 , Aˆ 2 (α) = −C(α)eˆr +7 + D(α)eˆr +2 , Eˆ1 = col(eˆ1 , eˆ2 , . . . , eˆr ), Eˆ2 = col(eˆr +7 , eˆr +8 , . . . , eˆ2r +6 ), Eˆ1r = col(eˆ2 , eˆ3 , . . . , eˆr +1 ), Eˆ2r = col(eˆr +8 , eˆr +9 , . . . , eˆ2r +7 ).

88

4 Stability Analysis for Delayed Switching GRNs

Theorem 4.5 For given an integer r  1 and constants μ  1 and β > 0, the switched GRN (4.2) subject to (4.3) and (4.21) is globally exponentially stable for any switching signal with the average dwell time satisfying Ta > Ta∗ := lnβμ , if there exist symmetric positive definite matrices Pi (k) ∈ Rn×n , Ri j (k) ∈ Rn×n (i, j = 1, 2), Q11 (k), Q21 (k) ∈ Rr n×r n , Qil (k) ∈ Rr n×r n (i = 1, 2; l = 2, 3), Λ := diag (λ1 , λ2 , . . . , λn ) and matrices X i (k) ∈ R2n×2n (i = 1, 2) such that for k, m ∈ N ,   ˜ i2 (k) X i (k) R Θi (k) := (4.25) ˜ i2 (k)  0, i = 1, 2, X iT (k) R Γˆ (k) := Γˆ0 + Γˆ1 (k) + Γˆ2 (k) + Γˆ3 (k) < 0, (4.26) Pi (m)  μPi (k), Ri j (m)  μRi j (k), i, j = 1, 2, Qil (m)  μQil (k), i = 1, 2; l = 1, 2, 3, where τ12 = τ2 − τ1 , σ12 = σ2 − σ1 , T T Γˆ0 = sym(eˆ2r +13 ΛK eˆr +7 + eˆ2r +14 ΛK eˆ2r +7 T T +eˆ2r +15 ΛK eˆ2r +8 + eˆ2r +16 ΛK eˆ2r +9 ) T T −2eˆ2r +13 Λeˆ2r +13 − 2eˆ2r +14 Λeˆ2r +14 T T −2eˆ2r +15 Λeˆ2r +15 − 2eˆ2r +16 Λeˆ2r +16 ,   Γˆ1 (k) = sym eˆ1T P1 (k)Aˆ 1 (k) + eˆrT+7 P2 (k)Aˆ 2 (k)

+β eˆ1T P1 (k)eˆ1 + β eˆrT+7 P2 (k)eˆr +7 , Γˆ2 (k) = Eˆ1T Q11 (k)Eˆ1 − e−

βτ1 r

T Eˆ1r Q11 (k)Eˆ1r

+e−βτ1 eˆrT+1 Q12 (k)eˆr +1 − (1 − τd )e−βτ2 eˆrT+2 Q12 (k)eˆr +2 +eˆ1T Q13 (k)eˆ1 + Eˆ2T Q21 (k)Eˆ2 − e−βτ2 eˆrT+3 Q13 (k)eˆr +3 −e−

βσ1 r

T T Eˆ2r Q21 (k)Eˆ2r + e−βσ1 eˆ2r +7 Q22 (k)eˆ2r +7

T T −(1 − σd )e−βσ2 eˆ2r +8 Q22 (k)eˆ2r +8 + eˆr +7 Q23 (k)eˆr +7 T −e−βσ2 eˆ2r +9 Q23 (k)eˆ2r +9 ,

Γˆ3 (k) = Γˆ31 (k) + Γˆ32 (k) + Γˆ33 (k), τ  1 2 R11 (k) + τ12 R12 (k) Aˆ 1 (k) Γˆ31 (k) = Aˆ T1 (k) r σ  1 T 2 ˆ +A2 (k) R22 (k) Aˆ 2 (k), R21 (k) + σ12 r T ˜ ˆ Γ32 (k) = −G 1 R11 (k)G 1 − Ω1T Θ1 (k)Ω1 ,

4.2 Stability Criteria

89

˜ 21 (k)G 4 − Ω2T Θ2 (k)Ω2 , Γˆ33 (k) = −G T4 R ˜ i j (k) = diag(Ri j (k), 3Ri j (k)), i, j = 1, 2, R Ω1 = col(G 2 , G 3 ), Ω2 = col(G 5 , G 6 ),

G1 = √

G2 =

√

G3 =

G4 = G5 = G6 =

√

r − βτ 1 e r col(eˆ1 − eˆ2 , eˆ1 + eˆ2 − 2eˆr +4 ), τ1

e−βτ2 col(eˆr +1 − eˆr +2 , eˆr +1 + eˆr +2 − 2eˆr +5 ),

e−βτ2 col(eˆr +2 − eˆr +3 , eˆr +2 + eˆr +3 − 2eˆr +6 ),

r − βσ1 e r col(eˆr +7 − eˆr +8 , eˆr +7 + eˆr +8 − 2eˆ2r +10 ), σ1

e−βσ2 col(eˆ2r +7 − eˆ2r +8 , eˆ2r +7 + eˆ2r +8 − 2eˆ2r +11 ),

√

e−βσ2 col(eˆ2r +8 − eˆ2r +9 , eˆ2r +8 + eˆ2r +9 − 2eˆ2r +12 ).

Proof Choose the following LKF: Z (t, α) = Z 1 (t, α) + Z 2 (t, α) + Z 3 (t, α) with ˜ + p˜ T (t)P2 (α) p(t), ˜ Z 1 (t, α) = m˜ T (t)P1 (α)m(t)  t Z 2 (t, α) = eβ(s−t) ξ1T (s)Q11 (α)ξ1 (s)ds t−

τ1 r



+ + +

t−τ (t)  t

eβ(s−t) m˜ T (s)Q12 (α)m(s)ds ˜

eβ(s−t) m˜ T (s)Q13 (α)m(s)ds ˜

t−τ2  t σ1 r

eβ(s−t) ξ2T (s)Q21 (α)ξ2 (s)ds



t−



t−σ(t) t

+ +

t−τ1

t−σ1

eβ(s−t) p˜ T (s)Q22 (α) p(s)ds ˜

eβ(s−t) p˜ T (s)Q23 (α) p(s)ds, ˜

t−σ2

90

4 Stability Analysis for Delayed Switching GRNs

 Z 3 (t, α) =



0

−

τ1 r

+



−τ1



−τ2



0 σ − r1



+τ12

˙˜ eβ(s−t) m˙˜ T (s)R11 (α)m(s)dsdθ

t+θ

+τ12 

t

t

˙˜ eβ(s−t) m˙˜ T (s)R12 (α)m(s)dsdθ

t+θ t

˙˜ eβ(s−t) p˙˜ T (s)R21 (α) p(s)dsdθ

t+θ −σ1

−σ2



t

˙˜ eβ(s−t) p˙˜ T (s)R22 (α) p(s)dsdθ,

t+θ

where Pi (α) > 0, Ri j (α) > 0 and Qil (α) > 0 (i, j = 1, 2, l = 1, 2, 3) are real matrices satisfying the preconditions of Theorem 4.5. Then, for a fixed α(t) = k, along the solution of system (4.2), we have from (4.22)–(4.24) that ˙˜ ˙˜ + 2 p˜ T (t)P2 (k) p(t) Z˙ 1 (t, k) = 2m˜ T (t)P1 (k)m(t) = −β Z 1 (t, k) + ζ T (t)Γˆ1 (k)ζ(t),

(4.27)

Z˙ 2 (t, k)  −β Z 2 (t, k) + ξ1T (t)Q11 (k)ξ1 (t)   βτ1 τ1  τ1  Q11 (k)ξ1 t − −e− r ξ1T t − r r +e−βτ1 m˜ T (t − τ1 )Q12 (k)m(t ˜ − τ1 ) −(1 − τd )e−βτ2 m˜ T (t − τ (t))Q12 (k)m(t ˜ − τ (t)) +m˜ T (t)Q13 (k)m(t) ˜ + ξ2T (t)Q21 (k)ξ2 (t) −e−βτ2 m˜ T (t − τ2 )Q13 (k)m(t ˜ − τ2 ) βσ1 σ σ1 1 −e− r ξ2T (t − )Q21 (k)ξ2 (t − ) r r +e−βσ1 p˜ T (t − σ1 )Q22 (k) p(t ˜ − σ1 ) −(1 − σd )e−βσ2 p˜ T (t − σ(t))Q22 (k) p(t ˜ − σ(t)) ˜ − e−βσ2 p˜ T (t − σ2 )Q23 (k) p(t ˜ − σ2 ) + p˜ T (t)Q23 (k) p(t) = −β Z 2 (t, k) + ζ T (t)Γˆ2 (k)ζ(t),

(4.28)

4.2 Stability Criteria

91

Z˙ 3 (t, k)  −β Z 3 (t, k) + ζ T (t)Γˆ31 (k)ζ(t)  t βτ1 ˙˜ m˙˜ T (s)R11 (k)m(s)ds −e− r t−



−τ12 e−βτ2



−e−βτ2 τ12 −e

−

βσ1 r



τ1 r

t−τ1 t−τ (t)

t−τ (t)

˙˜ m˙˜ T (s)R12 (k)m(s)ds

t−τ2 t

σ t− r1

−σ12 e

˙˜ m˜˙ T (s)R12 (k)m(s)ds

−βσ2

−e−βσ2 σ12

 

T ˙˜ p˙˜ (s)R21 (k) p(s)ds t−σ1

T ˙˜ p˙˜ (s)R22 (k) p(s)ds

t−σ(t) t−σ(t)

T ˙˜ p˙˜ (s)R22 (k) p(s)ds.

(4.29)

t−σ2

By Lemma 1.16, we have − e−

βτ1 r



t t−

τ1 r

˙˜ ˜ 11 (k)G 1 ζ(t)  −ζ T (t)G T1 R m˙˜ T (s)R11 (k)m(s)ds

(4.30)

and −τ12 e

−βτ2



t−τ1

t−τ (t)

−τ12 e−βτ2



T ˙˜ m˙˜ (s)R12 (k)m(s)ds

t−τ (t)

T ˙˜ m˙˜ (s)R12 (k)m(s)ds

t−τ2

τ12 ˜ 12 (k)G 2 ζ(t) ζ T (t)G T2 R − τ (t) − τ1 τ12 ˜ 12 (k)G 3 ζ(t) − ζ T (t)G T3 R τ2 − τ (t)   τ12 ˜ 0 T T τ (t)−τ1 R12 (k) = −ζ (t)Ω1 τ12 ˜ (k) Ω1 ζ(t). R 0 τ2 −τ (t) 12

(4.31)

Similarly, − e−

βσ1 r



t

σ t− r1

˙˜ ˜ 21 (k)G 4 ζ(t)  −ζ T (t)G T4 R p˜˙ T (s)R21 (k) p(s)ds

(4.32)

92

4 Stability Analysis for Delayed Switching GRNs

and −σ12 e−βσ2 −σ12 e−βσ2



t−σ1

˙˜ p˜˙ T (s)R22 (k) p(s)ds

t−σ(t)  t−σ(t)

˙˜ p˙˜ T (s)R22 (k) p(s)ds

t−σ2

σ12 ˜ 22 (k)G 5 ζ(t) ζ T (t)G T5 R σ(t) − σ1 σ12 ˜ 22 (k)G 6 ζ(t) − ζ T (t)G T6 R σ2 − σ(t)   σ12 ˜ 0 T T σ(t)−σ1 R22 (k) = −ζ (t)Ω2 σ12 ˜ 22 (k) Ω2 ζ(t). R 0 −

(4.33)

σ2 −σ(t)

Applying Lemma 1.9 to (4.31) and (4.33), respectively, one can derive from (4.25) that  t−τ1 ˙˜ m˙˜ T (s)R12 (k)m(s)ds −τ12 e−βτ2 −τ12 e−βτ2

t−τ (t) t−τ (t)



˙˜ m˙˜ T (s)R12 (k)m(s)ds

t−τ2

 −ζ T (t)Ω1T Θ1 (k)Ω1 ζ(t)

(4.34)

and −σ12 e

−βσ2

−σ12 e−βσ2



t−σ1

˙˜ p˙˜ T (s)R22 (k) p(s)ds

t−σ(t)  t−σ(t)

˙˜ p˙˜ T (s)R22 (k) p(s)ds

t−σ2

 −ζ T (t)Ω2T Θ2 (k)Ω2 ζ(t).

(4.35)

The combination of (4.29), (4.30), (4.32), (4.34) and (4.35) gives Z˙ 3 (t, k)  −β Z 3 (t, k) + ζ T (t)Γˆ3 (k)ζ(t).

(4.36)

It follows from (4.3) that ˜ p(t) ˜ − 2 f T ( p(t))Λ ˜ f ( p(t)) ˜  0, 2 f T ( p(t))ΛK

(4.37a)

˜ − σ1 ))ΛK p(t ˜ − σ1 ) − 2 f T ( p(t ˜ − σ1 ))Λ f ( p(t ˜ − σ1 ))  0, 2 f T ( p(t

(4.37b)

˜ − σ2 ))ΛK p(t ˜ − σ2 ) − 2 f T ( p(t ˜ − σ2 ))Λ f ( p(t ˜ − σ2 ))  0, 2 f T ( p(t

(4.37c)

4.2 Stability Criteria

93

2 f T ( p(t ˜ − σ(t)))ΛK p(t ˜ − σ(t)) − 2 f T ( p(t ˜ − σ(t)))Λ f ( p(t ˜ − σ(t)))  0.

(4.37d)

Due to (4.27), (4.28), (4.36) and (4.37), we have Z˙ (t, k) + β Z (t, k)  ζ T (t)Γˆ (k)ζ(t). Using (4.26), we have

Z˙ (t, k) + β Z (t, k) < 0.

The rest of the proof is the same as that in Theorem 4.3, thus we omit the details.

4.3 Numerical Examples In this section we will present two numerical examples to illustrate the theoretical results in the previous section. Example 4.6 When N = 2, consider the switched GRN (4.2) with the following parameters: Subsystem #1         20 20 10 1 −2 A1 = , C1 = , D1 = , W1 = ; 02 02 01 0.8 0 Subsystem #2  A2 =

       30 20 10 −1 0 , C2 = , D2 = , W2 = . 03 02 01 1 2

In addition, assume that K = diag(0.65, 0.65). Choose β = 0.575 and σ = 0.3. By Theorem 4.3, we can get the results for different r , which are listed in Table 4.1. From which, it is easy to see that when r becomes larger, τmax will become larger. That is, the maximum allowable delay will become larger when the partitioning becomes thinner. Furthermore, set μ = 1.2 > 1, then Ta > Ta∗ = lnβμ = 0.3171. Solving the inequalities of (4.7) and (4.8) in Theorem 4.3 with r = 2, we can obtain the feasible solutions. Because of the space limit, here we only list the matrix variables P1 (1), P2 (1), R1 (1) and R2 (1) as follows:

94

4 Stability Analysis for Delayed Switching GRNs

Table 4.1 Maximum allowable delays (Example 4.6)

P1 (1) P2 (1) R1 (1) R2 (1)

τmax

Methods Theorem 4.3, r Theorem 4.3, r Theorem 4.3, r Theorem 4.3, r Theorem 4.3, r Theorem 4.3, r

=1 =2 =3 =5 =7 = 10

2.7317 2.7563 2.7671 2.7759 2.7794 2.7816

 0.3234 −0.0030 = 10 × , −0.0030 0.4882   0.1972 −0.0460 = 10−4 × , −0.0460 0.4364   0.8776 0.3054 = 10−5 × , 0.3054 0.7502   0.2036 0.0763 = 10−4 × . 0.0763 0.5229 −4



Therefore, it is easy to know that under the above conditions the considered switched GRN is globally exponentially stable. Taking Ta = 0.5 > Ta∗ and considering (4.9) yields λ1 = 1.8871 × 10−5 , λ2 = 5.8048e × 10−4 , η = 5.5462 and λ = 0.1052, thus z(t) 2  5.5462e−0.1052t z 0 . Example 4.7 Consider the switched GRN (4.2) with time-varying delays and N = 2. The parameters are as follows: Subsystem #1         60 50 1 0 0 −5 A1 = , C1 = , D1 = , W1 = ; 06 05 0 1.2 0 0 Subsystem #2         70 60 10 0 −5.5 A2 = , C2 = , D2 = , W2 = . 07 06 01 −5.5 0 Assume that diag(0.65, 0.65). For given τ (t) = 3.25 + 2.95 cos(4t/59) and σ(t) = 0.3 + 0.2 sin(t/2), it is easy to know that τ1 = 0.3, τd = 0.2, σ1 = 0.1, σd = 0.1 and σ2 = 0.5. When μ = 1.2, β = 0.5 and r = 2, by solving the LMIs in Theorem 4.5, the corresponding admissible upper bounds of τ2 is 6.2454.

4.3 Numerical Examples

95 2.5

Fig. 4.1 Switching signal (Example 4.7)

2

1.5

1

0.5

0

1

2

3

4

5

6

7

8

9

10

Let the switching signal be given as Fig. 4.1, where the ‘1’ and ‘2’ represent the first and second subsystem, respectively. Then all of the states of mRNAs and proteins of the considered switched GRN, starting from 100 pairs of initial functions φ and ϕ, go to zero, and partial simulation results are shown in Figs. 4.2, 4.3 and 4.4. From which, we can see that through choosing the appropriate average dwell time, the considered switched system is globally exponentially stable when the parameters switch from one subsystem to the other.

30

12

25

10

20

8

15

6

10

4

5

2

0

0

-5

0

1

2

3

4

5

6

7

8

9

10

-2

0

1

2

3

4

Fig. 4.2 Trajectories of mRNA and protein concentrations (Example 4.7)

5

6

7

8

9

10

96

4 Stability Analysis for Delayed Switching GRNs 45 40 35 30 25 20 15 10 5 0 -5

40 35 30 25 20 15 10 5 0 0

1

2

3

4

5

6

7

8

9

10

-5

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

Fig. 4.3 Trajectories of mRNA and protein concentrations (Example 4.7) 30 25 20 15 10 5 0 -5

0

1

2

3

4

5

6

7

8

9

10

45 40 35 30 25 20 15 10 5 0 -5

0

1

2

3

4

Fig. 4.4 Trajectories of mRNA and protein concentrations (Example 4.7)

4.4 Remarks and Notes This chapter investigated the problem of global exponential stability for a class of delayed switching GRNs. By employing the average dwell time approach and the piecewise Lyapunov function technique, sufficient conditions have been established to guarantee the global exponential stability for the considered systems on the two cases: constant time delays and time-varying delays. This chapter is taken from [9]. We end this chapter by introducing the following items, which are related to this chapter: 1. In [3], a hybrid GRNs model was proposed based on the Markov chain, and the stochastic stability of Markovian switching GRNs was studied. 2. Yin [8] addressed the finite-time stability problem for switching GRNs with unbounded continuous distributed delays and interval time-varying delays. 3. By utilizing an average dwell time approach, Yao et al. [7] investigated the exponential stability for switched GRNs with time delays.

4.4 Remarks and Notes

97

4. By utilizing the discrete Wirtinger-based inequality and a newly constraint condition on the feedback regulatory function, the robust stability of discrete-time randomly switched delayed GRNs with time-varying delays is investigated in [5]. 5. In [4], the delayed switched GRNs are modeled from a real biological system. Global asymptotical stability for the proposed switched GRNs is addressed by employing the LKF method and the matrix inequality techniques.

References 1. Hasty, J., Pradines, J., Dolnik, M., Collins, J.: Noise-based switches and amplifiers for gene expression. Proc. Natl. Acad. Sci. 97(5), 2075–2080 (2000) 2. Liberzon, D.: Switching in Systems and Control. Springer, New York (2003) 3. Sun, Y., Feng, G., Cao, J.: Stochastic stability of Markovian switching genetic regulatory networks. Phys. Lett. A 373(18), 1646–1652 (2009) 4. Wang, L., Luo, Z.P., Yang, H.L., Cao, J.: Stability of genetic regulatory networks based on switched systems and mixed time-delays. Math. Biosci. 278, 94–99 (2016) 5. Wang, X., Ren, C., An, J.: Robust stability of discrete-time randomly switched delayed genetic regulatory networks with known sojourn probabilities. J. Adv. Comput. Intell. Intell. Inf. 20(7), 1094–1102 (2016) 6. Wu, L., Feng, Z., Zheng, W.X.: Exponential stability analysis for delayed neural networks with switching parameters: average dwell time approach. IEEE Trans. Neural Netw. 21(9), 1396–1407 (2010) 7. Yao, Y., Liang, J., Cao, J.: Stability analysis for switched genetic regulatory networks: an average dwell time approach. J. Franklin Inst. 348(10), 2718–2733 (2011) 8. Yin, L.: Finite-time stability analysis of switched genetic regulatory networks. J. Appl. Math. (Article ID 768483, 11 pages, 2014) 9. Yu, T.T., Liu, J.X., Zeng, Y., Zhang, X., Zeng, Q.S., Wu, L.G.: Stability analysis of genetic regulatory networks with switching parameters and time delays. IEEE Trans. Neural Netw. Learn. Syst. 29(7), 3047–3058 (2018)

Chapter 5

Stability Analysis for Delayed Stochastic GRNs

In this chapter we will establish a robust asymptotic mean square stability criterion for delayed stochastic GRNs with parameter uncertainties. The theory of stochastic functional differential equations will be employed to analyze robust asymptotic mean square stability. An appropriate LKF is constructed based on the so-called “delay fractioning” idea [33]. To accurately estimate the time-varying delays, we partition the intervals [0, τ1 ], [τ1 , τ2 ], [0, σ1 ] and [σ1 , σ2 ] into several equal components, that is, [(i − 1)τ1 /r, iτ1 /r ] (i ∈ r ), [( j − 1)(τ2 − τ1 )/ p, j (τ2 − τ1 )/ p] ( j ∈  p), [(u − 1)σ1 /m, uσ1 /m] (u ∈ m) and [(v − 1)(σ2 − σ1 )/l, v(σ2 − σ1 )/l] (v ∈ l), where r , p, m and l are pre-specified positive integers denoting the numbers of fractions. This will result in a less conservative robust asymptotic mean square stability criterion in the form of LMIs, which can be easily checked by the toolbox YALMIP of MATLAB.

5.1 Model Description and Problem Formulation Consider the delayed GRN model (1.8), that is, ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ (t))), t  0,

(5.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(5.1b)

The genetic regulatory process is subject to intracellular and extracellular noise perturbations and environment fluctuations, which bring not only stochastic fluctuations and parameter uncertainties into the genetic regulatory process. When we take the Itô-type stochastic disturbance and norm-bounded parameter uncertainties into

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_5

99

100

5 Stability Analysis for Delayed Stochastic GRNs

model 5.1, a class of uncertain stochastic GRN models with time-varying delays can be described as follows: dm(t) ˜ = [−(A + ΔA(t))m(t) ˜ + (W + ΔW (t)) f ( p(t ˜ − σ (t)))]dt +ρ(t, m(t), ˜ p(t ˜ − σ (t)))dω1 (t), t  0,

(5.2a)

d p(t) ˜ = [−(C + ΔC(t)) p(t) ˜ + (D + ΔD(t))m(t ˜ − τ (t))]dt +ρ(t, p(t), ˜ m(t ˜ − τ (t)))dω2 (t), t  0, m(t) ˜ = φ(t), p(t) ˜ = ϕ(t), −  t  0,

(5.2b) (5.2c)

where = max{τ2 , σ2 }, ω1 (t) and ω2 (t) are mutually uncorrelated one dimensional Brownian motions satisfying E{dωi (t)} = 0, E{dωi2 (t)} = dt, i = 1, 2, φ(t) and ϕ(t) are the initial conditions, ρ : [0, +∞) × Rn × Rn → Rn×n is a continuous function representing the noise intensity, ΔA(t), ΔW (t), ΔC(t) and ΔD(t) are the parameter uncertainties. In the rest of this chapter, we make the following assumptions: Assumption 5.1 

     ΔA(t) ΔW (t) M1 H (t) N1 N2 , = M2 ΔC(t) ΔD(t)

(5.3)

where Mi and Ni , i = 1, 2 are known real matrices of appropriate sizes, and H (t) is an unknown time-varying matrix satisfying H T (t)H (t)  I for all t  0. Assumption 5.2 There exist matrices U  0 and V  0 such that ρ T (t, u, v)ρ(t, u, v)  u T U u + v T V v, ∀u, v ∈ Rn , t  0. Assumption 5.3 The time-varying delays τ (t) and σ (t) satisfy 0 < τ1  τ (t)  τ2 , 0 < σ1  σ (t)  σ2 ,

(5.4a)

τ˙ (t)  τd < 1, σ˙ (t)  σd < 1,

(5.4b)

where τ1 , τ2 , σ1 , σ2 , τd and σd are known scalars. Assumption 5.4 The nonlinear feedback regulation function f (·) in GRN (5.2) satisfies the sector condition: 0  f i (s)/s  ki (∀s = 0) and f i (0) = 0, i ∈ n, which is equivalent to the following condition f T (y)[ f (y) − K y]  0 with K = diag(k1 , k2 , . . . , kn ). Definition 5.1 When the uncertainties ΔA(t), ΔW (t), ΔC(t) and ΔD(t) vanish, we say that the trivial solution of GRN (5.2) is asymptotically stable in the sense of mean square, if the following (i) and (ii) are satisfied:

5.1 Model Description and Problem Formulation

101

(i) for any ε > 0, there exists a δ(ε) > 0 such that sup E{ φ(s) 22 + ϕ(s) 22 } < δ

− s0

2 2 ⇒ E{ m(t) ˜ ˜ 2 + p(t) 2 } < ε, ∀t  0;

(ii) there exists a δ˜ > 0 such that sup E{ φ(s) 22 + ϕ(s) 22 } < δ˜

− s0

⇒ E m(t) ˜ ˜ 2 → 0, E p(t) 2 → 0 as t → +∞. Definition 5.2 The trivial solution of delayed uncertain stochastic GRN (5.2) is said to be robustly asymptotically stable in the sense of mean square, if it is asymptotically stable in the sense of mean square for all uncertainties ΔA(t), ΔW (t), ΔC(t) and ΔD(t) satisfying Assumption 5.1. The aim of this chapter is to establish a robust asymptotic mean square stability criterion for the delayed uncertain stochastic GRN (5.2).

5.2 Main Results In this section, one criterion is presented to guarantee the robust asymptotic mean square stability of the delayed uncertain stochastic GRN (5.2). Theorem 5.3 For given positive integers r , p, m and l, and positive scalars τ1 , τ2 , σ1 , σ2 , τd and σd , the uncertain stochastic GRN (5.2) subject to Assumptions 5.1–5.4 is robustly asymptotically mean square stable if there exist positive   P1 P3 (r ) ( p) , Q , Q , Q (m) , Q (l) , R1 , R2 , R3 , S1 , S2 , S3 , S4 , definite matrices P3T P2 Γi := diag(γi1 , γi2 , . . . , γin ) (i = 1, 2) and positive constants l1 , l2 , εk (k ∈ 12) such that the following LMIs: P1  l1 I, P2  l2 I, ⎡ Φ11 Φ12 Φ13 Φ14 Φ15 ⎢ T ⎢Φ12 Φ22 0 0 0 ⎢ ⎢Φ T 0 Φ33 0 0 ⎢ 13 Φ(r, p, m, l) := ⎢ T ⎢Φ14 0 0 Φ44 0 ⎢ ⎢ T ⎣Φ15 0 0 0 Φ55 T 0 Φ16

0

0

(5.5) Φ16

⎤

⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ < 0, 0 ⎥ ⎥ ⎥ 0 ⎦ 0 Φ66

(5.6)

102

5 Stability Analysis for Delayed Stochastic GRNs

where Φ11 = Φ1 + Φ2 + Φ3 + Φ4 + Φ5 , Φ1 =

3 i=0

Ψi , Φ4 =

10

Ψi , Φ5 = Ψ11 + Ψ12 ,

i=4

Ψ0 = sym(−e1T P1 Ae1 + e1T P1 W er + p+m+l+6 − erT+ p+3 P2 Cer + p+3 +erT+ p+3 P2 Der + p+2 − e1T P3 Cer + p+3 + e1T P3 Der + p+2 −erT+ p+3 P3T Ae1 + erT+ p+3 P3T W er + p+m+l+6 ), Ψ1 = l1 (e1T U1 e1 + erT+ p+m+l+4 V1 er + p+m+l+4 ), Ψ2 = l2 (erT+ p+3 U2 er + p+3 + erT+ p+2 V2 er + p+2 ), Ψ3 = ε1 e1T N1T N1 e1 + ε2 erT+ p+m+l+6 N2T N2 er + p+m+l+6 +ε3 erT+ p+3 N1T N1 er + p+3 + ε4 erT+ p+2 N2T N2 er + p+2 +ε5 erT+ p+3 N1T N1 er + p+3 + ε6 erT+ p+2 N2T N2 er + p+2 +ε7 e1T N1T N1 e1 + ε8 erT+ p+m+l+6 N2T N2 er + p+m+l+6 , Φ2 = E 1T Q (r ) E 1 − E 2T Q (r ) E 2 + E 3T Q ( p) E 3 − E 4T Q ( p) E 4 + e1T R1 e1 −(1 − τd )erT+ p+2 R1 er + p+2 , Φ3 = E 5T Q (m) E 5 − E 6T Q (m) E 6 + E 7T Q (l) E 7 −E 8T Q (l) E 8 + erT+ p+3 R2 er + p+3 −(1 − σd )erT+ p+m+l+4 R2 er + p+m+l+4 +erT+ p+m+l+5 R3 er + p+m+l+5 −(1 − σd )erT+ p+m+l+6 R3 er + p+m+l+6 , 1 [(er + p+2 − er + p+1 )T (S1 + S2 )(er + p+2 − er + p+1 )], τ12 1 Ψ5 = − [(er +1 − er + p+2 )T (S1 + S2 )(er +1 − er + p+2 )], τ12 1 Ψ6 = − [(e1 − er +1 )T S1 (e1 − er +1 )], τ1

Ψ4 = −

1 [(er + p+m+l+4 − er + p+m+l+3 )T (S3 + S4 ) σ12 ×(er + p+m+l+4 − er + p+m+l+3 )],

Ψ7 = −

5.2 Main Results

103

1 [(er + p+m+3 − er + p+m+l+4 )T (S3 + S4 ) σ12 ×(er + p+m+3 − er + p+m+l+4 )],

Ψ8 = −

Ψ9 = −

1 [(er + p+3 − er + p+m+3 )T S3 (er + p+3 − er + p+m+3 )], σ1

Ψ10 = τ2 ε9 N˜ 1T N˜ 1 + τ12 ε10 N˜ 1T N˜ 1 + σ2 ε11 N˜ 2T N˜ 2 + σ12 ε12 N˜ 2T N˜ 2 , Ψ11 = −2erT+ p+m+l+6 Γ1 er + p+m+l+6 + 2erT+ p+m+l+6 Γ1 K er + p+m+l+4 , Ψ12 = −2erT+ p+m+l+5 Γ2 er + p+m+l+5 + 2erT+ p+m+l+5 Γ2 K er + p+3 , Φ22 = −diag(ε1 In , ε2 In , ε3 In , ε4 In , ε5 In , ε6 In , ε7 In , ε8 In ),     −S1 S1 M1 −S2 S2 M1 Φ33 = , Φ , = 44 M1T S1 −ε9 In M1T S2 −ε10 In     −S3 S3 M2 −S4 S4 M2 Φ55 = , Φ66 = , M2T S3 −ε11 In M2T S4 −ε12 In Φ12 = [e1T P1 M1 e1T P1 M1 erT+ p+3 P2 M2 erT+ p+3 P2 M2 e1T P3 M2 e1T P3 M2 erT+ p+3 P3T M1 erT+ p+3 P3T M1 ], √ T √ T Φ13 = [ τ2 A S1 0(r + p+m+l+6)n,n ], Φ14 = [ τ12 A S2 0(r + p+m+l+6)n,n ], √ √ B T S3 0(r + p+m+l+6)n,n ], Φ16 = [ σ12 B T S4 0(r + p+m+l+6)n,n ], Φ15 = [ σ2 ei = [0n,(i−1)n In 0n,(r + p+m+l+6−i)n ], i ∈ r + p + m + l + 6, E 1 = col(e1 , e2 , . . . , er ), E 2 = col(e2 , e3 , . . . , er +1 ), E 3 = col(er +1 , er +2 , . . . , er + p ), E 4 = col(er +2 , er +3 , . . . , er + p+1 ), E 5 = col(er + p+3 , er + p+4 , . . . , er + p+m+2 ), E 6 = col(er + p+4 , er + p+5 , . . . , er + p+m+3 ), E 7 = col(er + p+m+3 , er + p+m+4 , . . . , er + p+m+l+2 ), E 8 = col(er + p+m+4 , er + p+m+5 , . . . , er + p+m+l+3 ), A˜ = −Ae1 + W er + p+m+l+6 , B˜ = Der + p+2 − Cer + p+3 , N˜1 = N1 e1 − N2 er + p+m+l+6 , N˜2 = −N2 er + p+2 + N1 er + p+3 ,

104

5 Stability Analysis for Delayed Stochastic GRNs

τ12 = τ2 − τ1 , σ12 = σ2 − σ1 . Proof Set m(α) ¯ = −(A + ΔA(α))m(α) ˜ + (W + ΔW (α)) f ( p(α ˜ − σ (α))),

(5.7a)

p(α) ¯ = −(C + ΔC(α)) p(α) ˜ + (D + ΔD(α))m(t ˜ − τ (α)).

(5.7b)

Then, it follows from (5.2) that dm(t) ˜ = m(t)dt ¯ + ρ(t, m(t), ˜ p(t ˜ − σ (t)))dω1 (t),

(5.8a)

d p(t) ˜ = p(t)dt ¯ + ρ(t, p(t), ˜ m(t ˜ − τ (t)))dω2 (t),

(5.8b)

and hence for any a  b,  m(b) ˜ − m(a) ˜ =

b



a

 p(b) ˜ − p(a) ˜ =

b

m(t)dt ¯ +

ρ(t, m(t), ˜ p(t ˜ − σ (t)))dω1 (t),

(5.9a)

ρ(t, p(t), ˜ m(t ˜ − τ (t)))dω2 (t).

(5.9b)

a b



b

p(t)dt ¯ +

a

a

Choose an LKF candidate to be V (t, m(t), ˜ p(t)) ˜ =

4

Vi (t, m(t), ˜ p(t)) ˜

(5.10)

i=1

with  T    m(t) ˜ P1 P3 m(t) ˜ ˜ p(t)) ˜ = V1 (t, m(t), , p(t) ˜ P3T P2 p(t) ˜  ˜ p(t)) ˜ = V2 (t, m(t),

t t− r1 τ1



+

M (s)Q M(s)ds +

V3 (t, m(t), ˜ p(t)) ˜ =

t−τ1 t−τ1 −



t

1 p τ12

t− m1 σ1



+

P (s)Q

t−σ1

t−σ1 − 1l σ12  t T t−σ (t)

t t−τ (t)

m˜ T (s)R1 m(s)ds ˜

¯ T (s)Q ( p) M(s)ds, ¯ M

T

+



(r )

T

(m)

 P(s)ds +

t t−σ (t)

¯ P¯ T (s)Q (l) P(s)ds

f ( p(s))R ˜ ˜ 3 f ( p(s))ds,

p˜ T (s)R2 p(s)ds ˜

5.2 Main Results

105

 V4 (t, m(t), ˜ p(t)) ˜ =



0

−τ2



+ + +

m¯ T (α)S1 m(α)dαdβ ¯

t+β −τ1  t

−τ2  0



t



−σ2 t+β −σ1  t −σ2

m¯ T (α)S2 m(α)dαdβ ¯

t+β t

p¯ T (α)S3 p(α)dαdβ ¯ p¯ T (α)S4 p(α)dαdβ, ¯

t+β

where      r −1 1 , τ1 M(t) = col m(t), ˜ m˜ t − τ1 , . . . , m˜ t − r r      p−1 1 ¯ M(t) = col m(t ˜ − τ1 ), m˜ t − τ12 , . . . , m˜ t − τ12 , p p      1 m−1 P(t) = col p(t), ˜ p˜ t − σ1 , . . . , p˜ t − , σ1 m m      l −1 1 ¯ P(t) = col p(t ˜ − σ1 ), p˜ t − σ12 , . . . , p˜ t − . σ12 l l By Itô’s differential formula along the trajectory of (5.8), we can obtain that ˜ p(t)) ˜ LV1 (t, m(t), = 2m˜ T (t)P1 m(t) ¯ + 2 p˜ T (t)P2 p(t) ¯ + 2m˜ T (t)P3 p(t) ¯ + 2 p˜ T (t)P3T m(t) ¯ +ρ T (t, m(t), ˜ p(t ˜ − σ (t)))P1 ρ(t, m(t), ˜ p(t ˜ − σ (t))) +ρ T (t, p(t), ˜ m(t ˜ − τ (t)))P2 ρ(t, p(t), ˜ m(t ˜ − τ (t))) = ξ T (t)Ψ0 ξ(t) − 2m˜ T (t)P1 ΔA(t)m(t) ˜ + 2m˜ T (t)P1 ΔW (t) f ( p(t ˜ − σ (t))) −2 p˜ T (t)P2 ΔC(t) p(t) ˜ + 2 p˜ T (t)P2 ΔD(t)m(t ˜ − τ (t)) −2m˜ T (t)P3 ΔC(t) p(t) ˜ + 2m˜ T (t)P3 ΔD(t)m(t ˜ − τ (t)) −2 p˜ T (t)P3T ΔA(t)m(t) ˜ + 2 p˜ T (t)P3T ΔW (t) f ( p(t ˜ − σ (t))) +ρ T (t, m(t), ˜ p(t ˜ − σ (t)))P1 ρ(t, m(t), ˜ p(t ˜ − σ (t))) +ρ T (t, p(t), ˜ m(t ˜ − σ (t)))P2 ρ(t, p(t), ˜ m(t ˜ − σ (t))),

(5.11)

106

5 Stability Analysis for Delayed Stochastic GRNs

LV2 (t, m(t), ˜ p(t)) ˜

    1 1 = MT (t)Q (r ) M(t) − MT t − τ1 Q (r ) M t − τ1 r r T ( p) ¯ ¯ +M (t − τ1 )Q M(t − τ1 )     1 1 T ( p) ¯ ¯ −M t − τ1 − τ12 Q M t − τ1 − τ12 p p ˜ − (1 − τ˙ (t))m˜ T (t − τ (t))R1 m(t ˜ − τ (t)) +m˜ T (t)R1 m(t) T  ξ (t)Φ2 ξ(t),

(5.12)

LV3 (t, m(t), ˜ p(t)) ˜

    1 1 = P T (t)Q (m) P(t) − P T t − σ1 Q (m) P t − σ1 m m ¯ − σ1 ) +P¯ T (t − σ1 )Q (l) P(t     1 1 ˜ −P¯ T t − σ1 − σ12 Q (l) P¯ t − σ1 − σ12 + p˜ T (t)R2 p(t) l l + f T ( p(t))R ˜ ˜ − (1 − σ˙ (t)) p˜ T (t − σ (t))R2 p(t ˜ − σ (t)) 3 f ( p(t)) −(1 − σ˙ (t)) f T ( p(t ˜ − σ (t)))R3 f ( p(t ˜ − σ (t)))  ξ T (t)Φ3 ξ(t),

(5.13)

˜ p(t)) ˜ LV4 (t, m(t), = τ2 m¯ T (t)S1 m(t) ¯ + τ12 m¯ T (t)S2 m(t) ¯ + σ2 p¯ T (t)S3 p(t) ¯ + σ12 p¯ T (t)S4 p(t) ¯  t  t−τ1 − m¯ T (α)S1 m(α)dα ¯ − m¯ T (α)S2 m(α)dα ¯  −

t−τ2 t t−σ2

 p¯ T (α)S3 p(α)dα ¯ −

t−τ2 t−σ1

p¯ T (α)S4 p(α)dα, ¯

(5.14)

t−σ2

where ¯ − σ1 ), ¯ − τ1 ), m(t ˜ − τ2 ), m(t ˜ − τ (t)), P(t), P(t ξ(t) = col(M(t), M(t ˜ − σ (t)), f ( p(t)), ˜ f ( p(t ˜ − σ (t)))). p(t ˜ − σ2 ), p(t From Assumption 5.2 and the inequality (5.5), we obtain that ˜ p(t ˜ − σ (t)))P1 ρ(t, m(t), ˜ p(t ˜ − σ (t))) ρ T (t, m(t), T  l1 ρ (t, m(t), ˜ p(t ˜ − σ (t)))ρ(t, m(t), ˜ p(t ˜ − σ (t)))  l1 [m˜ T (t)U1 m(t) ˜ + p˜ T (t − σ (t))V1 p(t ˜ − σ (t))] T = ξ (t)Ψ1 ξ(t).

(5.15)

˜ m(t ˜ − τ (t)))P2 ρ(t, p(t), ˜ m(t ˜ − τ (t)))  ξ T (t)Ψ2 ξ(t). ρ T (t, p(t),

(5.16)

Similarly,

5.2 Main Results

107

By using Lemma 1.11 and Assumption 5.1, we have ˜ −2m˜ T (t)P1 ΔA(t)m(t) −1 T T  m˜ (t)[ε1 P1 M1 M1 P1 + ε1 N1T N1 ]m(t) ˜ = ξ T (t)[Θ1 + ε1 e1T N1T N1 e1 ]ξ(t)

(5.17)

with Θ1 = ε1−1 e1T P1 M1 M1T P1 e1 . Similarly, ˜ − σ (t))) 2m˜ T (t)P1 ΔW (t) f ( p(t  ξ T (t)[Θ2 + ε2 erT+ p+m+l+6 N2T N2 er + p+m+l+6 ]ξ(t), − 2 p˜ T (t)P2 ΔC(t) p(t) ˜  ξ T (t)[Θ3 + ε3 erT+ p+3 N1T N1 er + p+3 ]ξ(t), 2 p˜ T (t)P2 ΔD(t)m(t ˜ − τ (t))  ξ T (t)[Θ4 + ε4 erT+ p+2 N2T N2 er + p+2 ]ξ(t), − 2m˜ T (t)P3 ΔC(t) p(t) ˜  ξ T (t)[Θ5 + ε5 erT+ p+3 N1T N1 er + p+3 ]ξ(t),

(5.18) (5.19)

(5.20) (5.21)

2m˜ T (t)P3 ΔD(t)m(t ˜ − τ (t))  ξ T (t)[Θ6 + ε6 erT+ p+2 N2T N2 er + p+2 ]ξ(t), ˜  ξ T (t)[Θ7 + ε7 e1T N1T N1 e1 ]ξ(t), − 2 p˜ T (t)P3T ΔA(t)m(t) 2 p˜ T (t)P3T ΔW (t) f ( p(t ˜ − σ (t))) T T  ξ (t)[Θ8 + ε8 er + p+m+l+6 N2T N2 er + p+m+l+6 ]ξ(t), where Θ2 = ε2−1 e1T P1 M1 M1T P1 e1 , Θ3 = ε3−1 erT+ p+3 P2 M2 M2T P2 er + p+3 , Θ4 = ε4−1 erT+ p+3 P2 M2 M2T P2 er + p+3 , Θ5 = ε5−1 e1T P3 M2 M2T P3T e1 , Θ6 = ε6−1 e1T P3 M2 M2T P3T e1 , Θ7 = ε7−1 erT+ p+3 P3T M1 M1T P3 er + p+3 , Θ8 = ε8−1 erT+ p+3 P3T M1 M1T P3 er + p+3 .

(5.22) (5.23)

(5.24)

108

5 Stability Analysis for Delayed Stochastic GRNs

The combination of (5.11) and (5.15)–(5.24) gives  LV1 (t, m(t), ˜ p(t)) ˜  ξ T (t) Φ1 +

8

 Θi ξ(t).

i=1

This, together with

8 i=1

−1 T Θi = −Φ12 Φ22 Φ12 , yields

  −1 T ˜ p(t)) ˜  ξ T (t) Φ1 − Φ12 Φ22 Φ12 ξ(t). LV1 (t, m(t),

(5.25)

From Lemma 1.14 and (5.9), it readily follows that  −

t−τ (t)

m¯ T (α)(S1 + S2 )m(α)dα ¯

t−τ2

−

1 τ2 − τ (t)



t−τ (t)

 m¯ T (α)dα(S1 + S2 )

t−τ2

t−τ (t)

m(α)dα ¯

t−τ2

2  ξ T (t)Ψ4 ξ(t) + [m(t ˜ − τ (t)) − m(t ˜ − τ2 )]T (S1 + S2 ) τ12  t−τ (t) ρ(α, m(α), ˜ p(α ˜ − σ (α)))dω1 (α). ×

(5.26)

t−τ2

Similarly, we have  −

t−τ1 t−τ (t)

m¯ T (α)(S1 + S2 )m(α)dα ¯

2 [m˜ T (t − τ1 ) − m˜ T (t − τ (t))](S1 + S2 )  ξ (t)Ψ5 ξ(t) + τ12  t−τ1 × ρ(α, m(α), ˜ p(α ˜ − σ (α)))dω1 (α), T

t−τ (t)



−

t

(5.27)

m¯ T (α)S1 m(α)dα ¯

t−τ1

2  ξ (t)Ψ6 ξ(t) + [m(t) ˜ − m(t ˜ − τ1 )]T S1 τ1  t × ρ(α, m(α), ˜ p(α ˜ − σ (α)))dω1 (α), T

t−τ1

(5.28)

5.2 Main Results

109



t−σ (t)

−

p¯ T (α)(S3 + S4 ) p(α)dα ¯

t−σ2

2  ξ T (t)Ψ7 ξ(t) + [ p(t ˜ − σ (t)) − p(t ˜ − σ2 )]T (S3 + S4 ) σ12  t−σ (t) ρ(α, p(α), ˜ m(α ˜ − τ (α)))dω2 (α), ×  −

(5.29)

t−σ2 t−σ1 t−σ (t)

p¯ T (α)(S3 + S4 ) p(α)dα ¯

2 [ p(t ˜ − σ1 ) − p(t ˜ − σ (t))]T (S3 + S4 )  ξ T (t)Ψ8 ξ(t) + σ12  t−σ1 × ρ(α, p(α), ˜ m(α ˜ − τ (α)))dω2 (α), t−σ (t)

−



t

(5.30)

p¯ T (α)S3 p(α)dα ¯

t−σ1

2  ξ T (t)Ψ9 ξ(t) + [ p(t) ˜ − p(t ˜ − σ1 )]T S3 σ1  t × ρ(α, p(α), ˜ m(α ˜ − τ (α)))dω2 (α).

(5.31)

t−σ1

Substituting (5.3) into (5.7), we obtain that m(t) ¯ = −(A + M1 H (t)N1 )m(t) ˜ + (W + M1 H (t)N2 ) f ( p(t ˜ − σ (t))) ˜ ˜ = ( A − M1 H (t) N1 )ξ(t), p(t) ¯ = −(C + M2 H (t)N1 ) p(t) ˜ + (D + M2 H (t)N2 )m(t ˜ − τ (t)) = ( B˜ − M2 H (t) N˜ 2 )ξ(t). Then according to Lemma 1.10, we have ¯  ξ T (t)[ A˜ T (S1−1 − ε9−1 M1 M1T )−1 A˜ + ε9 N˜ 1T N˜ 1 ]ξ(t), m¯ T (t)S1 m(t)

(5.32)

−1 m¯ T (t)S2 m(t) ¯  ξ T (t)[ A˜ T (S2−1 − ε10 M1 M1T )−1 A˜ + ε10 N˜ 1T N˜ 1 ]ξ(t),

(5.33)

−1 p¯ T (t)S3 p(t) ¯  ξ T (t)[ B˜ T (S3−1 − ε11 M2 M2T )−1 B˜ + ε11 N˜ 2T N˜ 2 ]ξ(t),

(5.34)

−1 p¯ T (t)S4 p(t) ¯  ξ T (t)[ B˜ T (S4−1 − ε12 M2 M2T )−1 B˜ + ε12 N˜ 2T N˜ 2 ]ξ(t).

(5.35)

110

5 Stability Analysis for Delayed Stochastic GRNs

Noting that −1 Φ33

=

−ε9−1

 −1  S1−1 S˜1 M1 ε9 S1 S˜1 S1−1 M1T S˜1 S1−1 (In − ε9−1 M1T S1 M1 )−1

with S˜1 = (S1−1 − ε9−1 M1 M1T )−1 , one can easily derive −1 T ˜ Φ13 = τ2 A˜ T (S1−1 − ε9−1 M1 M1T )−1 A. Φ13 Φ33

(5.36)

−1 T −1 ˜ Φ14 = τ12 A˜ T (S2−1 − ε10 M1 M1T )−1 A, Φ14 Φ44

(5.37)

−1 T −1 ˜ Φ15 Φ55 Φ15 = σ2 B˜ T (S3−1 − ε11 M2 M2T )−1 B,

(5.38)

−1 T −1 ˜ Φ16 = σ12 B˜ T (S4−1 − ε12 M2 M2T )−1 B. Φ16 Φ66

(5.39)

Similarly,

The combination of (5.14) and (5.26)–(5.39) gives  E{LV4 (t, m(t), ˜ p(t))} ˜  E{ξ (t) Φ4 + T

6

 Φ1i Φii−1 Φ1iT

ξ(t)}.

(5.40)

i=3

From Assumption 5.4, one can see that −2 f T ( p(t ˜ − σ (t)))Γ1 f ( p(t ˜ − σ (t))) + 2 f T ( p(t ˜ − σ (t)))Γ1 K p(t ˜ − σ (t))  0, that is, ξ T (t)Ψ11 ξ(t)  0.

(5.41)

Using the same way, we can also get ξ T (t)Ψ12 ξ(t)  0.

(5.42)

Combining (5.10), (5.12), (5.13), (5.25), (5.40), (5.41) and (5.42), we obtain that  E{LV (t, x(t), y(t))}  E{ξ (t) Φ11 + T

6 i=2

 Φ1i Φii−1 Φ1iT

ξ(t)}.

5.2 Main Results

111

This, together with (5.6), derives E{LV (t, x(t), y(t))} < 0, which implies from the theory of stochastic functional differential equations that the delayed uncertain stochastic GRN (5.2) is robustly asymptotically stable in the sense of mean square. The proof is completed. Remark 5.4 Compared with the approach proposed in this chapter and ones in [33, 38, 46], there are the following two main differences: • The LKFs employed in [33, 38, 46] contains noise perturbations which has been removed in our approach;  t−τ (t) ¯ are estimated by (5.9) • In this chapter, these terms like − t−τ2 m¯ T (α)S1 m(α)dα and Jensen’s inequality instead of the so-called free-weighting matrices technique in [33, 38, 46]. This will reduce the computational complexity.

5.3 Numerical Examples In this section, we give two numerical examples to demonstrate the effectiveness of the method proposed in this chapter. Example 5.5 Consider the delayed uncertain stochastic GRN (5.2) with A = diag(1, 2, 3), C = diag(2.5, 2.5, 2.5), D = diag(2, 2, 2), ⎡

⎤ 0 0 −1 W = 0.8 × ⎣−1 0 0 ⎦ , 0 −1 0 where the coefficient 0.8 is the transcriptional rate, and Mi = Ni = diag(0.25, 0.25, 0.25), i = 1, 2, H (t) = diag(sin(2t), cos(5t), cos(1.5t)). Assume that K = diag(0.65, 0.65, 0.65). The noise intensity vectors satisfy Assumption 5.2 with U = V = diag(0.5, 0.5, 0.5). The time delays τ (t) and σ (t) are assumed to be τ (t) = 0.1 + 0.2 sin2 (3t/2), σ (t) = 0.2 + 0.6 sin2 (2t/3).

112

5 Stability Analysis for Delayed Stochastic GRNs 3

2

2

1.5

1

1

0 -1

0.5

-2

0

-3

-0.5

-4

-1

-5 -6

0

1

2

3

4

5

6

7

8

9

10

-1.5

0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Fig. 5.1 Brownian motions (Example 5.5) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

-0.1

0

Fig. 5.2 Trajectories of mRNA and protein concentrations (Example 5.5)

When r = 2, p = 2, m = 1, l = 1, τ1 = 0.1, τ2 = 0.3, σ1 = 0.2, σ2 = 0.8, τd = 0.3 and σd = 0.4, by using the LMI Toolbox of MATLAB, we can find that the LMIs (5.5) and (5.6) are feasible. According to Theorem 5.3, the considered GRN with both noise perturbations and parameter uncertainties is robustly asymptotically mean square stable. In the case, the Brownian motions ω1 (t) and ω2 (t) are given as in Fig. 5.1, and the simulation results of the trajectories of mRNAs and proteins are shown in Fig. 5.2. Example 5.6 Consider the delayed uncertain stochastic GRN (5.2) with A = diag(6, 6), C = diag(5, 5), 

   1 0 0 −5 D= , W = , 0 1.2 −5 0

5.3 Numerical Examples

113

Table 5.1 Maximum values of σ2 with different partitions (Example 5.6)

(r, p, m, l)

Maximum of σ2

(1, 1, 1, 1) (2, 1, 1, 1) (1, 2, 1, 1) (1, 1, 2, 1) (1, 1, 1, 2) (2, 2, 1, 1) (1, 2, 1, 2) (2, 1, 1, 2) (1, 1, 2, 2) (3, 1, 1, 1) (4, 1, 1, 1)

911.02 911.02 911.23 912.01 911.10 912.32 913.00 913.30 912.53 911.10 912.11

M1 =



   0.1 0.08 0.2 0.1 , M2 = , 0.08 0.04 0.1 −0.3

N1 =

    0.4 0.1 0.2 −0.3 , N2 = , 0.1 0.4 −0.1 0.2

K = diag(0.65, 0.65), and the noise intensity vectors satisfy Assumption 5.2 with U = V = diag(0.5, 0.5). When τ1 = 0.1, τ2 = 0.3, σ1 = 0.2, τd = 1.2 and σd = 0.94, by using the LMI Toolbox of MATLAB to solve the LMIs (5.5) and (5.6), we can see from Table 5.1 that σ2 has different maximum for different partition, and the more number of partition, the greater maximum of σ2 .

5.4 Remarks and Notes This chapter investigated the problem of robust asymptotic mean square stability for a class of delayed uncertain stochastic GRNs. An novel “delay fractioning” approach is proposed to derive a robust mean square asymptotic stability criterion. This chapter is taken from [34]. We end this chapter by introducing the following items, which are related to this chapter: 1. For delayed stochastic GRNs, the problem of stability analysis has been investigated in [4, 7, 10, 15, 16, 18, 27, 35, 39, 46]. The related robust stability problems have been studied in [11, 21, 26, 31, 33, 36–38, 43] 2. The problem of asymptotic stability analysis for a class of delayed stochastic GRNs with impulses has been addressed in [12, 25].

114

5 Stability Analysis for Delayed Stochastic GRNs

3. For uncertain stochastic GRNs with mixed time-varying delays, Rakkiyappan and Balasubramaniam [22] investigated robust global asymptotic stability in the mean square sense, and Wang and Zhong [28] presented the stochastic asymptotic stability. 4. For delayed switched stochastic GRNs, Zhang et al. [45] investigated the exponential stability criterion, and Krishnasamy and Balasubramaniam [8] established the stochastic stability criteria. 5. In [44], by introducing an LKF which takes into account the ranges of delays and employing some free-weighting matrices, some delay-probability-distributiondependent stability criteria are established to guarantee the GRNs to be asymptotically mean square stable. In [23, 29, 30], the problem of robust delayprobability-distribution-dependent stability for uncertain stochastic delayed GRNs has been addressed. 6. For delayed Markovian jumping GRNs, Wang et al. [32] addressed the problem of global robust power-rate stability in mean square sense, Gomez et al. [17], Zhang et al. [42] and Ratnavelu et al. [24] derived the mean square asymptotic stability criteria, Ma et al. [19] investigated the problems of passivity and passification, Zhang et al. [40] obtained the stochastic stability criteria, and the related problem of robust stability has been addressed in [6, 9, 47]. 7. Balasubramaniam et al. [3] investigated the asymptotic stability of fuzzy Markovian jumping delayed GRNs, while Balasubramaniam and Sathy [2] addressed the problem of robust asymptotic stability for fuzzy Markovian jumping delayed GRNs with uncertain parameters. 8. For delayed Markovian jumping stochastic GRNs, sufficient conditions are given to ensure the robust mean square asymptotic stability [1], the robust global mean square asymptotic stability [13, 20], the robust global μ-stability [14] and the passivity [41]. 9. The dynamics of systems with stochastically varying time delays are investigated in [5].

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26. Salimpour, A., Majd, V.J., Sojoodi, M.: Comment on: robust stability of stochastic genetic regulatory networks with discrete and distributed delays. Symp. (Int.) Combust. 15(4), 769– 770 (2010) 27. Wang, G., Cao, J.: Robust exponential stability analysis for stochastic genetic networks with uncertain parameters. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3369–3378 (2009) 28. Wang, W., Zhong, S.: Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays. Neurocomputing 82, 143–156 (2012) 29. Wang, W., Zhong, S., Liu, F., Cheng, J.: Robust delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays. Int. J. Robust Nonlinear Control 24(16), 2574–2596 (2014). https://doi.org/ 10.1002/rnc.3011 30. Wang, W., Zhong, S., Nguang, S.K., Liu, F.: Robust deley-probability-distribution-dependent stability of uncertain genetic regulatory networks with time-varying delays. Neurocomputing 119(SI), 153–164 (2013) 31. Wang, W.Q., Nguang, S.K., Zhong, S.M., Liu, F.: Robust stability analysis of stochastic delayed genetic regulatory networks with polytopic uncertainties and linear fractional parametric uncertainties. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1569–1581 (2014) 32. Wang, Y., Cao, J., Li, L.: Global robust power-rate stability of delayed genetic regulatory networks with noise perturbations. Cogn. Neurodyn. 4(1), 81–90 (2010) 33. Wang, Y., Wang, Z., Liang, J.: On robust stability of stochastic genetic regulatory networks with time delays: a delay fractioning approach. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40(3), 729–740 (2010) 34. Wang, Y.T., Yu, A.H., Zhang, X.: Robust stability of stochastic genetic regulatory networks with time-varying delays: a delay fractioning approach. Neural Comput. Appl. 23(5), 1217–1227 (2013) 35. Wang, Z., Liao, X., Guo, S., Wu, H.: Mean square exponential stability of stochastic genetic regulatory networks with time-varying delays. Inf. Sci. 181(4), 792–811 (2011) 36. Wang, Z., Liao, X., Mao, J., Liu, G.: Robust stability of stochastic genetic regulatory networks with discrete and distributed delays. Symp. (Int.) Combust. 13(12), 1199–1208 (2009) 37. Wang, Z.X., Liu, G.D., Sun, Y.H., Wu, H.L.: Robust stability of stochastic delayed genetic regulatory networks. Cogn. Neurodyn. 3(3), 271–280 (2009) 38. Wu, H., Liao, X., Guo, S., Feng, W., Wang, Z.: Stochastic stability for uncertain genetic regulatory networks with interval time-varying delays. Neurocomputing 72(13–15), 3263– 3276 (2009) 39. Yu, T.T., Wang, J., Zhang, X.: A less conservative stability criterion for delayed stochastic genetic regulatory networks. Math. Prob. Eng. 2014 (Article ID 768483, 11 pages, 2014) 40. Zhang, B., Xu, S., Chu, Y., Zong, G.: Delay-dependent stability for Markovian genetic regulatory networks with time-varying delays. Asian J. Control 14(5), 1403–1406 (2012) 41. Zhang, D., Yu, L.: Passivity analysis for stochastic Markovian switching genetic regulatory networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 16(8), 2985–2992 (2011) 42. Zhang, W., Fang, J., Tang, Y.: Stochastic stability of Markovian jumping genetic regulatory networks with mixed time delays. Appl. Math. Comput. 217(17), 7210–7225 (2011) 43. Zhang, W., Tang, T., Fang, J.A., Wu, X.: Stochastic stability of genetic regulatory networks with a finite set of delay characterization. Chaos 22(2) (Article ID 023106, 2012) 44. Zhang, W.B., Fang, J.A., Tang, Y.: New robust stability analysis for genetic regulatory networks with random discrete delays and distributed delays. Neurocomputing 74(14–15), 2344–2360 (2011) 45. Zhang, W.B., Tang, Y., Wu, X.T., Fang, J.A.: Stochastic stability of switched genetic regulatory networks with time-varying delays. IEEE Trans. Nanobiosci. 13(3), 336–342 (2014) 46. Zhou, Q., Xu, S.Y., Chen, B., Li, H.Y., Chu, Y.M.: Stability analysis of delayed genetic regulatory networks with stochastic disturbances. Phys. Lett. A 373(41), 3715–3723 (2009) 47. Zhu, Y., Zhang, Q., Wei, Z., Zhang, L.: Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing 110, 44–50 (2013)

Chapter 6

Stability Analysis for Delayed Reaction-Diffusion GRNs

This chapter addresses the problems of asymptotic stability analysis and finite-time stability analysis for delayed reaction-diffusion GRNs, respectively.

6.1 Problem Formulation Consider the delayed GRN model (1.8), that is, ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(6.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(6.1b)

In this chapter, we will always assume that the time-varying delays satisfy ¯ σ(t) ˙  σ¯ d , 0  τ (t)  τ¯ , τ˙ (t)  τ¯d , 0  σ(t)  σ,

(6.2)

where τ¯ , σ, ¯ τ¯d and σ¯ d are nonnegative real numbers. In some mathematical models of GRNs, it is implicitly assumed that genetic regulatory systems are spatially homogeneous; namely, the concentrations of mRNAs and proteins are homogenous in space at all times. However, in some cases it is necessary to take into account the diffusion of gene products, including mRNAs and proteins [1]. Hence, it is imperative to introduce reaction-diffusions into any functional differential equation model describing GRNs.

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_6

117

118

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

When we take the reaction-diffusions into GRN model (6.1), a delayed reactiondiffusion GRN model can be described as follows:   l  ∂ m(t, ˜ x) ∂ ∂ m(t, ˜ x) Dk − Am(t, ˜ x) = ∂t ∂xk ∂xk k=1 +W f ( p(t ˜ − σ(t), x)), t  0,

(6.3a)

  l  ∂ p(t, ˜ x) ∂ ∂ p(t, ˜ x) Dk∗ − C p(t, ˜ x) = ∂t ∂xk ∂xk k=1 +D m(t ˜ − τ (t), x), t  0,

(6.3b)

where x = col(x1 , x2 , . . . , xl ) ∈ R ⊂ Rl , R = {x : |xk |  L k , k ∈ l}, A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ), D = diag(d1 , d2 , . . . , dn ), Dk = diag(D1k , D2k , . . . , Dnk ), ∗ ∗ ∗ Dk∗ = diag(D1k , D2k , . . . , Dnk ),

m(t, ˜ x) = col(m˜ 1 (t, x), m˜ 2 (t, x), . . . , m˜ n (t, x)), p(t, ˜ x) = col( p˜ 1 (t, x), p˜ 2 (t, x), . . . , p˜ n (t, x)), f ( p(t ˜ − σ(t), x)) = col( f 1 ( p˜ 1 (t − σ(t), x)), f 2 ( p˜ 2 (t − σ(t), x)), . . . , f n ( p˜ n (t − σ(t), x))), ∗ > 0 denote the diffusion rate matrices. and L k is a constant, Dik > 0 and Dik The function f in (6.3) satisfies:

f (0) = 0, f T (y)( f (y) − K y)  0, ∀y ∈ Rn , where K = diag(k1 , k2 , . . . , kn ) > 0.

(6.4)

6.1 Problem Formulation

119

The initial conditions associated with GRN (6.3) are given as follows: m˜ i (s, x) = φi (s, x), x ∈ R, s ∈ [−d, 0], i ∈ n, p˜ i (s, x) = φi∗ (s, x), x ∈ R, s ∈ [−d, 0], i ∈ n, where d = max{σ, ¯ τ¯ }, and φi (s, x), φi∗ (s, x) ∈ C 1 ([−d, 0] × R, R). The following two types of boundary conditions are considered: (i) Dirichlet boundary conditions m˜ i (t, x) = 0, x ∈ ∂R, t ∈ [−d, +∞), i ∈ n, p˜ i (t, x) = 0, x ∈ ∂R, t ∈ [−d, +∞), i ∈ n; (ii) Neumann boundary conditions ∂ m˜ i (t, x) = 0, x ∈ ∂R, t ∈ [−d, +∞), i ∈ n, ∂n ∂ p˜ i (t, x) = 0, x ∈ ∂R, t ∈ [−d, +∞), i ∈ n, ∂n where n is the outer normal vector of ∂R. Definition 6.1 [3] The trivial solution of GRN (6.3) is said to be stable, if for any ε > 0, there exists a δ(ε) > 0, such that φ(t, x) 2d  δ(ε), φ∗ (t, x) 2d  δ(ε) ⇒ m(t, ˜ x) 2  ε, p(t, ˜ x) 2  ε, ∀t  0. Definition 6.2 [3] The trivial solution of GRN (6.3) is said to be asymptotically stable, if it is stable and there exists a δ˜ > 0, such that ˜ φ∗ (t, x) 2  δ˜ φ(t, x) 2d  δ, d ⇒ m(t, ˜ x) → 0, p(t, ˜ x) → 0 as t → ∞. Definition 6.3 [10] The trivial solution of GRN (6.3) is said to be finite-time stable with respect to positive real numbers c1 , c2 and T , if φ(t, x) 2d + φ∗ (t, x) 2d  c1 ⇒ m(t, ˜ x) 2 + p(t, ˜ x) 2  c2 , ∀t ∈ [0, T ]. The aim of this chapter is to establish less conservative asymptotic stability criteria and finite-time stability criteria for GRN (6.3).

120

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

6.2 Infinite-Time Case In this section we will establish delay-dependent asymptotic stability criteria for delayed reaction-diffusion GRNs under Dirichlet boundary conditions and Neumann boundary conditions, respectively. By constructing an appropriate LKF and employing Green’s second identity and the reciprocally convex technique, delaydependent asymptotic stability criteria for a class of delayed reaction-diffusion GRNs are derived. The obtained stability criteria are diffusion-dependent under Dirichlet boundary conditions, and are diffusion-independent under Neumann boundary conditions. The main contributions of this section can be summarized as the following three aspects: • Compared with the results in [6, 8], we remove the restriction on the upper bounds of the delay derivatives being less than 1. This is realized by introducing the term ˜ := V0 (t, m)

l   k=1

R

∂ m˜ T (t, x) ∂ m(t, ˜ x) N 1 Dk dx ∂xk ∂xk

into the LKF; • Green’s second identity and the reciprocally convex technique are used to estimate ˜ which will obtain less conservative asymptotic stability the derivative of V0 (t, m), criteria; • It is theoretically and numerically shown that the stability criteria established in this section are less conservative than the corresponding criteria in [6, 8] (see Sects. 6.2.2 and 6.2.3 below).

6.2.1 Asymptotic Stability Criteria In this subsection, we will investigate asymptotic stability criteria for GRN (6.3) under Dirichlet boundary  conditions  and Neumann boundary conditions, respectively. l ∂ ∂ Note that k=1 ∂xk Dk ∂xk can be regarded as a Laplacian operator which is formally self-adjoint and differential under the L 2 inner product for functions with a Dirichlet boundary or a Neumann boundary. As a result, the following lemma follows directly from Lemma 1.18. Lemma 6.4 Let N1 and N2 be a pair of positive diagonal matrices. Then   l ∂m T (t, x)  ∂ ∂m(t, x) N1 Dk dx ∂t ∂xk ∂xk R k=1  

 l  ∂m(t, x) ∂ ∂ T = Dk dx, m (t, x)N1 ∂xk ∂xk ∂t R k=1 

6.2 Infinite-Time Case

 R

 =

R

121

  l ∂ p T (t, x)  ∂ ∂ p(t, x) Dk∗ dx N2 ∂t ∂xk ∂xk k=1  

l  ∂ p(t, x) ∂ ∂ Dk∗ dx. p T (t, x)N2 ∂xk ∂xk ∂t k=1

First, we give an asymptotic stability criterion for GRN (6.3) under Dirichlet boundary conditions. Theorem 6.5 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (6.2), the trivial solution of GRN (6.3) under Dirichlet boundary conditions is asymptotically stable if there exist matrices Q iT = Q i > 0 (i ∈ 5) and R Tj = R j > 0 ( j = 1, 2), diagonal matrices P j > 0, Λ j > 0, N j > 0 ( j = 1, 2), and matrices G 1 and G 2 of appropriate sizes, such that the following LMIs hold:



R2 G 2 R1 G 1  0,  0, G T1 R1 G T2 R2 ⎡ ⎤ Ξ11 Ξ12 Ξ13 T Ξ22 Ξ23 ⎦ < 0, Ξ := ⎣Ξ12 T T Ξ13 Ξ23 Ξ33

where ⎤ Φ1 G T1 R1 − G T1 −Q 2 − R1 R1 − G 1 ⎦ , = ⎣ G1 R1 − G 1 R1 − G T1 Φ2 ⎡

Ξ11

⎡

⎤ 0 0⎦ , 0

⎤ 0 0 ⎦, DN2

⎤ G T2 R2 − G T2 K Λ1 Φ3 ⎥ ⎢ G2 −Q 4 − R2 R2 − G 2 0 ⎥, =⎢ ⎦ ⎣ R2 − G 2 R2 − G T2 Φ4 0 0 0 Q 5 − 2Λ1 Λ1 K ⎡ ⎤ 0 0 −C N 2 ⎢ 0 0 0 ⎥ ⎥ Ξ23 = ⎢ ⎣ K Λ2 0 0 ⎦ , 0 0 0 ⎡

Ξ22

0 00 Ξ12 = ⎣ 0 0 0 D P2 0 0 ⎡ P1 W −AN 1 0 Ξ13 = ⎣ 0 0 0

(6.5)

(6.6)

122

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

⎡

Ξ33

⎤ Φ5 W T N1 0 ⎦, 0 = ⎣ N1 W τ¯ 2 R1 − 2N1 2 0 0 σ¯ R2 − 2N2

1 Φ1 = − π 2 P1 D L − 2P1 A + Q 1 + Q 2 − R1 , 2 Φ2 = (τ¯d − 1)Q 1 − 2R1 + G 1 + G T1 , 1 Φ3 = − π 2 P2 D L∗ − 2P2 C + Q 3 + Q 4 − R2 , 2 Φ4 = (σ¯ d − 1)Q 3 − 2R2 + G 2 + G T2 , Φ5 = (σ¯ d − 1)Q 5 − 2Λ2 , l l   1 1 ∗ ∗ D , D = D , k L 2 Lk L 2k k k=1 k=1

DL =

and L k , Dk , Dk∗ , A, C, W , D and K are the same with previous ones. Proof Choose an LKF candidate for GRN (6.3) as V (t, m, ˜ p) ˜ =

4 

Vi (t, m, ˜ p), ˜

i=1

where

 ˜ p) ˜ = V1 (t, m,

 m˜ (t, x)P1 m(t, ˜ x)dx + T

R

+

l   k=1

R

R

p˜ T (t, x)P2 p(t, ˜ x)dx

∂ m˜ T (t, x) ∂ m(t, ˜ x) N 1 Dk dx ∂xk ∂xk

l  

∂ p˜ T (t, x) ∂ p(t, ˜ x) N2 Dk∗ dx, ∂x ∂xk k k=1 R   t V2 (t, m, ˜ p) ˜ = m˜ T (s, x)Q 1 m(s, ˜ x)dsdx +

R

t−τ (t) t

 

+  +  +

m˜ T (s, x)Q 2 m(s, ˜ x)dsdx

R

t−τ¯  t

R

t−σ(t)  t

R

t−σ¯

p˜ T (s, x)Q 3 p(s, ˜ x)dsdx p˜ T (s, x)Q 4 p(s, ˜ x)dsdx,

6.2 Infinite-Time Case

123

  V3 (t, m, ˜ p) ˜ =

R

t

f T ( p(s, ˜ x))Q 5 f ( p(s, ˜ x))dsdx,

t−σ(t)

 



∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdθdx ∂s ∂s R −τ¯ t+θ   0 t ∂ p˜ T (s, x) ∂ p(s, ˜ x) R2 dsdθdx. +σ¯ ∂s ∂s R −σ¯ t+θ

˜ p) ˜ = τ¯ V4 (t, m,

0

t

˜ p) ˜ (i = 1, 2, 3, 4) along the solution of Then, computing the derivatives of Vi (t, m, GRN (6.3), we can get   l  ∂ ∂ m(t, ˜ x) Dk ∂xk ∂xk R k=1  −Am(t, ˜ x) + W f ( p(t ˜ − σ(t), x)) dx    l  ∂ ˜ x) T ∗ ∂ p(t, Dk p˜ (t, x)P2 +2 ∂xk ∂xk R k=1  −C p(t, ˜ x) + D m(t ˜ − τ (t), x) dx

∂ V1 (t, m, ˜ p) ˜ =2 ∂t



+2

m˜ T (t, x)P1

l   k=1

+2

l   k=1

∂ V2 (t, m, ˜ p) ˜ = ∂t

R

R

∂ m˜ T (t, x) ∂ ∂ m(t, ˜ x) dx N 1 Dk ∂xk ∂xk ∂t ∂ p˜ T (t, x) ∂ ∂ p(t, ˜ x) dx, N2 Dk∗ ∂xk ∂xk ∂t

(6.7)

 m˜ T (t, x)(Q 1 + Q 2 )m(t, ˜ x)dx R  − m˜ T (t − τ¯ , x)Q 2 m(t ˜ − τ¯ , x)dx R  −(1 − τ˙ (t)) m˜ T (t − τ (t), x)Q 1 m(t ˜ − τ (t), x)dx R  + p˜ T (t, x)(Q 3 + Q 4 ) p(t, ˜ x)dx R  − p˜ T (t − σ, ¯ x)Q 4 p(t ˜ − σ, ¯ x)dx R  −(1 − σ(t)) ˙ p˜ T (t − σ(t), x)Q 3 p(t ˜ − σ(t), x)dx, (6.8) R

124

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

 ∂ V3 (t, m, ˜ p) ˜ = −(1 − σ(t)) ˙ f T ( p(t ˜ − σ(t), x))Q 5 f ( p(t ˜ − σ(t), x))dx ∂t R  + f T ( p(t, ˜ x))Q 5 f ( p(t, ˜ x))dx, (6.9) R

 

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s R t−τ¯  ∂ m˜ T (t, x) ∂ m(t, ˜ x) 2 R1 dx +τ¯ ∂t ∂t R   t ∂ p˜ T (s, x) ∂ p(s, ˜ x) −σ¯ R2 dsdx ∂s ∂s R t−σ¯  ∂ p˜ T (t, x) ∂ p(t, ˜ x) R2 dx. +σ¯ 2 ∂t ∂t R

∂ ˜ p) ˜ = −τ¯ V4 (t, m, ∂t

t

(6.10)

From Green’s formula, Dirichlet boundary conditions and Lemma 1.13, we have l  

=

k=1

R

k=1

R

 ∂ m(t, ˜ x) dx ∂xk   ∂ ∂ m(t, ˜ x) m˜ T (t, x)P1 Dk dx ∂xk ∂xk

m˜ T (t, x)P1

l  

∂ ∂xk



Dk

l  

∂ m˜ T (t, x) ∂ m(t, ˜ x) P1 Dk dx ∂xk ∂xk k=1 R   l   ∂ m(t, ˜ x) l T m˜ (t, x)P1 Dk = · nds ∂xk k=1 k=1 ∂R l   ∂ m˜ T (t, x) ∂ m(t, ˜ x) − P1 Dk dx ∂x ∂xk k k=1 R l   ∂ m˜ T (t, x) ∂ m(t, ˜ x) =− P1 Dk dx ∂x ∂xk k k=1 R  π2 − m˜ T (t, x)P1 D L m(t, ˜ x)dx, 4 R −

where   ∂ m(t, ˜ x) l m˜ T (t, x)P1 Dk ∂xk k=1   ∂ m(t, ˜ x) ∂ m(t, ˜ x) T . = m˜ (t, x)P1 D1 , . . . , m˜ T (t, x)P1 Dl ∂x1 ∂xl

(6.11)

6.2 Infinite-Time Case

125

Similarly, l  

∂ p˜ (t, x)P2 ∂xk T

R

k=1

π2 − 4



R



∂ p(t, ˜ x) Dk∗ ∂xk

 dx

p˜ T (t, x)P2 D L∗ p(t, ˜ x)dx.

(6.12)

The combination of (6.7), (6.11) and (6.12) gives ∂ V1 (t, m, ˜ p) ˜ ∂t

 π2 T 2 m˜ (t, x)P1 − D L m(t, ˜ x) − Am(t, ˜ x) + W f ( p(t ˜ − σ(t), x)) dx 4 R

 π2 ∗ T p˜ (t, x)P2 − D L p(t, ˜ x) − C p(t, ˜ x) + D m(t ˜ − τ (t), x) dx +2 4 R l   ∂ m˜ T (t, x) ∂ ∂ m(t, ˜ x) +2 dx N 1 Dk ∂xk ∂xk ∂t k=1 R l   ∂ p˜ T (t, x) ∂ ∂ p(t, ˜ x) +2 dx. (6.13) N2 Dk∗ ∂x ∂x ∂t k k k=1 R From (6.5) and Lemmas 1.14 and 1.9, it follows that   −τ¯

R

t

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s

t−τ¯ t−τ (t)

 

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s R t−τ¯   t ∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx −τ¯ ∂s ∂s R t−τ (t)

 R1 G 1 − ζ1 (t)dx, ζ1T (t) T G 1 R1 R

= −τ¯

(6.14)

where ζ1 (t) = col(m(t ˜ − τ (t), x) − m(t ˜ − τ¯ , x), m(t, ˜ x) − m(t ˜ − τ (t), x)). Similarly,  

t ∂ p˜ T (s, x) ∂ p(s, ˜ x) R2 dsdx −σ¯ ∂s ∂s R t−σ¯

 R2 G 2 ζ2 (t)dx, − ζ2T (t) T G 2 R2 R

where ζ2 (t) = col( p(t ˜ − σ(t), x) − p(t ˜ − σ, ¯ x), p(t, ˜ x) − p(t ˜ − σ(t), x)).

(6.15)

126

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

For the diagonal matrices Λ1 > 0 and Λ2 > 0, we obtain from (6.4) that ˜ x))Λ1 f ( p(t, ˜ x)) − 2 p˜ T (t, x)K Λ1 f ( p(t, ˜ x))  0, 2 f T ( p(t, 2 f T ( p(t ˜ − σ(t), x))Λ2 f ( p(t ˜ − σ(t), x)) − 2 p˜ T (t − σ(t), x)K Λ2 f ( p(t ˜ − σ(t), x))  0.

(6.16)

(6.17)

For diagonal matrices N1 > 0 and N2 > 0, it is easy to see from (6.3) that  l   ∂  ∂ m(t, ∂ m˜ T (t, x) ˜ x) 2 Dk N1 ∂t ∂xk ∂xk R k=1

∂ m(t, ˜ x) dx = 0 − Am(t, ˜ x) + W f ( p(t ˜ − σ(t), x) − ∂t 

(6.18)

and  l   ∂  ∂ p(t, ∂ p˜ T (t, x) x) ∗ ˜ 2 Dk N2 ∂t ∂xk ∂xk R k=1

∂ p(t, ˜ x) − C p(t, ˜ x) + D m(t ˜ − τ (t), x) − dx = 0. ∂t 

(6.19)

According to Lemma 6.4, Green’s formula and Dirichlet boundary conditions, we have   l ∂ m˜ T (t, x)  ∂ ∂ m(t, ˜ x) Dk dx N1 ∂t ∂xk ∂xk R k=1  

 l  ∂ m(t, ˜ x) ∂ ∂ T = Dk dx m˜ (t, x)N1 ∂xk ∂xk ∂t R k=1   l   ∂ m(t, ˜ x) ∂ m˜ T (t, x) ∂ =− dx. N 1 Dk ∂xk ∂xk ∂t k=1 R 

(6.20)

Similarly,   l ∂ p˜ T (t, x)  ∂ ˜ x) ∗ ∂ p(t, Dk dx N2 ∂t ∂xk ∂xk R k=1   l   ∂ p(t, ˜ x) ∂ p˜ T (t, x) ∗ ∂ =− dx. N 2 Dk ∂xk ∂xk ∂t k=1 R 

(6.21)

6.2 Infinite-Time Case

127

Combining (6.8)–(6.10) and (6.13)–(6.21) results in  ∂ ∂ V (t, m, ˜ p) ˜ = Vi (t, m, ˜ p) ˜ ∂t ∂t i=1   ς T (t, x)Ξ ς(t, x)dx 4

R

 −λmin (−Ξ )( m(t, ˜ x) 2 + p(t, ˜ x) 2 ), where ς(t, x) = col(m(t, ˜ x), m(t ˜ − τ¯ , x), m(t ˜ − τ (t), x), p(t, ˜ x), p(t ˜ − σ, ¯ x) ˜ x) ∂ m(t, ˜ x) ∂ p(t, , ). p(t ˜ − σ(t), x), f ( p(t, ˜ x)), f ( p(t ˜ − σ(t), x)), ∂t ∂t Furthermore, ∂ ˜ x) 2 , V (t, m, ˜ p) ˜  −λmin (−Ξ ) m(t, ∂t ∂ V (t, m, ˜ p) ˜  −λmin (−Ξ ) p(t, ˜ x) 2 . ∂t Integrating two sides of the above inequalities from 0 to t, we have 

t

V (t, m, ˜ p) ˜  −λmin (−Ξ )

m(s, ˜ x) 2 ds + V (0, m(0, ˜ x), p(0, ˜ x)), (6.22a)

0



t

V (t, m, ˜ p) ˜  −λmin (−Ξ )

p(s, ˜ x) 2 ds + V (0, m(0, ˜ x), p(0, ˜ x)). (6.22b)

0

Therefore, 

t

λmin (−Ξ )

m(s, ˜ x) 2 ds  V (0, m(0, ˜ x), p(0, ˜ x)),

0



t

λmin (−Ξ )

p(s, ˜ x) 2 ds  V (0, m(0, ˜ x), p(0, ˜ x)),

0

which implies that m(t, ˜ x) 2 → 0 and p(t, ˜ x) 2 → 0 as t → ∞. Namely, m(t, ˜ x) → 0 and p(t, ˜ x) → 0 as t → ∞. From (6.22), we get V (t, m, ˜ p) ˜  V (0, m(0, ˜ x), p(0, ˜ x)).

(6.23)

128

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

Noting that φi (s, x), φi∗ (s, x) ∈ C 1 ([−d, 0] × R, R), namely, there exist nonnegative real numbers α, α∗ , β and β ∗ such that    ∂φi (s, x)    α,    ∂t

   ∂φi (s, x)     α∗ ,  ∂x  k

  ∗  ∂φi (s, x)    β,    ∂t

 ∗   ∂φi (s, x)     β∗.  ∂x  k

So, there exist nonnegative real numbers M1 and M2 , such that n   i=1

R

N1i

k=1 0  0

 

+τ¯

−τ¯

R

l 

θ

 Dik

∂ m˜ i (0, x) ∂xk

2 dx

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdθdx ∂s ∂s

 M1 φ(t, x) 2d and n   i=1

R

N2i

 

+σ¯

R

l 

k=1 0  0

−σ¯

θ

∗ Dik



∂ p˜ i (0, x) ∂xk

2 dx

∂ p˜ T (s, x) ∂ p(s, ˜ x) R2 dsdθdx ∂s ∂s

 M2 φ∗ (t, x) 2d . Therefore, V (0, m(0, ˜ x), p(0, ˜ x))   m˜ T (0, x)P1 m(0, ˜ x)dx + = R

p˜ T (0, x)P2 p(0, ˜ x)dx

 ∂ m˜ i (0, x) 2 dx ∂xk i=1 R k=1   n  l   ∂ p˜ i (0, x) 2 ∗ + N2i Dik dx ∂xk i=1 R k=1   0   + m˜ T (s, x)Q 1 m(s, ˜ x)dsdx + +

n  

 +

R

N1i

−τ (0)  0

R

−σ(0) 0

R

−σ(0)

  +

l 



R

Dik

R

  p˜ T (s, x)Q 3 p(s, ˜ x)dsdx +

R

f T ( p(s, ˜ x))Q 5 f ( p(s, ˜ x))dsdx

0 −τ¯ 0

−σ¯

m˜ T (s, x)Q 2 m(s, ˜ x)dsdx p˜ T (s, x)Q 4 p(s, ˜ x)dsdx

6.2 Infinite-Time Case

 



129

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdθdx ∂s ∂s R −τ¯ θ   0 0 ∂ p˜ T (s, x) ∂ p(s, ˜ x) +σ¯ R2 dsdθdx ∂s ∂s R −σ¯ θ

+τ¯

0

0

 λ11 φ(t, x) 2d + λ12 φ∗ (t, x) 2d ,

(6.24)

where λ11 = λmax (P1 ) + τ¯ λmax (Q 1 ) + τ¯ λmax (Q 2 ) + M1 , λ12 = λmax (P2 ) + σλ ¯ max (Q 3 ) + σλ ¯ max (Q 4 ) +σλ ¯ max (Q 5 )λmax (K T K ) + M2 . On the other hand, it is clear that ˜ x) 2 , V (t, m, ˜ p) ˜  λmin (P1,2 ) m(t,

(6.25)

V (t, m, ˜ p) ˜  λmin (P1,2 ) p(t, ˜ x) 2 ,

(6.26)

where λmin (P1,2 ) denotes the minimum eigenvalue of diag(P1 , P2 ). It follows from (6.23) to (6.26) that we have λ11 φ(t, x) 2d + λ12 φ∗ (t, x) 2d , λmin (P1,2 ) λ11 φ(t, x) 2d + λ12 φ∗ (t, x) 2d p(t, ˜ x) 2  . λmin (P1,2 )

m(t, ˜ x) 2 

(P1,2 ) ελmin (P1,2 ) For any ε > 0, there exists δ := min{ ελmin , 2λ12 }, such that 2λ11

m(t, ˜ x) 2  ε, p(t, ˜ x) 2  ε when φ(t, x) 2d  δ(ε), φ∗ (t, x) 2d  δ(ε). Hence, by Definition 6.2, the trivial solution of GRN (6.3) under Dirichlet boundary conditions is asymptotically stable. The proof is completed. ˜ p) ˜ are introRemark 6.6 It should be emphasized that the last two terms in V1 (t, m, duced to remove the restrictions τ¯d < 1 and σ¯ d < 1 required in [6, 8]. Remark 6.7 In order to establish a less conservative stability criterion for GRN (6.3), Lemmas 1.13 and 1.14 are used to estimate the integral terms

130

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

 R

m˜ T (t, x)P1

  l  ∂ ∂ m(t, ˜ x) Dk dx ∂xk ∂xk k=1

and  

t

R

t−τ¯

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx, ∂s ∂s

respectively. Remark 6.8 The so-called reciprocally convex technique (see [9]) is used to deal with the term   t ∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx, ∂s ∂s R t−τ¯ which reduces the conservativeness of the resulting stability criterion. The stability criterion for the trivial solution of GRN (6.3) under Neumann boundary conditions is given by the following theorem. Theorem 6.9 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (6.2), the trivial solution of GRN (6.3) under Neumann boundary conditions is asymptotically stable if there exist matrices Q iT = Q i > 0 (i ∈ 5) and R Tj = R j > 0 ( j = 1, 2), diagonal matrices P j > 0, Λ j > 0, N j > 0 ( j = 1, 2), and matrices G 1 and G 2 of appropriate sizes, such that



R1 G 1 R2 G 2  0,  0, (6.27) G T1 R1 G T2 R2 ⎡

11 Ξ T ⎣  Ξ := Ξ12 T Ξ13

⎤ Ξ12 Ξ13 22 Ξ23 ⎦ < 0, Ξ T Ξ23 Ξ33

where ⎡

⎤ 1 Φ G T1 R1 − G T1 11 = ⎣ G 1 −Q 2 − R1 R1 − G 1 ⎦ , Ξ R1 − G 1 R1 − G T1 Φ2 ⎤ 3 G T2 R2 − G T2 K Λ1 Φ ⎥ ⎢ −Q 4 − R2 R2 − G 2 0 ⎥, 22 = ⎢ G 2 Ξ T ⎦ ⎣ R2 − G 2 R2 − G 2 Φ4 0 0 0 Q 5 − 2Λ1 Λ1 K T ⎡

(6.28)

6.2 Infinite-Time Case

131

1 = −2P1 A + Q 1 + Q 2 − R1 , Φ 3 = −2P2 C + Q 3 + Q 4 − R2 , Φ and Ξ12 , Ξ13 , Ξ23 , Ξ33 , A, C and K are the same with ones in Theorem 6.5. Proof We take the LKF, V (t, m, ˜ p), ˜ in the proof of Theorem 6.5. From Green’s formula and Neumann boundary conditions, we have l  

=

k=1

R

k=1

R

 ∂ m(t, ˜ x) dx ∂xk   ∂ ∂ m(t, ˜ x) m˜ T (t, x)P1 Dk dx ∂xk ∂xk

m˜ T (t, x)P1

l  

∂ ∂xk



Dk

l  

∂ m˜ T (t, x) ∂ m(t, ˜ x) P1 Dk dx ∂xk ∂xk k=1 R   l   ∂ m(t, ˜ x) l T m˜ (t, x)P1 Dk = · n ds ∂xk k=1 k=1 ∂R l   ∂ m˜ T (t, x) ∂ m(t, ˜ x) − P1 Dk dx ∂x ∂xk k k=1 R l   ∂ m˜ T (t, x) ∂ m(t, ˜ x) =− P1 Dk dx ∂x ∂xk k k=1 R −

 0.

(6.29)

Similarly, 

  l  ∂ ˜ x) ∗ ∂ p(t, Dk dx  0. p˜ (t, x)P2 ∂xk ∂xk k=1 T

R

(6.30)

Based on (6.29) and (6.30), the proof follows in a manner similar to that of Theorem 6.5. Remark 6.10 The numbers of decision variables of the LMI conditions in [8, Theorem 2] and Theorem 6.9 are 4.5n 2 + 10.5n and 5.5n 2 + 9.5n, respectively, and hence the computational complexity of Theorem 6.9 is slightly greater than that of [8, Theorem 2]. However, the conservativeness of Theorem 6.9 is certainly less than [8, Theorem 2] (see Remark 6.18 below). Remark 6.11 Owing to Lemma 1.13, the information about the reaction-diffusion terms is remained in the stability criterion presented in Theorem 6.5. However,

132

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

because Lemma 1.13 is invalid under Neumann boundary conditions, the information about reaction-diffusion terms cannot be included into the stability criterion given in Theorem 6.9. Remark 6.12 The stability conditions in Theorems 6.5 and 6.9 are given in the form of LMIs. Thus, they can be easily verified by using the Toolbox YALMIP of MATLAB [7].

6.2.2 Theoretical Comparisons In this section, we further illustrate that the stability criterion given in Theorem 6.5 is less conservative than those in [6, Theorem 1] and [8, Theorem 1]. For this end, we first state [6, Theorem 1] and [8, Theorem 1] as follows. Theorem 6.13 [6, Theorem 1] For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying ¯ σ(t) ˙  σ¯ d < 1, 0  τ (t)  τ¯ , τ˙ (t)  τ¯d < 1, 0  σ(t)  σ,

(6.31)

the trivial solution of GRN (6.3) under Dirichlet boundary conditions is asymptotically stable if there exist positive definite matrices Q˜ iT = Q˜ i (i ∈ 3) and diagonal matrices P j > 0 and Λ j > 0 ( j = 1, 2), such that Ξ := diag(Ξ1 , Ξ2 ) < 0,

(6.32)

where D L and D L∗ are the same with previous ones, and ⎡

⎤ Φ¯ 1 0 P1 W Ξ1 = ⎣ 0 Φ¯ 2 K Λ2 ⎦ , T W P1 K Λ2 Φ¯ 3 ⎡ ⎤ 0 Φ¯ 4 D P 2 K Λ1 ⎦ , Ξ2 = ⎣ P2 D Φ¯ 5 ˜ 0 Λ1 K Q 3 − 2Λ1 1 Φ¯ 1 = − π 2 P1 D L − 2P1 A + Q˜ 1 , Φ¯ 2 = (σ¯ d − 1) Q˜ 2 , 2 Φ¯ 3 = (σ¯ d − 1) Q˜ 3 − 2Λ2 , Φ¯ 4 = (τ¯d − 1) Q˜ 1 , 1 Φ¯ 5 = − π 2 P2 D L∗ − 2P2 C + Q˜ 2 . 2 Theorem 6.14 [8, Theorem 1] For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (6.31), the trivial solution of GRN (6.3) is asymptotically stable under Dirichlet boundary

6.2 Infinite-Time Case

133

conditions if there exist matrices Q iT = Q i > 0 (i ∈ 5), RiT = Ri > 0, TiT = Ti > 0 (i = 1, 2), and diagonal matrices P j > 0, Λ j > 0 ( j = 1, 2), such that ⎡

⎤ χ1 0 0 P1 W T1 ⎢ 0 −Q 3 0 0 −T1 ⎥ ⎢ ⎥ 0 0 ( σ ¯ − 1)Q K Λ2 0 ⎥ Ξ1 := ⎢ d 2 ⎢ T ⎥ < 0, ⎣W P1 0 Λ2 K T χ3 0 ⎦ 0 0 − τ1¯ R1 T1T −T1T ⎡ ⎤ χ4 (P2 D)T 0 0 0 ⎢ D P 2 χ2 0 K Λ1 T2 ⎥ ⎢ ⎥ 0 0 −Q 0 −T2 ⎥ Ξ2 := ⎢ 4 ⎢ ⎥ < 0, ⎣ 0 Λ1 K 0 χ5 0 ⎦ 0 T2T −T2T 0 − σ1¯ R2

(6.33)

(6.34)

where D L and D L∗ are the same with previous ones, and χ1 = −2P1 D L − 2P1 A + Q 1 + Q 3 + τ¯ R1 , χ2 = −2P2 D L∗ − 2P2 C + Q 2 + Q 4 + σ¯ R2 , χ3 = (σ¯ d − 1)Q 5 − 2Λ2 , χ4 = (τ¯d − 1)Q 1 , χ5 = Q 5 − 2Λ1 . To show that the stability criterion given in Theorem 6.13 is less conservative than that in Theorem 6.14, Han and Zhang [6] investigated the following theorem. Theorem 6.15 [6, Theorem 3] If the LMIs (6.33) and (6.34) are feasible, then so is the LMI (6.32). Now we are in a position to state the following theorem, which claims that the stability criterion obtained in Theorem 6.5 is less conservative than that in Theorem 6.13, and hence also it is less conservative than that in Theorems 6.14 and 6.15. Theorem 6.16 If the LMI (6.32) is feasible, then so are the LMIs (6.5) and (6.6). Proof The LMI (6.32) is equivalent to Ξ :=

Ξ11 Ξ12 < 0, T Ξ12 Ξ22

where ⎡

Ξ11

⎡ ⎤ ⎤ Φ¯ 1 0 0 0 0 P1 W 0 ⎦, = ⎣ 0 Φ¯ 4 D P2 ⎦ , Ξ12 = ⎣0 0 0 K Λ1 0 0 P2 D Φ¯ 5

134

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

⎤ Φ¯ 2 0 K Λ2 = ⎣ 0 Q˜ 3 − 2Λ1 0 ⎦ . 0 Φ¯ 3 K Λ2 ⎡

Ξ22

Then there exist sufficiently small positive numbers εi (i = 1, 2, 5, 6) such that ⎡

Ξˆ 11 ⎢Ξˆ T ⎢ 12 ⎢ Ξˆ := ⎢ 0 ⎢ T ⎣Ξˆ 14 T Ξˆ 15

Ξˆ 12 Ξˆ 22 T Ξˆ 23

0 Ξˆ 23 Ξˆ 33 T 0 Ξˆ 34 T Ξˆ 25 0

Ξˆ 14 0 ˆ Ξ34 Φ¯ 3 T Ξˆ 45

⎤ Ξˆ 15 Ξˆ 25 ⎥ ⎥ ⎥ 0 ⎥ < 0, ⎥ Ξˆ 45 ⎦ Ξˆ 55

where ⎡

Ξˆ 11 = diag(Φ¯ 1 + ε1 I, −ε1 I, Φ¯ 4 ), Ξˆ 12 ⎡

⎡ ⎤ ⎤ 0 P1 W = ⎣ 0 ⎦ , Ξˆ 14 = ⎣ 0 ⎦ , D P2 0

⎤ −ε5 A 0   0 ⎦ , Ξˆ 22 = Φ¯ 5 + ε2 I, Ξˆ 23 = 0 0 K Λ1 , Ξˆ 15 = ⎣ 0 0 ε6 D   Ξˆ 25 = 0 −ε6 C , Ξˆ 33 = diag(−ε2 I, Φ¯ 2 , Q˜ 3 − 2Λ1 ), ⎡

Ξˆ 34

⎤

0   0 −2ε5 I T ⎣ ⎦ ˆ ˆ . = K Λ2 , Ξ45 = ε5 W 0 , Ξ55 = 0 −2ε6 I 0

Furthermore, there exist sufficiently small positive numbers εi (i = 3, 4) such that ⎡

Ξˆ 11 ⎢Ξˆ T ⎢ 12 ⎢ Ξˇ := ⎢ 0 ⎢ T ⎣Ξˆ 14 T Ξˆ 15

Ξˆ 12 0 Ξˆ 14 Ξˆ 22 Ξˆ 23 0 T ˆ Ξ33 Ξˆ 34 Ξˆ 23 T ¯ Φ3 0 Ξˆ 34 T T ˆ Ξ25 0 Ξˆ 45

⎤ Ξˆ 15 Ξˆ 25 ⎥ ⎥ ⎥ 0 ⎥ < 0, ⎥ Ξˆ 45 ⎦ Ξˇ 55

where Ξˇ 55 = diag(−2ε5 I + τ¯ 2 ε3 I, −2ε6 I + σ¯ 2 ε4 I ). Due to ⎡

⎤ −ε3 I 0 ε3 I ⎣ 0 −ε3 I ε3 I ⎦  0 ε3 I ε3 I −2ε3 I

(6.35)

6.2 Infinite-Time Case

135

and ⎡ ⎤ −ε4 I 0 ε4 I ⎣ 0 −ε4 I ε4 I ⎦  0, ε4 I ε4 I −2ε4 I we have from (6.35) that ⎤ Ξ¯ 11 Ξ¯ 12 Ξ¯ 13 T ¯ Ξ¯ := ⎣Ξ¯ 12 Ξ22 Ξ¯ 23 ⎦ < 0, T ¯T ¯ Ξ¯ 13 Ξ23 Ξ33 ⎡

(6.36)

where ⎡

⎤ −ε3 I 0 ε3 I Ξ¯ 11 = Ξˆ 11 + ⎣ 0 −ε3 I ε3 I ⎦ , ε3 I ε3 I −2ε3 I     Ξ¯ 12 = Ξˆ 12 0 , Ξ¯ 13 = Ξˆ 14 Ξˆ 15 , ⎡ ⎤ ε4 I 0 −ε4 I 0

⎢ 0 −ε4 I ε4 I 0⎥ Ξˆ Ξˆ 23 ⎥ +⎢ Ξ¯ 22 = ˆ 22 T ˆ ⎣ ε4 I ε4 I −2ε4 I 0⎦ , Ξ23 Ξ33 0 0 0 0



Φ¯ 3 Ξˆ 45 0 Ξˆ 25 ¯ ¯ , Ξ33 = ˆ T ˇ Ξ23 = ˆ . Ξ34 0 Ξ45 Ξ55 Set G 1 = 0, G 2 = 0, Q 2 = ε1 I , Q 4 = ε2 I , N1 = ε5 I , N2 = ε6 I , R1 = ε3 I , R2 = ε4 I , Q 1 = Q˜ 1 , Q 3 = Q˜ 2 and Q 5 = Q˜ 3 . Then the LMI (6.36) becomes (6.6), and hence the LMIs (6.5) and (6.6) are feasible. This completes the proof. Remark 6.17 It follows from Theorems 6.15 and 6.16 that Theorem 6.5 is less conservative than those in [6, 8]. This will be illustrated by a numerical example in the next subsection. Remark 6.18 As the previous discussion, one can easily show that the stability criterion presented in Theorem 6.9 is less conservative than that given in [8, Theorem 2].

6.2.3 Numerical Examples To demonstrate the effectiveness of the results obtained in this section, we give two numerical examples in this subsection.

136

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

Example 6.19 Consider the delayed reaction-diffusion GRN (6.3) with the following parameters: A = diag(7, 7), D = diag(3, 3), C = diag(6, 6), L 1 = L 2 = 1, K = diag(0.65, 0.65),

0 −5.5 W = , −5.5 0 D1 = D2 = diag(0.3, 0.3), D1∗ = D2∗ = diag(0.6, 0.6). Case 1: Dirichlet boundary conditions. In Table 6.1, when τ¯d =σ¯ d ∈ {0.77, 0.78, 0.80, 0.81, 0.82, 0.93, 1}, we list the maximum delay upper bounds obtained by applying Theorems 6.14, 6.13 and 6.5 to the considered GRN. From Table 6.1, it is clear that: (i) when τ¯d = σ¯ d  0.82, the LMI conditions presented in Theorems 6.14 and 6.13 are infeasible, while the LMI conditions in Theorem 6.5 are feasible; and (ii) when τ¯d = σ¯ d = 0.77, all LMI conditions discussed here are feasible, and all delay upper bounds from Theorems 6.14, 6.13 and 6.5 are larger than 104 . This is because the obtained stability conditions in Theorems 6.13 and 6.14 are in essence delay-independent, although τ¯ and σ¯ appear in the stability conditions in Theorem 6.14. In addition, we conclude from Theorems 6.15 and 6.16 that the maximum values obtained from Theorem 6.5 is certainly not smaller than those obtained from Theorems 6.13 and 6.14. Therefore, Theorem 6.5 reduces the conservatism of the stability criteria in Theorems 6.14 and 6.13. This illustrates the theoretical results presented previously. When τ¯ = σ¯ = 0.2 and τ¯d = σ¯ d = 1, by using the Toolbox YALMIP in MATLAB to solve the LMIs (6.5) and (6.6), we obtain the following feasible solution:



1.4820 0.0997 1.7562 0.0279 Q1 = , Q2 = , 0.0997 1.4820 0.0279 1.7562





1.8577 0.0886 1.9021 0.0279 1.5748 0.0046 , Q4 = , Q5 = , Q3 = 0.0886 1.8577 0.0279 1.9021 0.0046 1.5748



1.3523 −0.0136 1.3602 −0.0243 R1 = , R2 = , −0.0136 1.3523 −0.0243 1.3602

Table 6.1 Maximum values of τ¯ = σ¯ with different τ¯d = σ¯ d = μ (Example 6.19) μ 0.77 0.78 0.80 0.81 0.82 0.93 Theorem 6.14 Theorem 6.13 Theorem 6.5

>104 >104 >104

– >104 >104

– >104 >104

– >104 >104

– – 1.5457

– – 0.5654

1 – – 0.5496

6.2 Infinite-Time Case

137





0.0617 0.0226 0.1009 0.0427 G1 = , G2 = , 0.0226 0.0617 0.0427 0.1009 P1 = diag(0.4925, 0.4925), P2 = diag(0.6352, 0.6352), Λ1 = diag(1.8294, 1.8294), Λ2 = diag(1.8770, 1.8770), N1 = diag(0.0998, 0.0998), N2 = diag(0.1582, 0.1582). Case 2: Neumann boundary conditions. when τ¯ = σ¯ = 0.1 and τ¯d = σ¯ d = 1, by using the Toolbox YALMIP in MATLAB to solve the LMIs (6.27) and (6.28),we obtain the following feasible solution:



0.3833 0.0208 0.4405 0.0130 , Q2 = , 0.0208 0.3833 0.0130 0.4405





0.4437 0.0429 0.4691 0.0221 0.4483 −0.0048 Q3 = , Q4 = , Q5 = , 0.0429 0.4437 0.0221 0.4691 −0.0048 0.4483



0.4558 0.0027 0.4396 −0.0014 R1 = , R2 = , 0.0027 0.4558 −0.0014 0.4396



0.0787 −0.0018 0.0986 0.0062 G1 = , G2 = , −0.0018 0.0787 0.0062 0.0986 Q1 =

P1 = diag(0.1228, 0.1228), P2 = diag(0.1727, 0.1727), Λ1 = diag(0.5206, 0.5206), Λ2 = diag(0.4982, 0.4982), N1 = diag(0.0254, 0.0254), N2 = diag(0.0388, 0.0388). However, the LMI conditions presented in [8, Theorem 2] are infeasible. Therefore, the stability criterion obtained in Theorem 6.9 is less conservative than that in [8, Theorem 2]. On the other hand, the numbers of decision variables in [8, Theorem 2] and Theorem 6.9 are 4.5n 2 + 8.5n and 5.5n 2 + 9.5n, respectively. Hence the computational complexity of Theorem 6.9 is slightly greater than that of [8, Theorem 2]. To further demonstrate the effectiveness of our results, we now give another example. Example 6.20 When l = n = 1, GRN (6.3) simplifies to ∂ m(t, ˜ x) = ∂t

∂ ∂x



 ˜ D1 ∂ m(t,x) − Am(t, ˜ x) + W f ( p(t ˜ − σ(t), x)), ∂x

(6.37a)

138

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

∂ p(t, ˜ x) = ∂t

∂ ∂x



 ˜ D1∗ ∂ p(t,x) − C p(t, ˜ x) + D m(t ˜ − τ (t), x). ∂x

(6.37b)

We choose the values of parameters in (6.37) as follows, A = 0.3, D = 1.5, C = 0.2, L 1 = 1, W = −0.5, D1 = 0.2, D1∗ = 0.1. When τ¯d = 3, σ¯ d = 2, K = 0.65, τ¯ = 1 and σ¯ = 1.2, for Dirichlet boundary conditions, by using the Toolbox YALMIP in MATLAB to solve the LMIs (6.5) and (6.6), we obtain a feasible solution as follows: Q 1 = 1.9205, Q 2 = 2.2344 × 103 , Q 3 = 0.3215, Q 4 = 204.5217, Q 5 = 1.0840, R1 = 2.4531 × 103 , R2 = 516.4443, G 1 = −1.9857 × 103 , G 2 = −313.5628, P1 = 3.3111 × 103 , P2 = 704.6399, Λ1 = 7.3551, Λ2 = 954.1017, N1 = 2.4485 × 103 , N2 = 744.0726. However, the LMI conditions presented in Theorems 6.14 and 6.13 are infeasible. Further, when σ(t) ≡ 1.2 or 6, and τ (t) ≡ 1, the state responses of GRN (6.37) are given in Figs. 6.1 and 6.2. It is seen from Figs. 6.1 and 6.2 that GRN (6.37) is stable when σ(t) ≡ 1.2 and τ (t) ≡ 1, and is instable when σ(t) ≡ 6 and τ (t) ≡ 1. Thus, larger delays may lead to instability of GRNs.

0.2

0.6

0.1

0.4

0

0.2

-0.1

0

-0.2 20

-0.2

15

10

5

0 0

0.5

1

1.5

2

20

15

10

5

0 0

0.5

1

1.5

2

Fig. 6.1 Trajectories of mRNA and protein concentrations when σ(t) ≡ 1.2 and τ (t) ≡ 1 (Example 6.20)

6.3 Finite-Time Case

139

0.2

0.6 0.4

0.1

0.2

0

0

-0.1 -0.2 20

-0.2 15

10

5

1

0.5

0 0

1.5

2

-0.4 20

15

10

5

0 0

0.5

1

1.5

2

Fig. 6.2 Trajectories of mRNA and protein concentrations when σ(t) ≡ 6 and τ (t) ≡ 1 (Example 6.20)

6.3 Finite-Time Case In this section we study the problem of finite-time stability analysis for the delayed reaction-diffusion GRN (6.3). Sufficient conditions ensuring the finite-time stability are presented by using the LKF method. An important feature of the results reported here is that all the conditions are diffusion-dependent as well as delay-dependent. In order to illustrate the effectiveness of the obtained results, a numerical example is given. It should be emphasized that the LKF method proposed in this section is essentially different from the existing ones, because: (i) the new quad-slope integration like  

0



R

−τ¯

0 s

 α

0



t t+θ

∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudθdαdsdx ∂u ∂u

is introduced into the LKF, which is different from previous results; and (ii) the Wirtinger-type integral inequality, Gronwall’s inequality, convex technique, and reciprocally convex technique are employed to estimate the derivative of the LKF.

6.3.1 Finite-Time Stability Criteria Define ei = col(0(i−1)n×n , In , 0(18−i)n×n ), i ∈ 18, Δ1 = [e3 − e2 e3 + e2 − 2e11 e3 − e2 + 6e11 − 12e12 ], Δ2 = [e1 − e3 e1 + e3 − 2e13 e1 − e3 + 6e13 − 12e14 ],

140

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

Δ3 = [e6 − e5 e6 + e5 − 2e15 e6 − e5 + 6e15 − 12e16 ], Δ4 = [e4 − e6 e4 + e6 − 2e17 e4 − e6 + 6e17 − 12e18 ], Θ1 = [e1 − e13 e1 + 2e13 − 6e14 ], Θ2 = [e3 − e11 e3 + 2e11 − 6e12 ], Θ3 = [e4 − e17 e4 + 2e17 − 6e18 ], Θ4 = [e6 − e15 e6 + 2e15 − 6e16 ], ς(t, x) = col(m(t, ˜ x), m(t ˜ − τ¯ , x), m(t ˜ − τ (t), x), p(t, ˜ x), p(t ˜ − σ, ¯ x), p(t ˜ − σ(t), x), f ( p(t, ˜ x)), ˜ x) ∂ m(t, ˜ x) ∂ p(t, , , f ( p(t ˜ − σ(t), x)), ∂t ∂t  t−τ (t) 1 m(s, ˜ x)ds, τ¯ − τ (t) t−τ¯  t−τ (t)  t−τ (t) 1 m(s, ˜ x)dsdα, (τ¯ − τ (t))2 t−τ¯ α  t  t  t 1 1 m(s, ˜ x)ds, 2 m(s, ˜ x)dsdα, τ (t) t−τ (t) τ (t) t−τ (t) α  t−σ(t) 1 p(s, ˜ x)ds, σ¯ − σ(t) t−σ¯  t−σ(t)  t−σ(t) 1 p(s, ˜ x)dsdα, (σ¯ − σ(t))2 t−σ¯ α  t  t  t 1 1 p(s, ˜ x)ds, 2 p(s, ˜ x)dsdα). σ(t) t−σ(t) σ (t) t−σ(t) α Theorem 6.21 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (6.2), and positive constants α, c1 , c2 and T , the trivial solution of GRN (6.3) under Dirichlet boundary conditions is finite-time stable if there exist matrices Q iT = Q i > 0 (i ∈ 5), R Tj = R j > 0, W jT = W j > 0 and H jT = H j > 0, diagonal matrices P j > 0, Λ j > 0, N j > 0 ( j = 1, 2) and matrices Gˆ 1 and Gˆ 2 of appropriate sizes, such that the following inequalities hold for τ ∈ {0, τ¯ } and σ ∈ {0, σ}: ¯



Rˆ 1 Gˆ 1 Rˆ 2 Gˆ 2  0,  0, Gˆ T1 Rˆ 1 Gˆ T2 Rˆ 2

(6.38)

6.3 Finite-Time Case

141

Ξ (τ , σ) :=

4 

Ξi +

i=0

6 

Ξi (τ , σ) − αe1 P1 e1T − αe4 P2 e4T < 0,

c1 eαT (λ11 + λ12 ) − c2 λmin (P)  0, where     π2 π2 Ξ1 = 2e1 − P1 D L − P1 A e1T + 2e4 − P2 D L∗ − P2 C e4T 4 4 T −2e9 N1 e9T − 2e10 N2 e10 + sym(−e9 N1 Ae1T + e9 N1 W e8T +e4 P2 De3T − e10 N2 Ce4T + e10 N2 De3T + e1 P1 W e8T ),

Ξ2 = e1 (Q 1 + Q 2 )e1T − e2 Q 2 e2T + (τ¯d − 1)e3 Q 1 e3T +e4 (Q 3 + Q 4 )e4T − e5 Q 4 e5T + (σ¯ d − 1)e6 Q 3 e6T , Ξ3 = (σ¯ d − 1)e8 Q 5 e8T + e7 Q 5 e7T , T Ξ4 = τ¯ 2 e9 R1 e9T + σ¯ 2 e10 R2 e10

Rˆ Gˆ −[Δ1 Δ2 ] ˆ 1T ˆ 1 [Δ1 Δ2 ]T G 1 R1

Rˆ 2 Gˆ 2 −[Δ3 Δ4 ] ˆ T ˆ [Δ3 Δ4 ]T , G 2 R2

Ξ5 (τ , σ) = Ξ50 + Ξ51 (τ ) + Ξ52 (σ), Ξ50 =

τ¯ 2 σ¯ 2 T e9 W1 e9T + e10 W2 e10 , 2 2

Ξ51 (τ ) = −Θ1 W˜ 1 Θ1T − Θ2 W˜ 1 Θ2T −

τ¯ − τ Δ2 Wˆ 1 ΔT2 , τ¯

Ξ52 (σ) = −Θ3 W˜ 2 Θ3T − Θ4 W˜ 2 Θ4T −

σ¯ − σ Δ4 Wˆ 2 ΔT4 , σ¯

Ξ6 (τ , σ) = Ξ60 + Ξ61 (τ ) + Ξ62 (σ) −(τ¯ − τ )Θ1 H˜ 1 Θ1T − (σ¯ − σ)Θ3 H˜ 2 Θ3T , Ξ60 =

(6.39)

i=5

τ¯ 3 σ¯ 3 T e9 H1 e9T + e10 H2 e10 , 6 6

(6.40)

142

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

3 Ξ61 (τ ) = − τ (e1 − 2e14 )H1 (e1 − 2e14 )T 2 3 − (τ¯ − τ )(e3 − 2e12 )H1 (e3 − 2e12 )T , 2 3 Ξ62 (σ) = − σ(e4 − 2e18 )H2 (e4 − 2e18 )T 2 3 − (σ¯ − σ)(e6 − 2e16 )H2 (e6 − 2e16 )T , 2 Ξ0 = −2e7 Λ1 e7T + e4 K Λ1 e7T + e7 Λ1 K e4T −2e8 Λ2 e8T + e6 K Λ2 e8T + e8 Λ2 K e6T , W˜ i = diag(2Wi , 4Wi ), H˜ i = diag(2Hi , 4Hi ), i = 1, 2, Wˆ i = diag(Wi , 3Wi , 5Wi ), i = 1, 2, Rˆ i = diag(Ri , 3Ri , 5Ri ), i = 1, 2, P = diag(P1 , P2 ), λ11 = λmax (P1 ) + τ¯ λmax (Q 1 ) + τ¯ λmax (Q 2 ) l  1 + τ¯ 3 λmax (W1 ) + λmax (N1 )λmax (Dk ) 6 k=1 1 1 + τ¯ 3 λmax (R1 ) + τ¯ 4 λmax (H1 ), 2 24 λ12 = λmax (P2 ) + σλ ¯ max (Q 3 ) + σλ ¯ max (Q 4 ) 1 +σλ ¯ max (Q 5 )λmax (K T K ) + σ¯ 3 λmax (W2 ) 6 l  1 1 + λmax (N2 )λmax (Dk∗ ) + σ¯ 3 λmax (R2 ) + σ¯ 4 λmax (H2 ), 2 24 k=1 where A, B, C, W , D L , D L∗ and K are the same with previous ones. Proof Construct an LKF for GRN (6.3) as follows: V (t, m, ˜ p) ˜ =

6  i=1

Vi (t, m, ˜ p), ˜

6.3 Finite-Time Case

143

where  V1 (t, m, ˜ p) ˜ =

 m˜ (t, x)P1 m(t, ˜ x)dx + T

R

+

l   k=1

+

p˜ T (t, x)P2 p(t, ˜ x)dx

R

∂ m˜ T (t, x) ∂ m(t, ˜ x) N 1 Dk dx ∂xk ∂xk

R

∂ p˜ T (t, x) ∂ p(t, ˜ x) N2 Dk∗ dx, ∂xk ∂xk

l   k=1

R

 

V2 (t, m, ˜ p) ˜ =

t

t−τ (t) t

R

m˜ T (s, x)Q 1 m(s, ˜ x)dsdx

 

+  +  +   V3 (t, m, ˜ p) ˜ =

R

m˜ T (s, x)Q 2 m(s, ˜ x)dsdx

R

t−τ¯  t

R

t−σ(t)  t

R

t−σ¯

t

p˜ T (s, x)Q 3 p(s, ˜ x)dsdx p˜ T (s, x)Q 4 p(s, ˜ x)dsdx,

f T ( p(s, ˜ x))Q 5 f ( p(s, ˜ x))dsdx,

t−σ(t)

 



∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdθdx ∂s ∂s R −τ¯ t+θ   0 t ∂ p˜ T (s, x) ∂ p(s, ˜ x) +σ¯ R2 dsdθdx, ∂s ∂s R −σ¯ t+θ   0  0 t ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudθdsdx ˜ p) ˜ = V5 (t, m, ∂u ∂u t+θ R −τ¯ s   0  0 t ∂ p˜ T (u, x) ∂ p(u, ˜ x) W2 dudθdsdx, + ∂u ∂u t+θ R −σ¯ s   0  0 0 t ∂ m˜ T (u, x) ∂ m(u, ˜ x) V6 (t, m, H1 dudθdαdsdx ˜ p) ˜ = ∂u ∂u α t+θ R −τ¯ s   0  0 0 t ∂ p˜ T (u, x) ∂ p(u, ˜ x) H2 dudθdαdsdx. + ∂u ∂u R −σ¯ s α t+θ V4 (t, m, ˜ p) ˜ = τ¯

0

t

Then, calculating the derivatives of Vi (t, m, ˜ p) ˜ (i ∈ 6) along the solution of GRN (6.3), we have ∂ ˜ p) ˜  Vk (t, m, ∂t

 R

ς T (t, x)Ξk ς(t, x)dx, k ∈ 3,

(6.41)

144

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

(These inequalities can be obtained from the proof of Theorem 6.5.)  

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s R t−τ¯  ∂ m˜ T (t, x) ∂ m(t, ˜ x) R1 dx +τ¯ 2 ∂t ∂t R   t ∂ p˜ T (s, x) ∂ p(s, ˜ x) −σ¯ R2 dsdx ∂s ∂s R t−σ¯  ∂ p˜ T (t, x) ∂ p(t, ˜ x) R2 dx, (6.42) +σ¯ 2 ∂t ∂t R  ∂ ∂ m˜ T (t, x) ∂ m(t, ˜ x) τ¯ 2 V5 (t, m, W1 dx ˜ p) ˜ = ∂t 2 R ∂t ∂t   0 t ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx − ∂u ∂u R −τ¯ t+s  σ¯ 2 ∂ p˜ T (t, x) ∂ p(t, ˜ x) + W2 dx 2 R ∂t ∂t   0 t ∂ p˜ T (u, x) ∂ p(u, ˜ x) − W2 dudsdx, (6.43) ∂u ∂u R −σ¯ t+s  ∂ ∂ m˜ T (t, x) ∂ m(t, ˜ x) τ¯ 3 V6 (t, m, H1 dx ˜ p) ˜ = ∂t 6 R ∂t ∂t   0  0 t ∂ m˜ T (u, x) ∂ m(u, ˜ x) − H1 dudsdαdx ∂u ∂u t+s R −τ¯ α  σ¯ 3 ∂ p˜ T (t, x) ∂ p(t, ˜ x) + H2 dx 6 R ∂t ∂t   0  0 t ∂ p˜ T (u, x) ∂ p(u, ˜ x) − H2 dudsdαdx. (6.44) ∂u ∂u R −σ¯ α t+s ∂ ˜ p) ˜ = −τ¯ V4 (t, m, ∂t

t

From (6.38) and Lemmas 1.9 and 1.16, it follows that   −τ¯

R

 

t

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s

t−τ¯ t−τ (t)

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx ∂s ∂s R t−τ¯   t ∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdx −τ¯ ∂s ∂s R t−τ (t)

 Rˆ Gˆ − ς T (t, x)[Δ1 Δ2 ] ˆ 1T ˆ 1 [Δ1 Δ2 ]T ς(t, x)dx. G 1 R1 R

= −τ¯

(6.45)

6.3 Finite-Time Case

145

Similarly,  

∂ p˜ T (s, x) ∂ p(s, ˜ x) R2 dsdx ∂s ∂s R t−σ¯

 Rˆ Gˆ ς T (t, x)[Δ3 Δ4 ] ˆ 2T ˆ 2 [Δ3 Δ4 ]T ς(t, x)dx. − G 2 R2 R t

−σ¯

(6.46)

The combination of (6.42), (6.45) and (6.46) gives ∂ V4 (t, m, ˜ p) ˜  ∂t

 R

ς T (t, x)Ξ4 ς(t, x)dx.

(6.47)

The second term on the right of (6.43) can be divided into three parts:  



∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx ∂u ∂u R −τ¯ t+s   0  t ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx =− ∂u ∂u R −τ (t) t+s   −τ (t)  t−τ (t) ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx − ∂u ∂u t+s R −τ¯   t ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudx. −(τ¯ − τ (t)) ∂u ∂u R t−τ (t) 0

−

t

By Lemma 1.16, we can give the following estimation:   −  −

  − −

R

−τ (t)

R

R



0

t t+s

∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx ∂u ∂u

ς T (t, x)Θ1 W˜ 1 Θ1T ς(t, x)dx, −τ (t)

−τ¯



t−τ (t) t+s

∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx ∂u ∂u

ς (t, x)Θ2 W˜ 1 Θ2T ς(t, x)dx, T

R

  t ∂ m˜ T (u, x) ∂ m(u, ˜ x) −(τ¯ − τ (t)) W1 dudx ∂u ∂u R t−τ (t)  τ¯ − τ (t) − ς T (t, x)Δ2 Wˆ 1 ΔT2 ς(t, x)dx. τ¯ R

(6.48)

146

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

This, together with (6.48), implies that   −  



−τ¯

R

R

0

t

t+s

∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx ∂u ∂u

ς T (t, x)Ξ51 (τ (t))ς(t, x)dx.

(6.49)

Similarly, the last term on the right of (6.43) satisfies   −  

R

0



−σ¯

t t+s

∂ p˜ T (u, x) ∂ p(u, ˜ x) W2 dudsdx ∂u ∂u

ς (t, x)Ξ52 (σ(t))ς(t, x)dx. T

R

(6.50)

The combination of (6.43), (6.49) and (6.50) implies that ∂ V5 (t, m, ˜ p) ˜  ∂t

 R

ς T (t, x)Ξ5 (τ (t), σ(t))ς(t, x)dx.

(6.51)

The second term on the right of (6.44) can be written as:  





∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdαdx ∂u ∂u t+s R −τ¯ α   0  0 t ∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdαdx =− ∂u ∂u R −τ (t) α t+s   −τ (t)  −τ (t)  t−τ (t) ∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdαdx − ∂u ∂u R −τ¯ α t+s   0  t ∂ m˜ T (u, x) ∂ m(u, ˜ x) −(τ¯ − τ (t)) H1 dudsdx ∂u ∂u R −τ (t) t+s   t ∂ m˜ T (u, x) ∂ m(u, ˜ x) (τ¯ − τ (t))2 H1 dudx. (6.52) − 2 ∂u ∂u R t−τ (t) −

0

0

t

By Lemma 1.14, we have  





∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdαdx ∂u ∂u R −τ (t) α t+s   −τ (t)  −τ (t)  t−τ (t) ∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdαdx − ∂u ∂u α t+s  R −τ¯  ς T (t, x)Ξ61 (τ (t))ς(t, x)dx; −

R

0

0

t

(6.53)

6.3 Finite-Time Case

147

and by Lemma 1.16, we derive   −(τ¯ − τ (t))

−τ (t)

R

  −(τ¯ − τ (t))

R



0

t t+s

∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudsdx ∂u ∂u

ς T (t, x)Θ1 H˜ 1 Θ1T ς(t, x)dx.

(6.54)

Similarly,  





∂ p˜ T (u, x) ∂ p(u, ˜ x) H2 dudsdαdx ∂u ∂u t+s R −σ(t) α   −σ(t)  −σ(t)  t−σ(t) ∂ p˜ T (u, x) ∂ p(u, ˜ x) − H2 dudsdαdx ∂u ∂u α t+s  R −σ¯  ς T (t, x)Ξ62 (σ(t))ς(t, x)dx, −

0

0

t

(6.55)

R

  −(σ¯ − σ(t))

R

  −(σ¯ − σ(t))



0 −σ(t)

t

t+s

∂ p˜ T (u, x) ∂ p(u, ˜ x) H2 dudsdx ∂u ∂u

ς (t, x)Θ3 H˜ 2 Θ3T ς(t, x)dx. T

R

(6.56)

The combination of (6.44) and (6.52)–(6.56), implies that ∂ V6 (t, m, ˜ p) ˜  ∂t

 R

ς T (t, x)Ξ6 (τ (t), σ(t))ς(t, x)dx.

(6.57)

For given diagonal matrices Λ1 > 0 and Λ2 > 0, one can derive from (6.4) that: ς T (t, x)Ξ0 ς(t, x)  0. The combination of (6.41), (6.47), (6.51), (6.57) and (6.58) results in  ∂ ∂ V (t, m, ˜ p) ˜ = Vi (t, m, ˜ p) ˜ ∂t ∂t i=1   ς T (t, x)Ξ (τ (t), σ(t))ς(t, x)dx R  +α [m˜ T (t, x)P1 m(t, ˜ x) + p˜ T (t, x)P2 p(t, ˜ x)]dx R   ς T (t, x)Ξ (τ (t), σ(t))ς(t, x)dx + αV (t, m, ˜ p). ˜ 6

R

(6.58)

148

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

Since Ξ (τ (t), σ(t)) depends affinely on τ (t) and σ(t), one can derive that Ξ (τ (t), σ(t)) < 0, ∀0  τ (t)  τ¯ , 0  σ(t)  σ¯ if and only if Ξ (τ , σ) < 0, ∀τ ∈ {0, τ¯ }, σ ∈ {0, σ}. ¯ So, by (6.39), we have ∂ V (t, m, ˜ p) ˜  αV (t, m, ˜ p). ˜ ∂t

(6.59)

Integrating two sides of the inequality (6.59) from 0 to t, t ∈ [0, T ], we have 

t

V (t, m, ˜ p) ˜  V (0, m(0, ˜ x), p(0, ˜ x)) +

αV (s, m, ˜ p)ds. ˜

0

Using Lemma 1.12, we get ˜ x), p(0, ˜ x)). V (T, m, ˜ p) ˜  eαT V (0, m(0, This, together with V (0, m(0, ˜ x), p(0, ˜ x))   T m˜ (0, x)P1 m(0, ˜ x)dx + = R

+

l  

R

p˜ T (0, x)P2 p(0, ˜ x)dx

l   ∂ m˜ (0, x) ∂ m(0, ˜ x) ∂ p˜ T (0, x) ∂ p(0, ˜ x) N 1 Dk dx + N2 Dk∗ dx ∂x ∂x ∂x ∂xk k k k k=1 R k=1 R   0   0 + m˜ T (s, x)Q 1 m(s, ˜ x)dsdx + m˜ T (s, x)Q 2 m(s, ˜ x)dsdx T

R

−τ (0) 0

+

R

−σ(0)  0

 +

R

p˜ T (s, x)Q 3 p(s, ˜ x)dsdx +

−τ¯ 0

R

−σ¯

p˜ T (s, x)Q 4 p(s, ˜ x)dsdx

f T ( p(s, ˜ x))Q 5 f ( p(s, ˜ x))dsdx

−σ(0) 0  0

 

 

R

 

∂ m˜ T (s, x) ∂ m(s, ˜ x) R1 dsdθdx ∂s ∂s R −τ¯ θ   0 0 ∂ p˜ T (s, x) ∂ p(s, ˜ x) R2 dsdθdx +σ¯ ∂s ∂s R −σ¯ θ   0  0 0 ∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudθdsdx + ∂u ∂u R −τ¯ s θ

+τ¯

6.3 Finite-Time Case

 



149



∂ p˜ T (u, x) ∂ p(u, ˜ x) W2 dudθdsdx ∂u ∂u θ R −σ¯ s   0  0 0 0 ∂ m˜ T (u, x) ∂ m(u, ˜ x) + H1 dudθdαdsdx ∂u ∂u α θ R −τ¯ s   0  0 0 0 ∂ p˜ T (u, x) ∂ p(u, ˜ x) H2 dudθdαdsdx + ∂u ∂u α θ R −σ¯ s

+

0

0

0

 λ11 φ(t, x) 2d + λ12 φ∗ (t, x) 2d  (λ11 + λ12 )( φ(t, x) 2d + φ∗ (t, x) 2d ), implies that V (t, m, ˜ p) ˜  eαT (λ11 + λ12 )( φ(t, x) 2d + φ∗ (t, x) 2d ).

(6.60)

Note that ˜ x) 22 + λmin (P2 ) p(t, ˜ x) 22 V (t, m, ˜ p) ˜  λmin (P1 ) m(t,  λmin (P)( m(t, ˜ x) 22 + p(t, ˜ x) 22 ), t ∈ [0, T ]. where λmin (P) is the minimum eigenvalue of diag(P1 , P2 ). Now, from (6.60), it is easy to see that m(t, ˜ x) 22 + p(t, ˜ x) 22 

eαT (λ11 + λ12 )( φ(t, x) 2d + φ∗ (t, x) 2d ) . λmin (P)

Hence, by Definition 6.3 and (6.40), it is easy to see that the trivial solution of GRN (6.3) under Dirichlet boundary conditions is finite-time stable. The proof is completed. Remark 6.22 Compared with Theorem 6.5, in order to reduce conservativeness of the resulted finite-time stability criterion, we employ Lemma 1.16 instead of Jensen’s inequalities to estimate the integrals like   R

t

t−τ (t)

∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudx ∂u ∂u

and   R

0

−τ¯



t

t+s

∂ m˜ T (u, x) ∂ m(u, ˜ x) W1 dudsdx. ∂u ∂u

150

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

Remark 6.23 We introduce new integral items like  

0

R

−τ¯



0 s

 α

0



t t+θ

∂ m˜ T (u, x) ∂ m(u, ˜ x) H1 dudθdαdsdx ∂u ∂u

into LKF, which will results in less conservative finite-time stability criteria. Remark 6.24 Compare the method proposed in this section with one in [10], there are the following disparities: (i) The LKF employed in this section utilizes more information of GRNs; (ii) The so-called convex technique and reciprocally convex technique are used simultaneously; (iii) some effective integral inequalities, including Wirtinger-type inequalities and Gronwall’s inequality are employed. This will yield less conservative finite-time stability criteria. Specially, the finite-time stability criteria obtained in this section remove the condition, required in [10], that the upper bounds of delays’ derivatives are restricted to be less than 1. Noting that the inequality (6.40) in Theorem 6.21 is not an LMI, we can not solve it by employing the Toolbox YALMIP of MATLAB. In order to deal with the nonlinear problem, we investigate the following theorem for transforming the nonlinear problem into a linear one. Theorem 6.25 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (6.2), and positive constants α, c1 , c2 and T , the trivial solution of GRN (6.3) under Dirichlet boundary conditions is finite-time stable if there exist real numbers λqi > 0, (i ∈ 5), λ pj > 0, λn j > 0, λr j > 0, λwj > 0, λh j > 0, ( j = 1, 2) and λ p > 0, matrices Q iT = Q i > 0 (i ∈ 5), RkT = Rk > 0, WkT = Wk > 0 and HkT = Hk > 0, diagonal matrices Pk > 0, Λk > 0, Nk > 0 (k = 1, 2), and matrices Gˆ 1 and Gˆ 2 of appropriate sizes, such that (6.38), (6.39) and the following LMIs hold: 0  Q i  λqi I, i ∈ 5,

(6.61)

0  P j  λ pj I, j = 1, 2,

(6.62)

0  N j  λn j I, j = 1, 2,

(6.63)

0  R j  λr j I, j = 1, 2,

(6.64)

0  W j  λwj I, j = 1, 2,

(6.65)

0  H j  λh j I, j = 1, 2,

(6.66)

λ p I  P,

(6.67)

6.3 Finite-Time Case

151

1 1 c1 eαT (λ p1 + τ¯ λq1 + τ¯ λq2 + τ¯ 3 λw1 + τ¯ 3 λr 1 6 2 l  1 + λn1 λmax (Dk ) + λ p2 + σλ ¯ q3 + σλ ¯ q4 + τ¯ 4 λh1 24 k=1 1 1 1 + σλ ¯ q5 λmax (K T K ) + σ¯ 3 λw2 + σ¯ 3 λr 2 + σ¯ 4 λh2 6 2 24 l  + λn2 λmax (Dk∗ ))  c2 λ p , k=1

where Dk , Dk∗ and K are defined as previously. Proof From (6.61)–(6.66) we can obtain n 1 := c1 eαT (λmax (P1 ) + τ¯ λmax (Q 1 ) + τ¯ λmax (Q 2 ) l  1 + τ¯ 3 λmax (W1 ) + λmax (N1 )λmax (Dk ) 6 k=1 1 1 + τ¯ 3 λmax (R1 ) + τ¯ 4 λmax (H1 )) 2 24 1 1  c1 eαT (λ p1 + τ¯ λq1 + τ¯ λq2 + τ¯ 3 λw1 + τ¯ 3 λr 1 6 2 l  1 4 + τ¯ λh1 + λn1 λmax (Dk )), 24 k=1 n 2 := c1 eαT (λmax (P2 ) + σλ ¯ max (Q 3 ) + σλ ¯ max (Q 4 ) +σλ ¯ max (Q 5 )λmax (K T K ) l  1 + σ¯ 3 λmax (W2 ) + λmax (N2 )λmax (Dk∗ ) 6 k=1 1 1 + σ¯ 3 λmax (R2 ) + σ¯ 4 λmax (H2 )) 2 24  c1 eαT (λ p2 + σλ ¯ q3 + σλ ¯ q4 + σλ ¯ q5 λmax (K T K ) 1 1 1 + σ¯ 3 λw2 + σ¯ 3 λr 2 + σ¯ 4 λh2 6 2 24 l  +λn2 λmax (Dk∗ )). k=1

Then using (6.67)–(6.68) we can get n 1 + n 2  λ p c2  c2 λmin (P),

(6.68)

152

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

i.e., the matrix inequality (6.40) is feasible. By Theorem 6.21 we know the trivial solution of GRN (6.3) under Dirichlet boundary conditions is finite-time stable respect to (c1 , c2 , T ). The proof is completed. Remark 6.26 Note that, due to Dirichlet boundary conditions, we can use the socalled Wirtinger’s inequality (i.e., Lemma 1.13) in the proof of Theorem 6.5, which results in an effective finite-time stability criterion. It should be emphasized that the Wirtinger’s inequality may not be available for the other boundary conditions, for example, Neumann boundary conditions. However, one can obtain finite-time stability criterion under Neumann boundary conditions by employing a method similar to ones in the proof of Theorem 6.9.

6.3.2

A Numerical Example

Example 6.27 When l = n = 1, GRN (6.3) simplifies into   ∂ ∂ m(t, ˜ x) ∂ m(t, ˜ x) = D1 − Am(t, ˜ x) ∂t ∂x ∂x +W f ( p(t ˜ − σ(t), x)),

(6.69a)

  ∂ p(t, ˜ x) ∂ ∂ p(t, ˜ x) = D1∗ − C p(t, ˜ x) ∂t ∂x ∂x +D m(t ˜ − τ (t), x).

(6.69b)

We choose the values of parameters in (6.69) are as follows: A = 0.3, D = 1.5, C = 0.2, L 1 = 1, W = −0.5, D1 = 0.2, D1∗ = 0.1, K = 0.65. When τ¯d = σ¯ d = 2, c1 = 1.3, c2 = 5, T = 10 and α = 0.002, it is easily be checked by using the Toolbox YALMIP of MATLAB that the LMIs (6.38), (6.39) and (6.61)–(6.68) are feasible for τ¯ = σ¯ ∈ (0, 0.4060]. By Theorem 6.25, we can conclude that the GRN under consideration is finite-time stable when τ¯ = σ¯ ∈ (0, 0.4060]. Furthermore, when σ(t) = τ (t) ≡ 0.4, we obtain a feasible solution of LMIs (6.38), (6.39) and (6.61)–(6.68) as follows: Q 1 = ×10−7 , Q 2 = 9.2159 × 10−4 , Q 3 = 7.1758 × 10−8 , Q 4 = 2.2272 × 10−6 , Q 5 = 5.3089 × 10−7 , R1 = 0.0034,

6.3 Finite-Time Case

153

R2 = 1.5648 × 10−4 , W1 = 0.0013, W2 = 7.3230 × 10−5 , P1 = 0.0015, P2 = 7.5419 × 10−4 , H1 = 4.8050 × 10−5 , H2 = 1.1805 × 10−5 , N1 = 4.5331 × 10−4 , N2 = 3.1107 × 10−5 , Λ1 = 3.5408 × 10−6 , Λ2 = 4.4381 × 10−4 , λq1 = 2.2639 × 10−6 , λq2 = 9.2267 × 10−4 , λq3 = 2.1831 × 10−6 , λq4 = 3.7268 × 10−6 , λq5 = 5.3527 × 10−6 , λr 1 = 0.0034, λr 2 = 1.7100 × 10−4 , λw1 = 0.0014, λw2 = 1.3102 × 10−4 , λ p1 = 0.0015, λ p2 = 7.5463 × 10−4 , λh1 = 8.2482 × 10−4 , λh2 = 8.0603 × 10−4 , λn1 = 4.5547 × 10−4 , λn2 = 3.5969 × 10−5 , λ p = 7.5407 × 10−4 , ⎡

⎤ −0.0033 0.0003 0.0000 Gˆ 1 = ⎣ 0.0000 −0.0001 0.0007⎦ , −0.0000 −0.0007 0.0011 ⎡ ⎤ −0.2066 −0.0171 0.0647 Gˆ 2 = 10−5 × ⎣−0.0117 −0.1228 0.1998⎦ . −0.0089 −0.1656 0.2830 The state responses of GRN (6.69) are given in Fig. 6.3 which show the effectiveness of the approach proposed in this section.

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 10 5 0 0

1

2

3

4

1 0.5 0 -0.5 -1 -1.5 -2 10 5 0 0

Fig. 6.3 Trajectories of mRNA and protein concentrations (Example 6.27)

1

2

3

4

154

6 Stability Analysis for Delayed Reaction-Diffusion GRNs

6.4 Remarks and Notes This chapter addresses the problems of asymptotic stability analysis and finite-time stability analysis for the trivial solution of the delayed reaction-diffusion GRN (6.3), respectively. Dirichlet boundaryconditions and Neumann boundary conditions are involved. By constructing LKFs and estimating its derivatives via appropriate techniques, we establish asymptotic stability criteria and finite-time stability criteria. It has been theoretically and numerically demonstrated that the obtained asymptotic stability criteria are less conservative than the corresponding criteria in [6, 8]. Moreover, in this chapter we remove the restrictions τ¯d < 1 and σ¯ d < 1 required in [6, 8, 10]. The results related to asymptotic stability and finite-time stability are taken from [5] and [4], respectively. Refer to [2] for the related work on a class of impulsive stochastic GRNs with time-varying delays and reaction-diffusion.

References 1. Busenberg, S., Mahaffy, J.: Interaction of spatial diffusion and delays in models of genetic control by repression. J. Mol. Biol. 22(3), 313–333 (1985) 2. Cao, B., Zhang, Q., Ye, M.: Exponential stability of impulsive stochastic genetic regulatory networks with time-varying delays and reaction-diffusion. Adv. Differ. Eqn. 2016 (Article No. 307, 2016) 3. Du, Y., Li, Y., Xu, R.: Stability analysis for impulsive stochastic reaction-diffusion differential system and its application to neural networks. J. Appl. Math. 2013 (Article ID 785141, 12 pages, 2013) 4. Fan, X., Zhang, X., Wu, L., Shi, M.: Finite-time stability analysis of reaction-diffusion genetic regulatory networks time-varying delays. IEEE/ACM Trans. Comput. Biol. Bioinf. 14(4), 868– 879 (2017) 5. Han, Y., Zhang, X., Wang, Y.: Asymptotic stability criteria for genetic regulatory networks with time-varying delays and reaction-diffusion terms. Circuits Syst. Signal Process. 34(10), 3161–3190 (2015) 6. Han, Y.Y., Zhang, X.: Stability analysis for delayed regulatory networks with reaction-diffusion terms (in Chinese). J. Nat. Sci. Heilongjiang Univ. 31(1), 32–40 (2014) 7. Löfberg, J.: YAPLMI: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design (CACSD), pp. 284–289 (2004) 8. Ma, Q., Shi, G.D., Xu, S.Y., Zou, Y.: Stability analysis for delayed genetic regulatory networks with reaction-diffusion terms. Neural Comput. Appl. 20(4), 507–516 (2011) 9. Park, P.G., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011) 10. Zhou, J.P., Xu, S.Y., Shen, H.: Finite-time robust stochastic stability of uncertain stochastic delayed reaction-diffusion genetic regulatory networks. Neurocomputing 74(17), 2790–2796 (2011)

Part II

Design of Delayed GRNs

Chapter 7

State Estimation for Delayed GRNs

Usually, not all states information (that is, mRNA and protein concentrations) of GRNs are measurable, we have to estimate unmeasured state information by making use of the effective network outputs. The chapter addresses the problem of state estimation for delayed GRNs.

7.1 Problem Formulation Consider the GRN model (1.8), that is ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(7.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(7.1b)

Here f : Rn → Rn is a continuous function satisfying the following sector conditions: f j (s)  k j , ∀0 = s ∈ R, j ∈ n, (7.2) f j (0) = 0, 0  s where k j is a nonnegative scalar, and f j (s) is the jth entry of f (s). Let K = diag(k1 , k2 , . . . , kn ). The main aim of the chapter is to design full- and reduced-order state observers for GRN (7.1). To this end, we require the network measurements of the following form: ˜ z p (t) = N p(t), ˜ z m (t) = M m(t),

(7.3)

where M and N are the full-row-rank constant matrices of appropriate sizes. © Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_7

157

158

7 State Estimation for Delayed GRNs

7.2 Full-Order State Observer The section will propose an approach to design a full-order state observer for GRNs with interval time-varying delays. By constructing an appropriate LKF and estimating its derivative by using Proposition 3.3 which has been investigated to be less conservative than Lemma 1.14(ii), we investigate a delay-range-dependent and delay-rate-dependent sufficient condition, represented by a set of LMIs, under which the resultant error system is asymptotically stable. Based on a feasible solution of these LMIs, a full-order state observer is designed. In the section, we construct the full-order state observer for (7.1) as follows: ˙ˆ ˆ m(t) = −Am(t) ˆ + W f ( p(t ˆ − σ(t))) + Km [z m (t) − M m(t)],

(7.4a)

˙ˆ = −C p(t) p(t) ˆ + D m(t ˆ − τ (t)) + K p [z p (t) − N p(t)], ˆ

(7.4b)

where Km and K p are observer gain matrices. In this section, we will always assume that the delays τ (t) and σ(t) are differentiable functions satisfying 0  τ1  τ (t)  τ2 , 0  σ1  σ(t)  σ2 ,

(7.5a)

τ˙ (t)  τd < ∞, σ(t) ˙  σd < ∞,

(7.5b)

where τ1 , τ2 , σ1 , σ2 , τd and σd are constants. ˜ − m(t) ˆ and e p (t) = p(t) ˜ − p(t), ˆ it Let the error state vectors be em (t) = m(t) follows from (7.1), (7.3) and (7.4) that the state error system is immediately obtained as follows: e˙m (t) = −(A + Km M)em (t) + W fˆ(e p (t − σ(t))),

(7.6a)

e˙ p (t) = −(C + K p N )e p (t) + Dem (t − τ (t)),

(7.6b)

where fˆ(e p (t)) = f ( p(t)) ˜ − f ( p(t)) ˆ and f satisfies the sector condition (7.2).

7.2.1 Observer Design Based on the previous preparation, one can investigate the following theorem, which provides a method to design a state observer (7.4) for GRN (7.1). Theorem 7.1 For given constants τ1 , τ2 , σ1 , σ2 , τd and σd with 0 < τ1 < τ2 and 0 < σ1 < σ2 , the state error system (7.6) subject to (7.2) and (7.5) is asymptotically

7.2 Full-Order State Observer

159

stable if there exist matrices P T = P > 0, Q iT = Q i > 0, RiT = Ri > 0 (i ∈ 7), Λ j := diag(λ1 j , λ2 j , . . . , λn j ) > 0, T j := diag(t1 j , t2 j , . . . , tn j ) > 0 ( j = 1, 2), U1 with det(U1 ) = 0, U2 with det(U2 ) = 0, X 1 , X 2 , Q˜ 5 and R˜ 5 of appropriate sizes, such that the following LMIs hold: 

   Q 5 Q˜ 5 R5 R˜ 5  0,  0, Q˜ T5 Q 5 R˜ 5T R5

Ψ := Ψ1 + Ψ1T +

6 

Ψi +

i=2

9 

(7.7)

(Ψi + ΨiT ) < 0,

i=7



e9 Ψ1 = Γ PΓd + K e2 − e9 T

T 

 Λ1 e , Λ2 18

Ψ2 = e1T (Q 1 + Q 3 )e1 + e3T (Q 2 − Q 1 )e3 − e5T Q 2 e5 − (1 − τd )e4T Q 3 e4 , Ψ3 = e2T R1 e2 + e6T (R2 − R1 )e6 − e8T R2 e8 +  −(1 − σd )

e7 e10

T

 R3

  T   e11 e11 e1 − Q4 e17 e1 − e3 e1 − e3 ⎡ ⎤T ⎤ e13 e13   ⎢e3 − e4 ⎥ Q 5 Q˜ 5 ⎢e3 − e4 ⎥ ⎢ ⎥ ⎥ −⎢ ⎣ e12 ⎦ Q˜ T Q 5 ⎣ e12 ⎦ , 5 e4 − e5 e4 − e5  T   e e 2 Ψ5 = 2 (σ12 R4 + σ12 R5 ) 2 e18 e18 ⎡ ⎡ ⎤T ⎤ e16 e16 T      ⎢e6 − e7 ⎥ R5 R˜ 5 ⎢e6 − e7 ⎥ e14 e14 ⎢ ⎥ ⎥ −⎢ − R4 ⎣ e15 ⎦ R˜ T R5 ⎣ e15 ⎦ , e2 − e6 e2 − e6 5 e7 − e8 e7 − e8 

Ψ4 =

e1 e17 ⎡

T

 e7 , e10

 T   e e2 R3 2 e9 e9



2 (τ12 Q 4 + τ12 Q5)

4 τ14 T σ1 Q 6 + τs2 Q 7 )e17 + e18 ( R6 + σs2 R7 )e18 4 4 −(τ1 e1 − e11 )T Q 6 (τ1 e1 − e11 ) 2τs − 2 (τ12 e3 − e12 − e13 )T Q 7 (τ12 e3 − e12 − e13 ) τ12 τs τ12 − (e1 − e3 )T Q 7 (e1 − e3 ) − (σ1 e2 − e14 )T R6 (σ1 e2 − e14 ) τ1

T Ψ6 = e17 (

(7.8)

160

7 State Estimation for Delayed GRNs

2σs (σ12 e6 − e15 − e16 )T R7 (σ12 e6 − e15 − e16 ) 2 σ12 σs σ12 − (e2 − e6 )T R7 (e2 − e6 ), σ1

−

Ψ7 = −(e9 − K e2 )T T1 e9 − (e10 − K e7 )T T2 e10 , Ψ8 = (e1 + e17 )T [(−U1 A − X 1 M)e1 + U1 W e10 − U1 e17 ], Ψ9 = (e2 + e18 )T [(−U2 C − X 2 N )e2 + U2 De4 − U2 e18 ], Γ = col(e1 , e2 , e11 , e12 + e13 , e14 , e15 + e16 ), Γd = col(e17 , e18 , e1 − e3 , e3 − e5 , e2 − e6 , e6 − e8 ), 1 2 1 (τ2 − τ12 ), σ12 = σ2 − σ1 , σs = (σ22 − σ12 ), 2 2

τ12 = τ2 − τ1 , τs =

ei = [ 0 · · · 0 In

0 · · · 0 ], i ∈ 18.

number i−1

number 18−i

Moreover, the required observer gains are given by Km = U1−1 X 1 and K p = U2−1 X 2 . Proof Set η(t) = col(em (t), e p (t), em (t − τ1 ), em (t − τ (t)), em (t − τ2 ), e p (t − σ1 ),  t e p (t − σ(t)), e p (t − σ2 ), fˆ(e p (t)), fˆ(e p (t − σ(t))), em (s)ds, 

t−τ (t)

t−τ2  t−σ1

 em (s)ds,

t−τ1

t−τ (t)

 em (s)ds,

t

 e p (s)ds,

t−σ1

t−τ1 t−σ(t)

e p (s)ds,

t−σ2

e p (s)ds, e˙m (t), e˙ p (t))

t−σ(t)

and  η1 (t) = col(em (t), e p (t), 

t t−σ1

 e p (s)ds,



t

em (s)ds,

t−τ1 t−σ1

e p (s)ds).

t−σ2

t−τ1

t−τ2

em (s)ds,

7.2 Full-Order State Observer

161

Then η1 (t) = Γ η(t) and η˙1 (t) = Γd η(t). Choose an LKF candidate as: V (t, em , e p ) =

6 

Vi (t, em , e p ),

(7.9)

i=1

where

 e (t) n λi1 0 pi fˆi (s)ds V1 (t, em , e p ) = η1T (t)Pη1 (t) + 2 i=1  e (t) n +2 i=1 λi2 0 pi (ki s − fˆi (s))ds, V2 (t, em , e p ) =

V3 (t, em , e p ) =

t



t−τ eT (s)Q e (s)ds + t−τ21 t−τ  1t m T 1 m + t−τ (t) em (s)Q 3 em (s)ds,

emT (s)Q 2 em (s)ds

t

 t−σ eTp (s)R1 e p (s)ds + t−σ21 eTp (s)R2 e p (s)ds T    t e p (s) e p (s) + t−σ(t) ˆ R3 ˆ ds, f (e p (s)) f (e p (s)) t−σ1

T    em (s) em (s) dsdθ Q4 V4 (t, em , e p ) = τ1 −τ1 t+θ e˙m (s) e˙ (s) T m    −τ  t e (s) e (s) dsdθ, +τ12 −τ21 t+θ m Q5 m e˙m (s) e˙m (s) 0 t

T    e p (s) e p (s) V5 (t, em , e p ) = σ1 −σ1 t+θ dsdθ R4 e˙ p (s) e˙ (s)  T p   −σ  t e (s) e (s) +σ12 −σ21 t+θ p R5 p dsdθ, e˙ p (s) e˙ p (s) 0 t

V6 (t, em , e p ) =

0 0t e˙mT (s)Q 6 e˙m (s)dsdλdθ −τ 1 θ  t+λ  −τ 0t 1 +τs −τ2 θ t+λ e˙mT (s)Q 7 e˙m (s)dsdλdθ σ2  0  0  t + 21 −σ1 θ t+λ e˙Tp (s)R6 e˙ p (s)dsdλdθ  −σ  0  t +σs −σ21 θ t+λ e˙Tp (s)R7 e˙ p (s)dsdλdθ, τ12 2

and the matrices P, Q i , Ri (i ∈ 7) and Λ j := diag(λ1 j , λ2 j , . . . , λn j ) > 0 ( j = 1, 2) are taken from a feasible solution to (7.7) and (7.8). Then, following the proof of Theorem 3.5, one can easily derive that V˙ (t, em , e p )  η T (t)(Ψ − Ψ8 − Ψ8T − Ψ9 − Ψ9T )η(t).

(7.10)

For any matrices U1 and U2 , one has from (7.6) that 0 = 2[em (t) + e˙m (t)]T U1 [−(A + Km M)em (t) + W fˆ(e p (t − σ(t))) − e˙m (t)] = η T (t)(Ψ8 + Ψ8T )η(t),

162

7 State Estimation for Delayed GRNs

0 = 2[e p (t) + e˙ p (t)]T U2 [−(C + K p N )e p (t) + Dem (t − τ (t)) − e˙ p (t)] = η T (t)(Ψ9 + Ψ9T )η(t). This, together with (7.10), implies that V˙ (t, em , e p )  η T (t)Ψ η(t), and hence V˙ (t, em , e p ) < 0 from (7.8). Therefore, the state error system (7.6) is asymptotically stable. The proof is completed. Remark 7.2 Compared with the LKFs used in [2, 4–6] which consider the design problem of state observers for GRNs, our LKF includes triple-integral terms (see V6 ) and the regulatory function-based items (see V1 ). Such an LKF not only fully utilizes the information of regulatory functions, but also is very effective to reduce conservativeness of conclusions [8]. Remark 7.3 When an LKF including triple-integral terms is employed, its derivative will include double-integral terms of the form  −

−τ1 −τ2



t t+θ

e˙mT (s)Q 7 e˙m (s)dsdθ.

One popular approach to deal with the type of double-integral terms is to enlarge them by using Lemma 1.14(ii). However, in this section the type of double-integral terms is enlarged by employing the inequality in Proposition 3.3, which has been investigated to be less conservative than Lemma 1.14(ii).

7.2.2 A Numerical Example Example 7.4 Consider GRN (7.1) with the following parameters: A = diag(7, 7), C = diag(6, 6), D = diag(1, 1), 

 0 −5.5 W = , K = diag(0.65, 0.65). −5.5 0 The network measurement parameters in (7.3) are assumed as:     M = 1 0.5 , N = 0.7 −0.25 . Assume that the time-varying delays τ (t) and σ(t) satisfy that τ1 = σ1 = 0.1, τ2 = σ2 = 0.8 and τd = σd = 1. By using the Toolbox YALMIP of MATLAB to solve the LMIs (7.7) and (7.8), the partial feasible solutions are listed as follows:

7.2 Full-Order State Observer

163

    278.1281 75.5875 168.1826 36.6480 , Q2 = , 75.5875 176.8886 36.6480 111.4685     428.3073 −12.2446 265.9105 −13.2724 R1 = , R2 = , −12.2446 339.4888 −13.2724 227.9433 Q1 =

Λ1 = diag(209.1275, 197.9765), Λ2 = diag(209.1151, 197.9479), T1 = diag(168.5886, 166.2046), T2 = diag(414.9787, 359.8430), 

   14.7727 −1.2507 59.3661 2.2300 U1 = , U2 = , −1.6039 18.7843 4.7293 59.5329     204.6455 110.7985 X1 = , X2 = . 104.7956 −62.5653 By Theorem 7.1, a full-order state observer for system (7.1) is given by (7.4) with gains:     14.4295 1.9115 , Kp = . Km = 6.8109 −1.2028 Furthermore, when the time-varying delays τ (t) and σ(t) are assumed to be τ (t) = 0.7sin2 (10t/7) + 0.1, σ(t) = 0.7cos2 (−10t/7) + 0.1. The trajectories of mRNAs and their estimates are given in Fig. 7.1 with the initial ˆ ≡ [3, −3]T , t ∈ [−0.8, 0], conditions m(t) ˜ ≡ [1.8, 0.2]T , t ∈ [−0.8, 0] and m(t) while the trajectories of proteins and their estimates are given in Fig. 7.2 with the iniˆ ≡ [3, 1]T , t ∈ [−0.8, 0]. tial conditions p(t) ˜ ≡ [0.7, 2.3]T , t ∈ [−0.8, 0] and p(t) The estimation errors are given in Fig. 7.3. 3

3

2.5

2

2

1

1.5

0

1

-1

0.5

-2

0 -0.5

0

0.5

1

1.5

2

2.5

3

-3

0

0.5

1

Fig. 7.1 The mRNA concentrations and their estimations (Example 7.4)

1.5

2

2.5

3

164

7 State Estimation for Delayed GRNs 3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0 -0.5 0

0.5

1

1.5

2

2.5

3

-0.5

0

0.5

1

1.5

2

2.5

3

1.5

2

2.5

3

Fig. 7.2 The protein concentrations and their estimations (Example 7.4) 1.5

6 5

1

4

0.5

3

0

2

-0.5

1

-1

0

-1.5

-1

-2

-2

-2.5 0

0

0.5

1

1.5

2

2.5

3

0.5

1

Fig. 7.3 Estimation errors (Example 7.4)

7.3 Reduced-Order State Observer The section will estimate the mRNA and protein concentrations via establish a reduced-order state observer based on available measurement outputs. An LKF is constructed as well as its derivative is estimated by using the Wirtinger-type integral inequalities and (reciprocally) convex technique. From which, a set of LMI conditions that ensure the existence of reduced-order observer are derived. Thereby, the observer gains of a reduced-order state observer can be represented by feasible solutions of the set of LMIs. In this section we will always assume that the delays τ (t) and σ(t) are differentiable functions satisfying ˙  σ¯ d < ∞, 0  τ (t)  τ¯ , 0  σ(t)  σ, ¯ τ˙ (t)  τ¯d < ∞, σ(t) where τ¯ , σ, ¯ τ¯d and σ¯ d are known constants.

(7.12)

7.3 Reduced-Order State Observer

165

7.3.1 Observer Design In this subsection we will present an approach to design reduced-order state observers for GRN (7.1). Since M and N are of full-row-rank, there exist nonsingular matrices Q m and Q p such that M Q m = [Iq 0] = N Q p . Let ˜ = col(m˜ 1 (t), m˜ 2 (t)), Q −1 ˜ = col( p˜ 1 (t), p˜ 2 (t)), Q −1 m m(t) p p(t)

(7.13)

   A¯ 11 A¯ 12 D¯ 11 D¯ 12 −1 , Q , D Q = m p A¯ 21 A¯ 22 D¯ 21 D¯ 22     C¯ 11 C¯ 12 Q p11 Q p12 −1 Qp CQp = ¯ ¯ , Qp = , Q p21 Q p22 C21 C22   W¯ 11 W¯ 12 , W = Q −1 m W¯ 21 W¯ 22

Q −1 m AQ m =



where A¯ 11 , D¯ 11 , C¯ 11 , W¯ 11 , Q p11 ∈ Rq×q and m˜ 1 (t), p˜ 1 (t) ∈ Rq . Then z m (t) = m˜ 1 (t), z p (t) = p˜ 1 (t) from (7.3), and further the delayed GRN (7.1) can be written as:

where

m˙˜ 2 (t) = − A¯ 21 z m (t) − A¯ 22 m˜ 2 (t) + W¯ 21 Φ1 ( p˜ 2 (t − σ(t))) +W¯ 22 Φ2 ( p˜ 2 (t − σ(t))),

(7.14a)

z¯ m (t) = − A¯ 12 m˜ 2 (t) + W¯ 11 Φ1 ( p˜ 2 (t − σ(t))) +W¯ 12 Φ2 ( p˜ 2 (t − σ(t))),

(7.14b)

p˙˜ 2 (t) = −C¯ 21 z p (t) − C¯ 22 p˜ 2 (t) + D¯ 21 z m (t − τ (t)) + D¯ 22 m˜ 2 (t − τ (t)),

(7.14c)

z¯ p (t) = −C¯ 12 p˜ 2 (t) + D¯ 12 m˜ 2 (t − τ (t)),

(7.14d)

166

7 State Estimation for Delayed GRNs

 z p (t − σ(t)) , p˜ 2 (t − σ(t))    z p (t − σ(t)) Φ2 ( p˜ 2 (t − σ(t))) = [0 In−q ] f Q p , p˜ 2 (t − σ(t)) 



Φ1 ( p˜ 2 (t − σ(t))) = [Iq 0] f

Qp

z¯ m (t) = z˙ m (t) + A¯ 11 z m (t),

(7.15a)

z¯ p (t) = z˙ p (t) + C¯ 11 z p (t) − D¯ 11 z m (t − τ (t)).

(7.15b)

Now, we make a state observer for (7.14) as follows: ˙ m˜ˆ 2 (t) = − A¯ 21 z m (t) − A¯ 22 mˆ˜ 2 (t) + W¯ 21 Φ1 ( pˆ˜ 2 (t − σ(t))) +W¯ 22 Φ2 ( pˆ˜ 2 (t − σ(t))) + Lm [¯z m (t) + A¯ 12 mˆ˜ 2 (t) −W¯ 11 Φ1 ( pˆ˜ 2 (t − σ(t))) − W¯ 12 Φ2 ( pˆ˜ 2 (t − σ(t)))],

(7.16a)

˙ pˆ˜ 2 (t) = −C¯ 21 z p (t) − C¯ 22 pˆ˜ 2 (t) + D¯ 21 z m (t − τ (t)) + D¯ 22 m˜ˆ 2 (t − τ (t)) + L p [¯z p (t) + C¯ 12 pˆ˜ 2 (t) − D¯ 12 mˆ˜ 2 (t − τ (t))],

(7.16b)

where Lm and L p are the observer gains. Let em (t) = m˜ 2 (t) − mˆ˜ 2 (t) and e p (t) = p˜ 2 (t) − pˆ˜ 2 (t) be the error state vectors. Then, from (7.14) and (7.16), we can obtain the error system: e˙m (t) = −( A¯ 22 − Lm A¯ 12 )em (t) +(W¯ 21 − Lm W¯ 11 )Φˆ 1 (e p (t − σ(t))) +(W¯ 22 − Lm W¯ 12 )Φˆ 2 (e p (t − σ(t))),

(7.17a)

e˙ p (t) = −(C¯ 22 − L p C¯ 12 )e p (t) +( D¯ 22 − L p D¯ 12 )em (t − τ (t)),

(7.17b)

where Φˆ j (e p (s)) = Φ j ( p˜ 2 (s)) − Φ j ( pˆ˜ 2 (s)), j = 1, 2. This, together with the relationship among Φˆ 1 , Φ1 , f and g, implies that Φˆ 1 (e p (s))        z p (s) z p (s) = [Iq 0] f Q p − f Qp ˆ p˜ 2 (s) p˜ 2 (s)

7.3 Reduced-Order State Observer

167

         z p (s) z p (s) ∗ ∗ + p − g Qp ˆ +p . = [Iq 0] g Q p p˜ 2 (s) p˜ 2 (s) By the help of (1.3), we obtain Φˆ 1 (0) = 0, Φˆ 1T (e p (s))(Φˆ 1 (e p (s)) − K 1 Q p12 e p (s))  0,

(7.18)

where K 1 = diag(k1 , k2 , . . . , kq ). Similarly, Φˆ 2 (0) = 0, Φˆ 2T (e p (s))(Φˆ 2 (e p (s)) − K 2 Q p22 e p (s))  0,

(7.19)

where K 2 = diag(kq+1 , kq+2 , . . . , kn ). In order to determine the observer gains Lm and L p , the following notations are required. ei = col(0(i−1)n×n , In , 0(20−i)n×n ), i ∈ 20, Δ1 = [e3 − e2 e3 + e2 − 2e13 e3 − e2 + 6e13 − 12e14 ], Δ2 = [e1 − e3 e1 + e3 − 2e15 e1 − e3 + 6e15 − 12e16 ], Δ3 = [e6 − e5 e6 + e5 − 2e17 e6 − e5 + 6e17 − 12e18 ], Δ4 = [e4 − e6 e4 + e6 − 2e19 e4 − e6 + 6e19 − 12e20 ], Δ5 = [e13 e13 − 2e14 ], Δ6 = [e15 e15 − 2e16 ], Δ7 = [e17 e17 − 2e18 ], Δ8 = [e19 e19 − 2e20 ], Θ1 = [e1 − e15 e1 + 2e15 − 6e16 ], Θ2 = [e3 − e13 e3 + 2e13 − 6e14 ], Θ3 = [e4 − e19 e4 + 2e19 − 6e20 ], Θ4 = [e6 − e17 e6 + 2e17 − 6e18 ], ς(t) = col(em (t), em (t − τ¯ ), em (t − τ (t)), e p (t), e p (t − σ), ¯ e p (t − σ(t)), Φˆ 1 (e p (t)), Φˆ 2 (e p (t)),

168

7 State Estimation for Delayed GRNs

Φˆ 1 (e p (t − σ(t))), Φˆ 2 (e p (t − σ(t))), e˙m (t),  t−τ (t) 1 em (s)ds, e˙ p (t), τ¯ − τ (t) t−τ¯  t−τ (t)  t−τ (t) 1 em (s)dsdα, (τ¯ − τ (t))2 t−τ¯ α  t  t  t 1 1 em (s)ds, 2 em (s)dsdα, τ (t) t−τ (t) τ (t) t−τ (t) α  t−σ(t) 1 e p (s)ds, σ¯ − σ(t) t−σ¯  t−σ(t)  t−σ(t) 1 e p (s)dsdα, (σ¯ − σ(t))2 t−σ¯ α  t  t  t 1 1 e p (s)ds, 2 e p (s)dsdα). σ(t) t−σ(t) σ (t) t−σ(t) α Theorem 7.5 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (7.12), the trivial solution of error system (7.17) is asymptotically stable if there exist matrices Q iT = Q i > 0 (i ∈ 6), R Tj = R j > 0, ( j ∈ 4), WkT = Wk > 0, HkT = Hk > 0 and PkT = Pk > 0 (k ∈ 2), diagonal matrices Λk > 0 (k ∈ 4) and matrices Gˆ 1 , Gˆ 2 , N1 and N2 of appropriate sizes, such that the following inequalities hold for τ ∈ {0, τ¯ } and σ ∈ {0, σ}: ¯ 

   Rˆ 2 Gˆ 2 Rˆ 1 Gˆ 1  0, ˆ T ˆ  0, Gˆ T1 Rˆ 1 G 2 R2

Ξ (τ , σ) :=

3  i=1

Ξi +

6  i=4

Ξi (τ , σ) +

4 

(7.20)

Ξ0i < 0,

i=1

where Ξ1 = sym(−e1 (P1 A¯ 22 − N1 A¯ 12 )e1T + e1 (P1 W¯ 21 − N1 W¯ 11 )e9T T +e1 (P1 W¯ 22 − N1 W¯ 12 )e10 + e4 (P2 D¯ 22 − N2 D¯ 12 )e3T −e4 (P2 C¯ 22 − N2 C¯ 12 )e4T − e11 (P1 A¯ 22 − N1 A¯ 12 )e1T T +e11 (P1 W¯ 21 − N1 W¯ 11 )e9T + e11 (P1 W¯ 22 − N1 W¯ 12 )e10 +e12 (P2 D¯ 22 − N2 D¯ 12 )e3T − e12 (P2 C¯ 22 − N2 C¯ 12 )e4T T T −e11 P1 e11 − e12 P2 e12 ),

Ξ2 = e1 (Q 1 + Q 2 )e1T − e2 Q 2 e2T + (τd − 1)e3 Q 1 e3T +e4 (Q 3 + Q 4 )e4T − e5 Q 4 e5T + (σd − 1)e6 Q 3 e6T ,

(7.21)

7.3 Reduced-Order State Observer

169

T Ξ3 = (σd − 1)e9 Q 5 e9T + e7 Q 5 e7T + (σd − 1)e10 Q 6 e10

+e8 Q 6 e8T , T Ξ4 (τ , σ) = τ¯ 2 (e11 R1 e11 + e1 R3 e1T ) − τ¯ τ Δ6 R¯ 3 ΔT6 T ¯ T +σ¯ 2 (e12 R2 e12 + e4 R4 e4T ) − σσΔ ¯ 8 R 4 Δ8   Rˆ Gˆ −[Δ1 Δ2 ] ˆ 1T ˆ 1 [Δ1 Δ2 ]T G 1 R1   Rˆ Gˆ − [ Δ3 Δ4 ] ˆ 2T ˆ 2 [Δ3 Δ4 ]T G 2 R2 −τ¯ (τ¯ − τ )Δ5 R¯ 3 ΔT5 − σ( ¯ σ¯ − σ)Δ7 R¯ 4 ΔT7 ,

Ξ5 (τ , σ) = Ξ50 + Ξ51 (τ ) + Ξ52 (σ), Ξ50 =

τ¯ 2 σ¯ 2 T T e11 W1 e11 + e12 W2 e12 , 2 2

Ξ51 (τ ) = −Θ1 W˜ 1 Θ1T − Θ2 W˜ 1 Θ2T −

τ¯ − τ Δ2 Wˆ 1 ΔT2 , τ¯

Ξ52 (σ) = −Θ3 W˜ 2 Θ3T − Θ4 W˜ 2 Θ4T −

σ¯ − σ Δ4 Wˆ 2 ΔT4 , σ¯

Ξ6 (τ , σ) = Ξ60 + Ξ61 (τ ) + Ξ62 (σ) − (τ¯ − τ )Θ1 H˜ 1 Θ1T −(σ¯ − σ)Θ3 H˜ 2 Θ3T , Ξ60 =

τ¯ 3 σ¯ 3 T T e11 H1 e11 + e12 H2 e12 , 6 6

3 Ξ61 (τ ) = − τ (e1 − 2e16 )H1 (e1 − 2e16 )T 2 3 − (τ¯ − τ )(e3 − 2e14 )H1 (e3 − 2e14 )T , 2 3 Ξ62 (σ) = − σ(e4 − 2e20 )H2 (e4 − 2e20 )T 2 3 − (σ¯ − σ)(e6 − 2e18 )H2 (e6 − 2e18 )T , 2 Ξ01 = sym(−e7 Λ1 e7T + e4 Q Tp12 K 1 Λ1 e7T ), Ξ02 = Sym(−e9 Λ2 e9T + e6 Q Tp12 K 1 Λ2 e9T ),

170

7 State Estimation for Delayed GRNs

Ξ03 = sym(−e8 Λ3 e8T + e4 Q Tp22 K 2 Λ3 e8T ), T T Ξ04 = Sym(−e10 Λ4 e10 + e6 Q Tp22 K 2 Λ4 e10 ),

P = diag(P1 , P2 ), Wˆ i = diag(Wi , 3Wi , 5Wi ), Hˆ i = diag(Hi , 3Hi , 5Hi ), W˜ i = diag(2Wi , 4Wi ), Rˆ i = diag(Ri , 3Ri , 5Ri ), H˜ i = diag(2Hi , 4Hi ), i ∈ 2, R¯ j = diag(R j , 3R j ), j = 3, 4, and A, D, C, W , K 1 and K 2 are defined as previously. Furthermore, if the inequalities in (7.20) and (7.21) are feasible, then the state observer of (7.14) is given by (7.16) with Lm = P1−1 N1 and L p = P2−1 N2 . Proof Construct an LKF for error system (7.17) as follows: V (t, em , e p ) =

6 

Vi (t, em , e p ),

i=1

where V1 (t, em , e p ) = emT (t)P1 em (t) + eTp (t)P2 e p (t),  V2 (t, em , e p ) =



t t−τ (t)  t

emT (s)Q 1 em (s)ds

+

t t−τ¯



+

t−σ(t)

 V3 (t, em , e p ) =

eTp (s)Q 3 e p (s)ds t

t−σ(t)  t

t−σ(t)



t−σ¯

eTp (s)Q 4 e p (s)ds,



+σ¯ 

Φˆ 2T (e p (s))Q 6 Φˆ 2 (e p (s))ds,



0

−τ¯

+τ¯

t

Φˆ 1T (e p (s))Q 5 Φˆ 1 (e p (s))ds

+

V4 (t, em , e p ) = τ¯

+

emT (s)Q 2 em (s)ds

t t+θ

0



e˙mT (s)R1 e˙m (s)dsdθ t

−σ¯ t+θ 0  t −τ¯

t+θ

e˙Tp (s)R2 e˙ p (s)dsdθ emT (s)R3 em (s)dsdθ

7.3 Reduced-Order State Observer

171

 +σ¯  V5 (t, em , e p ) =



0

−τ¯

 V6 (t, em , e p ) =



0 −τ¯

0



t+θ t

−σ¯

s



0

0

t+θ



−σ¯

s

α

eTp (s)R4 e p (s)dsdθ,

e˙mT (u)W1 e˙m (u)dudθds

t

e˙Tp (u)W2 e˙ p (u)dudθds,

e˙mT (u)H1 e˙m (u)dudθdαds

s α t+θ 0  0 0 t



+

−σ¯

t

s t+θ 0  0 t



+



0

t+θ

e˙Tp (u)H2 e˙ p (u)dudθdαds.

Then, calculating the derivatives of Vi (t, em , e p ) (i ∈ 6) along the solution of error system (7.17), we have V˙1 (t, em , e p ) = 2emT (t)P1 e˙m (t) + 2eTp (t)P2 e˙ p (t),

(7.22)

V˙2 (t, em , e p ) = emT (t)(Q 1 + Q 2 )em (t) − emT (t − τ¯ )Q 2 em (t − τ¯ ) −(1 − τ˙ (t))emT (t − τ (t))Q 1 em (t − τ (t)) +eTp (t)(Q 3 + Q 4 )e p (t) − eTp (t − σ)Q ¯ 4 e p (t − σ) ¯ T −(1 − σ(t))e ˙ p (t − σ(t))Q 3 e p (t − σ(t))

 ς T (t)Ξ2 ς(t),

(7.23)

V˙3 (t, em , e p ) = −(1 − σ(t)) ˙ Φˆ 1T (e p (t − σ(t)))Q 5 Φˆ 1 (e p (t − σ(t))) −(1 − σ(t)) ˙ Φˆ 2T (e p (t − σ(t)))Q 6 Φˆ 2 (e p (t − σ(t))) +Φˆ 1T (e p (t))Q 5 Φˆ 1 (e p (t)) + Φˆ 2T (e p (t))Q 6 Φˆ 2 (e p (t))  ς T (t)Ξ3 ς(t), V˙4 (t, em , e p )  t = −τ¯ e˙mT (s)R1 e˙m (s)ds + τ¯ 2 e˙mT (t)R1 e˙m (t) t−τ¯  t e˙Tp (s)R2 e˙ p (s)ds + σ¯ 2 e˙Tp (t)R2 e˙ p (t) −σ¯ t−σ¯  t −τ¯ emT (s)R3 em (s)ds + τ¯ 2 emT (t)R3 em (t) t−τ¯

(7.24)

172

7 State Estimation for Delayed GRNs

 −σ¯

t

t−σ¯

eTp (s)R4 e p (s)ds + σ¯ 2 eTp (t)R4 e p (t),

V˙5 (t, em , e p ) =

τ¯ 2 T e˙ (t)W1 e˙m (t) − 2 m



0



−τ¯

σ¯ 2 + e˙Tp (t)W2 e˙ p (t) − 2



t t+s

0



−σ¯

(7.25)

e˙mT (u)W1 e˙m (u)duds t

t+s

e˙Tp (u)W2 e˙ p (u)duds,

(7.26)

τ¯ 3 σ¯ 3 V˙6 (t, em , e p ) = e˙mT (t)H1 e˙m (t) + e˙Tp (t)H2 e˙ p (t) 6 6  0  0 t e˙mT (u)H1 e˙m (u)dudsdα −  −

−τ¯ 0

−σ¯

α



0 α

t+s t



t+s

e˙Tp (u)H2 e˙ p (u)dudsdα.

(7.27)

Firstly, from (7.17) we have 2e˙mT (t)P1 [−( A¯ 22 − Lm A¯ 12 )em (t) + (W¯ 21 − Lm W¯ 11 )Φˆ 1 (e p (t − σ(t))) + (W¯ 22 − Lm W¯ 12 )Φˆ 2 (e p (t − σ(t))) − e˙m (t)] = 0 (7.28) and 2e˙Tp (t)P2 [−(C¯ 22 − L p C¯ 12 )e p (t) + ( D¯ 22 − L p D¯ 12 )em (t − τ (t)) − e˙ p (t)] = 0.

(7.29)

Combining (7.22), (7.28) and (7.29) we get V˙1 (t, em , e p ) = ς T (t)Ξ1 ς(t)

(7.30)

by choosing Lm = P1−1 N1 and L p = P2−1 N2 . Secondly, it follows from (7.20), Lemma 1.9 and the first inequality of Lemma 1.9 that  t e˙mT (s)R1 e˙m (s)ds −τ¯ t−τ¯ t−τ (t)

 = −τ¯

t−τ¯

 e˙mT (s)R1 e˙m (s)ds

− τ¯

t t−τ (t)

e˙mT (s)R1 e˙m (s)ds

  Rˆ Gˆ  −ς T (t)[Δ1 Δ2 ] ˆ 1T ˆ 1 [Δ1 Δ2 ]T ς(t). G 1 R1

(7.31)

7.3 Reduced-Order State Observer

173

Similarly,  −σ¯

t

t−σ¯

e˙Tp (s)R2 e˙ p (s)ds

  Rˆ Gˆ  −ς T (t)[Δ3 Δ4 ] ˆ 2T ˆ 2 [Δ3 Δ4 ]T ς(t). G 2 R2

(7.32)

Again using the first inequality of Lemma 1.15, one can obtain  −τ¯

t

emT (s)R3 em (s)ds

t−τ¯ t−τ (t)

 = −τ¯

t−τ¯

 emT (s)R3 em (s)ds

− τ¯

 −τ¯ (τ¯ − τ (t))ς T (t)Δ5 R¯ 3 ΔT5 ς(t) −τ¯ τ (t)ς T (t)Δ6 R¯ 3 ΔT6 ς(t)

t t−τ (t)

emT (s)R3 em (s)ds

(7.33)

and  − σ¯

t

t−σ¯

eTp (s)R4 e p (s)ds  −σ( ¯ σ¯ − σ(t))ς T (t)Δ7 R¯ 4 ΔT7 ς(t) T −σσ(t)ς ¯ (t)Δ8 R¯ 4 ΔT8 ς(t).

(7.34)

The combination of (7.25) and (7.31)–(7.34) gives V˙4 (t, em , e p )  ς T (t)Ξ4 (τ (t), σ(t))ς(t).

(7.35)

Thirdly, the second term on the right of (7.26) can be divided into three parts:  −



0 −τ¯



=−  −

t t+s

0

e˙mT (u)W1 e˙m (u)duds 

t

e˙mT (u)W1 e˙m (u)duds

−τ (t) t+s −τ (t)  t−τ (t) −τ¯

t+s

−(τ¯ − τ (t))



e˙mT (u)W1 e˙m (u)duds

t

t−τ (t)

e˙mT (u)W1 e˙m (u)du.

(7.36)

174

7 State Estimation for Delayed GRNs

It follows from the first two inequalities of Lemma 1.16 that  −

 −



0 −τ (t)

t

t+s



−τ (t) −τ¯

e˙mT (u)W1 e˙m (u)duds  −ς T (t)Θ1 W˜ 1 Θ1T ς(t),

t−τ (t)

e˙mT (u)W1 e˙m (u)duds  −ς T (t)Θ2 W˜ 1 Θ2T ς(t)

t+s

and  −(τ¯ − τ (t))

t

t−τ (t)

e˙mT (u)W1 e˙m (u)du  −

τ¯ − τ (t) T ς (t)Δ2 Wˆ 1 ΔT2 ς(t). τ¯

By (7.36), it implies that  −

0



−τ¯

t

e˙mT (u)W1 e˙m (u)duds  ς T (t)Ξ51 (τ (t))ς(t).

t+s

(7.37)

Similarly,  −

0



−σ¯

t

t+s

e˙Tp (u)W2 e˙ p (u)duds  ς T (t)Ξ52 (σ(t))ς(t).

This, together with (7.26) and (7.37), implies that V˙5 (t, em , e p )  ς T (t)Ξ5 (τ (t), σ(t))ς(t).

(7.38)

Fourthly, the third term on the right of (7.27) can be written as:  − − −

0

−τ¯  0



0





t+s 0 t

α

t

e˙mT (u)H1 e˙m (u)dudsdα

−τ (t) α t+s  −τ (t)  −τ (t) −τ¯

α



−(τ¯ − τ (t))

e˙mT (u)H1 e˙m (u)dudsdα 

t−τ (t) t+s



0

−τ (t)

t

t+s

e˙mT (u)H1 e˙m (u)dudsdα

e˙mT (u)H1 e˙m (u)duds.

By the laster two inequalities of Lemma 1.16, we obtain  −

0 −τ (t)



0 α



t t+s

e˙mT (u)H1 e˙m (u)dudsdα

(7.39)

7.3 Reduced-Order State Observer

 −

−τ (t)

175



−τ¯

−τ (t)



α

t−τ (t) t+s

e˙mT (u)H1 e˙m (u)dudsdα

 ς T (t)Ξ61 (τ (t))ς(t)

(7.40)

and  −(τ¯ − τ (t))



0

−τ (t) T

t

e˙mT (u)H1 e˙m (u)duds

t+s

 −(τ¯ − τ (t))ς (t)Θ1 H˜ 1 Θ1T ς(t).

(7.41)

Similarly,  − −

0



0



t

−σ(t) α t+s  −σ(t)  −σ(t) −σ¯

e˙Tp (u)H2 e˙ p (u)dudsdα 

α

t−σ(t)

t+s

e˙Tp (u)H2 e˙ p (u)dudsdα

 ς T (t)Ξ62 (σ(t))ς(t)

(7.42)

and  −(σ¯ − σ(t))

0



−σ(t) T

t t+s

e˙Tp (u)H2 e˙ p (u)duds

 −(σ¯ − σ(t))ς (t)Θ3 H˜ 2 Θ3T ς(t).

(7.43)

By (7.27) and (7.39)–(7.43), it is derived that V˙6 (t, em , e p )  ς T (t)Ξ6 (τ (t), σ(t))ς(t).

(7.44)

Lastly, for given diagonal matrices Λi > 0, i ∈ 4, we get from (7.18) and (7.19) that ς T (t)Ξ0i ς(t)  0, i ∈ 4.

(7.45)

The combination of (7.23), (7.24), (7.30), (7.35), (7.38), (7.44) and (7.45) yields V˙ (t, em , e p ) =

6 

V˙i (t, em , e p )

i=1

 ς T (t)Ξ (τ (t), σ(t))ς(t). Using (7.21), it follows that V˙ (t, em , e p ) < 0 for all τ (t) and σ(t) satisfying (7.12). Therefore, the trivial solution of error system (7.17) is asymptotically stable.

176

7 State Estimation for Delayed GRNs

Due to Theorem 7.5, the observer gains Lm and L p can be obtained by solving the LMIs in (7.20) and (7.21). Combining (7.13), (7.15) and (7.16), and setting zˆ m (t) = mˆ˜ 2 (t) − Lm z m (t), zˆ p (t) = pˆ˜ 2 (t) − L p z p (t), one can easily obtain the following theorem. Theorem 7.6 For given scalars τ¯ , σ, ¯ τ¯d , and σ¯ d satisfying (7.12), let Q m , Q p , Lm and L p be defined as previously, then the desired reduced-order state observer of GRN (7.1) is given as follows: z˙ˆ m (t) = (W¯ 21 − Lm W¯ 11 )Φ1 (ˆz p (t − σ(t)) + L p z p (t − σ(t))) +(W¯ 22 − Lm W¯ 12 )Φ2 (ˆz p (t − σ(t)) + L p z p (t − σ(t))) +(Lm A¯ 12 − A¯ 22 )ˆz m (t) + (Lm A¯ 11 − A¯ 21 +Lm A¯ 12 Lm − A¯ 22 Lm )z m (t),

z˙ˆ p (t) = (L p C¯ 12 − C¯ 22 )ˆz p (t) + ( D¯ 22 − L p D¯ 12 )ˆz m (t − τ (t))) +(L p C¯ 11 − C¯ 21 + L p C¯ 12 L p − C¯ 22 L p )z p (t) +( D¯ 21 − L p D¯ 11 + D¯ 22 Lm − L p D¯ 12 Lm )z m (t − τ (t)),

(7.46a)

(7.46b)

m(t) ˆ = (Q m1 + Q m2 Lm )z m (t) + Q m2 zˆ m (t),

(7.46c)

p(t) ˆ = (Q p1 + Q p2 L p )z p (t) + Q p2 zˆ p (t),

(7.46d)

where m(t) ˆ and p(t) ˆ are the reconstructed states of m(t) ˜ and p(t), ˜ respectively, and     Q m = Q m1 Q m2 , Q p = Q p1 Q p2 . Remark 7.7 The reduced-order state observer (7.46) is n − q dimensional, which has more simpler structure than full-order ones. In addition, the observer gains can be obtained by using the MATLAB Toolbox LMI or YALMIP.

7.3.2 A Numerical Example Now we will present the availability of the reduced-order state observers proposed previously by the following numerical example. Example 7.8 Consider delayed GRN (7.1) with network measurements (7.3), where A = diag(2.8, 3.1, 3.2), D = diag(0.9, 0.9, 0.9),

7.3 Reduced-Order State Observer

177

⎡

⎤ 0 0 −1.4 0 ⎦, C = diag(2.6, 2.5, 2.6), W = ⎣−1.4 0 0 −1.4 0

M=

    0.8 −0.5 0.1 0.8 0.5 0.2 , N= . 1.3 0.4 −0.2 −0.3 0.7 −0.1

Set ⎡

⎤ 0.4336 0.4997 0.0592 Q m = ⎣−1.2376 0.7486 0.2859⎦ , 0.3432 −0.2547 0.9564 ⎡

⎤ 0.9230 −0.6493 −0.2584 Q p = ⎣0.4292 1.1210 0.0272 ⎦ . 0.2349 −0.2053 0.9657 Then M Q m = N Q p = [I2 0]. Take K 1 = diag(0.65, 0.65) and K 2 = 0.65. When τ¯ = σ¯ = 30 and τ¯d = σ¯ d = 30, by utilizing the Toolbox YALMIP of MATLAB, one can see that the LMIs in (7.20) and (7.21) are feasible. To save space, we only list partial solution matrices as follows: Q 1 = 4.2364 × 10−5 , Q 2 = 0.4039, R1 = 4.6589 × 10−4 , R2 = 0.0055, W1 = 7.2789 × 10−6 , W2 = 0.0014, H1 = 7.7074 × 10−6 , H2 = 5.6198 × 10−4 , P1 = 1.7770, P2 = 5.5429,     N1 = −0.6143 0.4555 , N2 = 103 × 1.8563 −8.1048 . By Theorem 7.5, the corresponding observer gains are   Lm = P1−1 N1 = −0.3457 0.2563 ,

178

7 State Estimation for Delayed GRNs 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0

2

4

6

8

10 12 14 16 18 20

60 40 20 0 -20 -40 -60 -80 -100 -120 -140

0

2

4

6

8

10 12 14 16 18 20

Fig. 7.4 The mRNA and protein concentrations and their estimations (Example 7.8)

  L p = P2−1 N2 = 103 × 0.3349 −1.4622 . Again applying Theorem 7.6, one can obtain a reduced-order state observer of GRN (7.1) as follows: z˙ˆ m (t) = −[0.0148 − 1.4591]Φ1 (ˆz p (t − σ(t)) + [0.3349 − 1.4622]z p (t − σ(t))) −0.0035Φ2 (ˆz p (t − σ(t)) + [0.3349 − 1.4622]z p (t − σ(t))) −3.1996ˆz m (t) + [−0.0010 0.0014]z m (t),

(7.47a)

z˙ˆ p (t) = −0.2712ˆz p (t) + 6.3210 × 10−8 zˆ m (t − τ (t)) −103 × [0.8166 − 3.3092]z p (t) +103 × [−1.2950 0.3076]z m (t − τ (t)), ⎡ ⎤ ⎤ 0.0592 0.4131 0.5149 m(t) ˆ = ⎣−1.3364 0.8219 ⎦ z m (t) + ⎣0.2859⎦ zˆ m (t), 0.9564 0.0126 −0.0096 ⎡ ⎤ ⎡ ⎤ −0.0856 0.3772 −0.2584 p(t) ˆ = 103 × ⎣ 0.0095 −0.0387⎦ z p (t) + ⎣ 0.0272 ⎦ zˆ p (t). 0.3236 −1.4122 0.9657

(7.47b)

⎡

(7.47c)

(7.47d)

When the initial function  T  T m(t) ˜ ≡ −0.5 0.3 0.3 , p(t) ˜ ≡ −0.2 −0.1 0.1 , zˆ m (t) ≡ 0, zˆ p (t) ≡ 0 for any t ∈ [−1, 0], and the delays σ(t) = 0.9 cos2 (5/9t) + 0.1, τ (t) = 0.9 sin2 (5/9t) + 0.1, t  0,

7.3 Reduced-Order State Observer 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 0

2

4

6

8

10 12 14 16 18 20

179 14 12 10 8 6 4 2 0 -2 -4 -6 0

2

4

6

8

10 12 14 16 18 20

Fig. 7.5 The mRNA and protein concentrations and their estimations (Example 7.8) 0.3

500

0.2

400

0.1

300

0

200

-0.1

100

-0.2 -0.3

0

-0.4

-100

-0.5 0

2

4

6

8

10 12 14 16 18 20

-200

0

2

4

6

8

10 12 14 16 18 20

Fig. 7.6 The mRNA and protein concentrations and their estimations (Example 7.8) 0.45

200

0.4

100

0.35

0

0.3 0.25

-100

0.2

-200

0.15

-300

0.1

-400

0.05 0 0

2

4

6

8

10 12 14 16 18 20

Fig. 7.7 Estimation errors (Example 7.8)

-500 0

2

4

6

8

10 12 14 16 18 20

180

7 State Estimation for Delayed GRNs

the state responses of considered GRN, the reconstructed states and corresponding error systems are given in Figs. 7.4, 7.5, 7.6 and 7.7, respectively, which shows the effectiveness of the approach proposed in this section. Remark 7.9 From the example, we have seen that the computation time of the reduceorder observer gains is more shorter than one of the full-order observer gains. This means the obtained reduce-order state observer is more effective than the full-order ones in practical applications. Importantly, due to the lower dimensions of the reduce observer, the cost for constructing the observer is lower in the practical engineering applications.

7.4 Remarks and Notes In this chapter, the problems of designing full- and reduced-order state observers for GRNs with time-varying delays have been investigated. The results related to fulland reduced-order state observers are taken from [12] and [13], respectively. We end this chapter by introducing the following items, which are related to this chapter: 1. Mohammadian et al. [7] proposed an LPV approach to designing a nonlinear state estimator for a class of delayed GRNs. By using an appropriate LKF, sufficient conditions guaranteeing the asymptotic stability of estimation errors was obtained. Furthermore, a sufficient condition for achieving the peak-to-peak state estimation was derived. In the delay-free case, the results in [7] improve the ones in [3] and [11]. 2. Anbuvithya et al. [1] investigated the sampled-data state estimation problem for a class of delayed GRNs by using the LKF method and LMI approach. They first converted the sampling period into a bounded time-varying delay, and then designed a state estimator to guarantee that the resulting error system, containing two different time delays, is globally asymptotically stable. 3. The problem of designing state observers for delayed GRNs has been addressed in [9, 10] 4. Liang et al. [5] investigated the state estimation problem for the uncertain delayed GRNs with Markovian jumping parameters by utilizing the LKF method and some stochastic analysis technique.

References 1. Anbuvithya, R., Mathiyalagan, K., Sakthivel, R., Prakash, P.: Sampled-data state estimation for genetic regulatory networks with time-varying delays. Neurocomputing 151, 737–744 (2015) 2. Lakshmanan, S., Park, H., Jung, H.Y., Balasubramaniam, P., Lee, S.M.: Design of state estimator for genetic regulatory networks with time-varying delays and randomly occurring uncertainties. Biosystems 111(1), 51–70 (2013)

References

181

3. Lee, T.H., Lakshmanan, S., Park, J.H., Balasubramaniam, P.: State estimation for genetic regulatory networks with mode-dependent leakage delays, time-varying delays, and Markovian jumping parameters. IEEE Trans. Nanobiosci. 12(4), 363–375 (2013) 4. Liang, J.L., Lam, J.: Robust state estimation for stochastic genetic regulatory networks. Int. J. Syst. Sci. 41(1), 47–63 (2010) 5. Liang, J.L., Lam, J., Wang, Z.D.: State estimation for Markov-type genetic regulatory networks with delays and uncertain mode transition rates. Phys. Lett. A 373(47), 4328–4337 (2009) 6. Lv, B., Liang, J.L., Cao, J.D.: Robust distributed state estimation for genetic regulatory networks with Markovian jumping parameters. Commun. Nonlinear Sci. Numer. Simul. 16(10), 4060– 4078 (2011) 7. Mohammadian, M., Momeni, H.R., Karimi, H.S., Shafikhani, I., Tahmasebi, M.: An LPV based robust peak-to-peak state estimation for genetic regulatory networks with time varying delay. Neurocomputing 160, 261–273 (2015) 8. Sun, J., Liu, G.P., Chen, J.: Delay-dependent stability and stabilization of neutral time-delay systems. Int. J. Robust Nonlinear Control 19(12), 1364–1375 (2009) 9. Tian, L.P., Wang, Z.J., Mohammadbagheri, A., Wu, F.X.: State observer design for delayed genetic regulatory networks. Comput. Math. Meth. Med. 2014 (Article ID 761562, 7 pages, 2014) 10. Vembarasan, V., Nagamani, G., Balasubramaniam, P., Park, J.H.: State estimation for delayed genetic regulatory networks based on passivity theory. Math. Biosci. 244(2), 165–175 (2013) 11. Wu, Y.W., Hu, L.J., Rong, C.G., Sheng, D.Z., He, Z.Q.: Estimating uncertain delayed genetic regulatory networks: an adaptive filtering approach. IEEE Trans. Autom. Control 54(4), 892– 897 (2009) 12. Xue, Y., Cui, S.C., Zhang, X.: Design of state observer for genetic regulatory networks with interval time-varying delays. In: Proceedings of the 34th Chinese Control Conference (CCC), pp. 2901–2906 (2015) 13. Zhang, X., Fan, X., Wu, L.: Reduced- and full-order observers for delayed genetic regulatory networks. IEEE Trans. Cybern. 48(7), 1989–2000 (2018)

Chapter 8

Guaranteed Cost Control for Delayed GRNs

This chapter addresses the problem of state feedback guaranteed cost control for uncertain GRNs with interval time-varying delays. The involved norm-bounded uncertainties are first transformed into external disturbances, and then an LKF approach combined with the convex technique and cone complementarity linearization technique is proposed to investigate a sufficient condition for the existence of expected controllers. Thereby, we design a state feedback guaranteed cost controller which guarantees the resultant closed-loop system is robustly asymptotically stable and its linear quadratic performance has an upper bound. A numerical example is provided to show the effectiveness of the proposed method.

8.1 Problem Formulation Consider the GRN model (1.8), that is ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(8.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(8.1b)

Here f : Rn → Rn is a continuous function satisfying the following sector condition: f j (0) = 0, 0 

f j (s)  k j , ∀0 = s ∈ R, j ∈ n, s

(8.2)

where k j is a nonnegative scalar, and f j (s) is the jth entry of f (s). Let K = diag(k1 , k2 , . . . , kn ). © Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_8

183

184

8 Guaranteed Cost Control for Delayed GRNs

When we add control inputs and parameter uncertainties into the model (8.1), the following delayed uncertain GRNs (8.3) can be obtained: ˙˜ m(t) = (−A + ΔA)m(t) ˜ + (W + ΔW ) f ( p(t ˜ − σ(t))) + u m (t), t  0, (8.3a) ˙˜ = (−C + ΔC) p(t) p(t) ˜ + (D + ΔD)m(t ˜ − τ (t)) + u p (t), t  0,

(8.3b)

˜ = Φ2 (s), s ∈ [−d, 0], m(s) ˜ = Φ1 (s), p(s)

(8.3c)

where d = max{τ2 , σ2 }, ΔA, ΔW , ΔC and ΔD are norm-bounded uncertainties satisfying the following Assumption 8.1, u m (t) and u p (t) are the control inputs, Φ1 (s) and Φ2 (s) are the continuous differential initial functions, σ : [0, +∞) → [σ1 , σ2 ] and τ : [0, +∞) → [τ1 , τ2 ] are continuous functions representing the feedback regulation delay and the translation delay, respectively, where 0 < τ1 < τ2 and 0 < σ1 < σ2 are known constants. Assumption 8.1 The uncertain matrices ΔA, ΔW , ΔC and ΔD possess the following form: [ΔA ΔW ] = G m Hm (t)[A0 W0 ],

(8.4)

[ΔC ΔD] = G p H p (t)[C0 D0 ],

(8.5)

where G m , G p , A0 , W0 , C0 and D0 are known constant matrices of appropriate sizes, and Hm (t) and H p (t) are uncertainties satisfying HmT (t)Hm (t)  I, H pT (t)H p (t)  I.

(8.6)

The objective of this chapter is to design the following state feedback guaranteed cost controller: ˜ u p (t) = K p p(t), ˜ u m (t) = Km m(t),

(8.7)

where Km and K p represent state feedback gains of appropriate sizes. By applying the state feedback controller (8.7) to GRN (8.3), we obtain the following closed-loop system: ˙˜ ˜ + (W + ΔW ) f ( p(t ˜ − σ(t))), t  0, (8.8a) m(t) = (−A + Km + ΔA)m(t) ˙˜ = (−C + K p + ΔC) p(t) ˜ + (D + ΔD)m(t ˜ − τ (t)), t  0, p(t)

(8.8b)

˜ = Φ2 (s), s ∈ [−d, 0]. m(s) ˜ = Φ1 (s), p(s)

(8.8c)

8.1 Problem Formulation

185

The closed-loop system (8.8) can be rewriten as the following form: ˙˜ ˜ + W f ( p(t ˜ − σ(t))) + G m u 1 (t), t  0, m(t) = (−A + Km )m(t)

(8.9a)

˙˜ = (−C + K p ) p(t) ˜ + D m(t ˜ − σ(t)) + G p u 2 (t), t  0, p(t)

(8.9b)

u 1 (t) = Hm (t)ν1 (t), u 2 (t) = H p (t)ν2 (t), t  0,

(8.9c)

˜ + W0 f ( p(t ˜ − σ(t))), t  0, ν1 (t) = A0 m(t)

(8.9d)

˜ + D0 m(t ˜ − σ(t)), t  0, ν2 (t) = C0 p(t)

(8.9e)

˜ = Φ2 (s), s ∈ [−d, 0]. m(s) ˜ = Φ1 (s), p(s)

(8.9f)

Definition 8.1 Define a performance index Jc of system (8.9) by 

∞

Jc =

X T AX dt

(8.10)

0

with ˜ u p (t)), A = diag(Q (1) , R (1) , Q (2) , R (2) ), X = col(m(t), ˜ u m (t), p(t), where (Q (i) )T = Q (i) > 0 and (R (i) )T = R (i) > 0 (i = 1, 2) are known real matrices. The state feedback controller (8.7) is said to be a guaranteed cost controller for GRN (8.3), if (i) The closed-loop system (8.9) is robustly asymptotically stable; (ii) There exists a positive scalar J ∗ such that Jc  J ∗ . The aim of this chapter is to design a state feedback guaranteed cost controller for GRN (8.3), that is, find state feedback gains Km and K p such that the closed-loop system (8.9) is robustly asymptotically stable and the performance index Jc has an upper bound J ∗ .

8.2 Design of Guaranteed Cost Controller In this section, we will first investigate Theorem 8.2 below which provides a sufficient condition for the existence of state feedback guaranteed cost controllers for GRN (8.3). Then we will present the procedure of designing a state feedback controller which ensures the closed-loop system (8.9) to be robustly asymptotically stable and the linear quadratic performance Jc has an upper bound J ∗ .

186

8 Guaranteed Cost Control for Delayed GRNs

8.2.1 Existence of Guaranteed Cost Controllers The following theorem offers a sufficient condition for the existence of state feedback guaranteed cost controllers. Theorem 8.2 For given scalars τ1 , τ2 , σ1 and σ2 , if there exist scalars λi > 0 (i = 1, 2), a diagonal matrix S > 0, and matrices Q iT = Q i > 0 (i = 1, 2), R Tj = R j > 0, Z Tj = Z j > 0 ( j ∈ 4), Km and K p such that ψi j := Σi j + ψ1 + Υ + ψ2 < 0, i, j = 1, 2 with ψ1 = e1T Q 1 Δ1 + ΔT1 Q 1 e1 + e7T Q 2 Δ2 + ΔT2 Q 2 e7 + e1T R1 e1 + e7T R3 e7 T −e3T (R1 − R2 )e3 − e4T R2 e4 − e9T (R3 − R4 )e9 − e10 R4 e10 ,

ψ2 = e1T (Q (1) + KmT R (1) Km )e1 + e7T (Q (2) + KTp R (2) K p )e7 , Σ11 = Σ − (e2 − e4 )T Z 2 (e2 − e4 ) − (e8 − e10 )T Z 4 (e8 − e10 ), Σ12 = Σ − (e2 − e4 )T Z 2 (e2 − e4 ) − (e9 − e8 )T Z 4 (e9 − e8 ), Σ21 = Σ − (e3 − e2 )T Z 2 (e3 − e2 ) − (e8 − e10 )T Z 4 (e8 − e10 ), Σ22 = Σ − (e3 − e2 )T Z 2 (e3 − e2 ) − (e9 − e8 )T Z 4 (e9 − e8 ), 2 2 Z 2 )Δ1 + ΔT2 (σ12 Z 3 + σ12 Z 4 )Δ2 Σ = ΔT1 (τ12 Z 1 + τ12 T T −(e1 − e3 ) Z 1 (e1 − e3 ) − (e2 − e4 ) Z 2 (e2 − e4 )

−(e3 − e2 )T Z 2 (e3 − e2 ) − (e7 − e9 )T Z 3 (e7 − e9 ) −(e9 − e8 )T Z 4 (e9 − e8 ) − (e8 − e10 )T Z 4 (e8 − e10 ), Υ = e5T S K e8 + e8T K Se5 − 2e5T Se5 + λ1 ΔT3 Δ3 T −λ1 e6T e6 + λ2 ΔT4 Δ4 − λ2 e11 e11 , Δ1 = (−A + Km )e1 + W e5 + G m e6 , Δ2 = (−C + K p )e7 + De2 + G p e11 , Δ3 = A0 e1 + W0 e5 , Δ4 = C0 e7 + D0 e2 , ei = [0n×(i−1)n In 0n×((9−i)n+l1 +l2 ) ], i ∈ 5, e6 = [0l1 ×5n Il1 0l1 ×(4n+l2 ) ],

(8.11)

8.2 Design of Guaranteed Cost Controller

187

e j = [0n×(( j−2)n+l1 ) In 0n×((9− j+1)n+l2 ) ], j = 7, 8, 9, 10, e11 = [0l2 ×(9n+l1 ) Il2 ], τ12 = τ2 − τ1 , σ12 = σ2 − σ1 , ˜ and u p (t) = K p p(t), ˜ is a guarthen the state feedback controller, u m (t) = Km m(t) anteed cost controller for GRN (8.3). Furthermore, if the inequalities in (8.11) are feasible, then the performance index Jc has an upper bound J ∗ defined by J ∗ = Φ1T (0)Q 1 Φ1 (0) + Φ2T (0)Q 2 Φ2 (0)  0  −τ1 T + Φ1 (s)R1 Φ1 (s)ds + Φ1T (s)R2 Φ1 (s)ds  +

−τ1 0

−σ1



+τ1

Φ2T (s)R3 Φ2 (s)ds + 0



0

−τ1 s −τ1  0

 +σ1 +σ12

−σ2

Φ2T (s)R4 Φ2 (s)ds

T Φ˙1 (θ)Z 1 Φ˙1 (θ)dθds

 +τ12

−τ2 −σ1



s −τ2 0  0

T Φ˙1 (θ)Z 2 Φ˙1 (θ)dθds

T Φ˙2 (θ)Z 3 Φ˙2 (θ)dθds

−σ1 s  −σ1  0

T Φ˙2 (θ)Z 4 Φ˙2 (θ)dθds.

(8.12)

V (t) = V1 (t) + V2 (t) + V3 (t)

(8.13)

−σ2

s

Proof Choose an LKF as follows:

with ˜ + p˜ T (t)Q 2 p(t), ˜ V1 (t) = m˜ T (t)Q 1 m(t)  t  t−τ1 m˜ T (s)R1 m(s)ds ˜ + m˜ T (s)R2 m(s)ds ˜ V2 (t) = t−τ1



+

t t−σ1

t−τ2  t−σ1

p˜ T (s)R3 p(s)ds ˜ +

t−σ2

p˜ T (s)R4 p(s)ds, ˜

188

8 Guaranteed Cost Control for Delayed GRNs

 V3 (t) = τ1

0 −τ1



+τ12  +σ1 +σ12



t

˙˜ m˙˜ T (θ)Z 1 m(θ)dθds

t+s −τ1  t

˙˜ m˙˜ T (θ)Z 2 m(θ)dθds

−τ2 t+s 0  t

˙˜ p˙˜ T (θ)Z 3 p(θ)dθds

−σ1 t+s  −σ1  t −σ2

˙˜ p˜˙ T (θ)Z 4 p(θ)dθds.

t+s

Set ˜ − τ2 ), f ( p(t ˜ − σ(t))), ξ(t) = col(m(t), ˜ m(t ˜ − τ (t)), m(t ˜ − τ1 ), m(t u 1 (t), p(t), ˜ p(t ˜ − σ(t)), p(t ˜ − σ1 ), p(t ˜ − σ2 ), u 2 (t)). Next we compute the derivatives of Vi (t)(i = 1, 2) along the trajectories of system (8.9) for any t  0. It is easy to see that ˜ + 2 p˙˜ T (t)Q 2 p(t) ˜ V˙1 (t) = 2m˙˜ T (t)Q 1 m(t) = ξ T (t)(e1T Q 1 Δ1 + ΔT1 Q 1 e1 + e7T Q 2 Δ2 + ΔT2 Q 2 e7 )ξ(t), ˜ − m˜ T (t − τ1 )(R1 − R2 )m(t ˜ − τ1 ) V˙2 (t) = m˜ T (t)R1 m(t) −m˜ T (t − τ2 )R2 m(t ˜ − τ2 ) + p˜ T (t)R3 p(t) ˜ T − p˜ (t − σ1 )(R3 − R4 ) p(t ˜ − σ1 ) − p˜ T (t − σ2 )R4 p(t ˜ − σ2 ) = ξ T (t)(e1T R1 e1 + e7T R3 e7 − e3T (R1 − R2 )e3 − e4T R2 e4 T −e9T (R3 − R4 )e9 − e10 R4 e10 )ξ(t). Then V˙1 (t) + V˙2 (t) = ξ T (t)ψ1 ξ(t).

(8.14)

Due to Lemma 1.14 and the method in [14], the derivative of V3 (t) along the trajectories of system (8.9) satisfies ˙˜ V˙3 (t) = τ12 m˙˜ T (t)Z 1 m(t) − τ1 2 ˙T ˙˜ +τ12 m˜ (t)Z 2 m(t)

˙˜ +σ12 p˙˜ T (t)Z 3 p(t)



t

˙˜ m˜˙ T (s)Z 1 m(s)ds

t−τ1



− τ12 

− σ1

2 ˙T ˙˜ − σ12 +σ12 p˜ (t)Z 4 p(t)

t−τ1

˙˜ m˙˜ T (s)Z 2 m(s)ds

t−τ2 t

˙˜ p˙˜ T (s)Z 3 p(s)ds

t−σ1  t−σ1 t−σ2

˙˜ p˙˜ T (s)Z 4 p(s)ds

8.2 Design of Guaranteed Cost Controller

189

 ξ T (t)[Σ − α(t)(e2 − e4 )T Z 2 (e2 − e4 ) −(1 − α(t))(e3 − e2 )T Z 2 (e3 − e2 ) −β(t)(e8 − e10 )T Z 4 (e8 − e10 ) −(1 − β(t))(e9 − e8 )T Z 4 (e9 − e8 )]ξ(t), 1 and β(t) = where α(t) = τ (t)−τ τ12 From (8.2), we obtain that

(8.15)

σ(t)−σ1 . σ12

˜ − σ(t)))S[ f ( p(t ˜ − σ(t))) − K p(t ˜ − σ(t))]  0. − 2 f T ( p(t

(8.16)

From Assumption 8.1 and (8.9), we obtain that u T1 u 1  ν1T ν1 and u T2 u 2  ν2T ν2 , and hence the following inequalities hold. λ1 ξ T (t)(ΔT3 Δ3 − e6T e6 )ξ(t)  0,

(8.17a)

T e11 )ξ(t)  0. λ2 ξ T (t)(ΔT4 Δ4 − e11

(8.17b)

Utilizing (8.14)–(8.17), we can derive the following inequality: V˙ (t)  ξ T (t)[ψ1 + Σ + Υ − α(t)(e2 − e4 )T Z 2 (e2 − e4 ) −(1 − α(t))(e3 − e2 )T Z 2 (e3 − e2 ) − β(t)(e8 − e10 )T Z 4 (e8 − e10 ) −(1 − β(t))(e9 − e8 )T Z 4 (e9 − e8 )]ξ(t). (8.18) Since 0  α(t)  1 and 0  β(t)  1, if the inequalities in (8.11) hold, then V˙ (t) < 0. Then according to Lemmas 1.19 and 1.20, it is easy to see that system (8.9) is robustly asymptotically stable, and hence V (∞) = 0. Furthermore, we can also get from (8.10), (8.11) and (8.18) that 

∞

Jc  −

V˙ (s)ds = V (0).

0

Now, using (8.12) and (8.13), we obtain that Jc  J ∗ . The proof is completed. Remark 8.3 In Theorem 8.2, in order to avoid large and complex computation in solving the inequalities in (8.11), we remove the free-weighting matrices which have been used in some literature, and use the convex representation of V˙ (t) to reduce the computational burden as well as the conservatism.

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8 Guaranteed Cost Control for Delayed GRNs

8.2.2 Design Method In order to obtain the expected state feedback guaranteed cost controller ˜ u p (t) = K p p(t) ˜ u m (t) = Km m(t), from the conclusion of Theorem 8.2, we must deal with the nonlinearity in (8.11). The following theorem gives a method to deal with the nonlinearity, and works out the state feedback guaranteed cost controller. Theorem 8.4 For given τ1 , τ2 , σ1 and σ2 , if there exist scalars λi > 0 (i = 1, 2), a diagonal matrix S > 0, matrices Q˜ iT = Q˜ i > 0, Q iT = Q i > 0 (i = 1, 2), Z˜ Tj = Z˜ j > 0, Z Tj = Z j > 0, R Tj = R j > 0 ( j ∈ 4), T˜lT = T˜l > 0, TlT = Tl > 0 (l ∈ 6), K˜ 1 and K˜ 2 such that ⎤ ⎡ ˜ ψi j τ1 Δ¯ T1 τ12 Δ¯ T1 σ1 Δ¯ T2 σ12 Δ¯ T2 ( K˜ 1 e1 )T ( K˜ 2 e7 )T ⎢ τ1 Δ¯ 1 −T1 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ τ12 Δ¯ 1 0 −T2 0 0 0 0 ⎥ ⎥ ⎢ ⎢ σ1 Δ¯ 2 0 0 −T3 0 0 0 ⎥ ⎥ < 0, ⎢ ⎥ ⎢σ12 Δ¯ 2 0 0 0 −T 0 0 4 ⎥ ⎢ ⎣ K˜ 1 e1 0 0 0 0 −T5 0 ⎦ K˜ 2 e7 0 0 0 0 0 −T6 i, j = 1, 2,



−T˜i Q˜ 1 −T˜ j Q˜ 2  0, i = 1, 2,  0, j = 3, 4, Q˜ T1 − Z˜ i Q˜ T − Z˜ j

2

−T˜5 Q˜ 1 Q˜ 2 −T˜6  0,  0, Q˜ T1 −(R (1) )−1 Q˜ T2 −(R (2) )−1 Q i Q˜ i = I (i = 1, 2), Z j Z˜ j = I ( j ∈ 4), Tl T˜l = I (l ∈ 6),

(8.19) (8.20) (8.21) (8.22)

where ψ˜i j = ψ1 + Σ˜ i j + Υ + e1T Q (1) e1 + e7T Q (2) e7 , 2 2 Σ˜ i j = Σi j − τ12 ΔT1 Z 1 Δ1 − τ12 ΔT1 Z 2 Δ1 − σ12 ΔT2 Z 3 Δ2 − σ12 ΔT2 Z 4 Δ2 ,

Δ¯ 1 = (−Q 1 A + K˜ 1 )e1 + Q 1 W e5 + Q 1 G m e6 , Δ¯ 2 = (−Q 2 C + K˜ 2 )e7 + Q 2 De2 + Q 2 G p e11 , then the state feedback controller, u m (t) = Q˜ 1 K˜ 1 m(t) ˜ and u p (t) = Q˜ 2 K˜ 2 p(t), ˜ is a guaranteed cost controller for GRN (8.3), and the performance index Jc has an upper bound J ∗ defined by (8.12).

8.2 Design of Guaranteed Cost Controller

191

Proof By Lemma 1.7, the inequalities (8.11) in Theorem 8.2 are equivalent to ⎡

⎤ ψ˜i j Δˆ T (Km e1 )T (K p e7 )T ⎢ Δˆ − Zˆ −1 ⎥ 0 0 ⎢ ⎥ < 0, i, j = 1, 2, ⎣Km e1 0 −(R (1) )−1 ⎦ 0 (2) −1 K p e7 0 0 −(R ) where Zˆ = diag(Z 1 , Z 2 , Z 3 , Z 4 ), Δˆ = col(τ1 Z 1 Δ1 , τ12 Z 2 Δ1 , σ1 Z 3 Δ2 , σ12 Z 4 Δ2 ). Then, we have ⎡ ˜ ⎤ ψi j τ1 Δ¯ T1 τ12 Δ¯ T1 σ1 Δ¯ T2 σ12 Δ¯ T2 ( K˜ 1 e1 )T ( K˜ 2 e7 )T ⎢ τ1 Δ¯ 1 −1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ τ12 Δ¯ 1 0 −2 0 0 0 0 ⎥ ⎢ ⎥ ⎢ σ1 Δ¯ 2 0 0 −3 0 0 0 ⎥ ⎢ ⎥ < 0, ⎢σ12 Δ¯ 2 0 0 0 −4 0 0 ⎥ ⎢ ⎥ ⎣ K˜ 1 e1 0 0 0 0 −5 0 ⎦ K˜ 2 e7 0 0 0 0 0 −6 i, j = 1, 2, where 1 = Q 1 Z 1−1 Q 1 , 2 = Q 1 Z 2−1 Q 1 , 3 = Q 2 Z 3−1 Q 2 , 4 = Q 2 Z 4−1 Q 2 , 5 = Q 1 (R (1) )−1 Q 1 , 6 = Q 2 (R (2) )−1 Q 2 . By (8.20)–(8.22), it is easy to see that (8.19) implies the inequalities (8.11) in Theorem 8.2. The proof is completed. Remark 8.5 The conditions in Theorem 8.4 are not all of LMI forms due to (8.22), so they cannot be solved directly using the Toolbox YALMIP of MATLAB. However, with the cone complementarity linearization technique in [3, 4], we can solve these nonconvex feasibility problems by formulating them into some sequential optimization problems subject to LMI constraints. In order to solve the state feedback guaranteed cost controller, a cone complementarity linearization algorithm is proposed based on Theorem 8.4 as follows: Algorithm 8.1 For given σ1 , σ2 and τ1 , find the maximum value of τ2 , which makes the LMIs (8.19)–(8.21) with constraint (8.22) are feasible. Step 1 Choose an initial τ2 > τ1 , such that there exists a feasible solution to (8.19)– (8.21) and

192

8 Guaranteed Cost Control for Delayed GRNs







Qi I Zj I Tl I  0,  0,  0, I Q˜ i I Z˜ j I T˜l

(8.23)

i = 1, 2, j ∈ 4, l ∈ 6. Set τmax = τ2 . Step 2 Find a feasible set of Q i0 , Q˜ i0 , Z j0 , Z˜ j0 , Tl0 , T˜l0 satisfying (8.19)–(8.21) and (8.23). Set k = 0. Step 3 Solve the following optimal problem for the variables K˜ 1 , K˜ 2 , Q i , Q˜ i (i = 1, 2), Z j , Z˜ j ( j ∈ 4), Tl and T˜l (l ∈ 6). min

s.t.(8.19)−(8.21),(8.23)

trΘk ,

where Θk =

2 4   (Q i Q˜ ik + Q ik Q˜ i ) + (Z j Z˜ jk + Z jk Z˜ j ) i=1

j=1

6  (Tl T˜lk + Tlk T˜l ). + l=1

Set Q i k+1 = Q i , Q˜ i T˜l k+1 = T˜l . Step 4 If the LMIs

k+1

= Q˜ i , Z j

k+1

= Z j , Z˜ j

k+1

= Z˜ j , Tl k+1 = Tl ,

⎤ Δˆ T Zˆ (K1 e1 )T R (1) (K2 e7 )T R (2) ψ˜i j ⎥ ⎢ Zˆ Δˆ − Zˆ 0 0 ⎥ < 0, i, j = 1, 2 (8.24) ⎢ ⎦ ⎣(R (1) )T K1 e1 0 −R (1) 0 0 −R (2) (R (1) )T K2 e7 0 ⎡

are feasible for the variables R j and Z j , and the matrices K1 = Q˜ 1 K˜ 1 and K2 = Q˜ 2 K˜ 2 obtained in Step 3, then set τmax = τ2 , increase τ2 by a small amount and return to Step 2. If the LMIs (8.24) are infeasible within a specified number of iteration, then stop; otherwise, set k = k + 1 and go to Step 3. Remark 8.6 From Theorem 8.4 and its proof, we can see that the controller obtained from Algorithm 8.1 ensures the closed-loop system to be robustly asymptotically stable, and the linear quadratic performance has an upper bound which depends on the initial functions of GRN (8.3). However, the dependency on the initial functions of GRN (8.3) can be eliminated under certain conditions. Indeed, assume that the initial functions of GRN (8.3) belong to the sets Yi = {Φi (s) : Φi (s) = U ωi (s), Φ˙ i (s) = V νi (s), ωiT (s)ωi (s)  1, νiT νi  1},

8.2 Design of Guaranteed Cost Controller

193

where i = 1, 2, U and V are given n × n matrices. Then the upper bound J ∗ defined by (8.12) satisfies J ∗ = λmax (U T Q 1 U ) + λmax (U T Q 2 U ) +τ1 λmax (U T R1 U ) + τ12 λmax (U T R2 U ) +σ1 λmax (U T R3 U ) + σ12 λmax (U T R4 U ) τ 2 (τ2 + τ1 ) τ3 λmax (V T Z 2 V ) + 1 λmax (V T Z 1 V ) + 12 2 2 σ 2 (σ2 + σ1 ) σ3 λmax (V T Z 4 V ). + 1 λmax (V T Z 3 V ) + 12 2 2

8.3 A Numerical Example In this section, we will illustrate the effectiveness and the superiority of our approach by a numerical example. Example 8.7 Consider GRN (8.3) with A = diag(6, 6), C = diag(5, 5), D = diag(1, 1.2),



0 −5 W = , Gm = Ga Gb , G p = Gc Gd , −5 0



0.04 −0.01 0 0 Ga = , Gb = , −0.01 0.03 −0.2 0



0.4 −0.1 0.03 −0.02 , Gd = , Gc = −0.1 0.3 −0.02 0.06 A0 = G a , C0 = G c , D0 = G d , W0 = diag(0.2, 0.2),

T

T Hm (t) = Ha (t) Hb (t) , H p (t) = Hc (t) Hd (t) , Ha (t) = diag(sint, cos(2t)), Hc (t) = −0.01Ha (t), Hb (t) = Hd (t) = diag(1, 1),

T

T Φ1 (s) = −2 3 , Φ2 (s) = 0.5 2 , ∀s ∈ [−d, 0]. Assume that the nonlinear function f satisfies (8.2) with K = 0.65I2 . Based on Algorithm 8.1, by solving the LMIs (8.19)–(8.21) with constraint (8.22), the maximums of τ2 for different τ1 are shown in Table 8.1, where σ1 = 0.5 and σ2 = 1. By Theorem 8.4, we know that system (8.9) is robustly asymptotically stable.

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8 Guaranteed Cost Control for Delayed GRNs

Table 8.1 The maximum values of τ2 (Example 8.7) τ1 τ2

0.1 1.7221

0.3 1.9221

0.5 2.1221

2

3

1.5

2

1

1

0

˜ 2 (t) m

˜ 1 (t) m

0.5 -0.5 -1

0 -1 -2

-1.5

-3

-2 -2.5

1 2.6221

0

10

20

30

40

50

-4

60

0

10

20

30

40

50

60

10

20

30

40

50

60

t

t

Fig. 8.1 Trajectories of mRNA concentrations (Example 8.7) 2

1.5

1.5

1

1

p˜2 (t)

p˜1 (t)

0.5 0 -0.5

0 -0.5 -1

-1 -1.5

0.5

-1.5 0

10

20

30

40

50

60

-2

0

t

t

Fig. 8.2 Trajectories of protein concentrations (Example 8.7)

When σ2 = 1, σ1 = τ1 = 0.5 and τ2 = 2, the state feedback gains Km and K p are as follows:



−13.4455 0.4413 −12.5327 0.6250 Km = , Kp = . 0.4241 −13.5764 0.6606 −12.8474 Furthermore, the upper bound, J ∗ , of the performance index Jc obtained by Theorem 8.4 is 89.8642. When U = V = I , according to Remark 8.6, the upper bound J ∗ = 0.4565 is derived. When σ(t) = 0.75 + 0.25 sin t and τ (t) = 1.25 + 0.75 cos t, the trajectories of the concentrations of mRNAs and proteins in the closed-loop are shown in Figs. 8.1 and 8.2. From which, it is clearly seen that the closed-loop trajectories of mRNA

8.3 A Numerical Example

195

and protein concentrations are convergent, which shows the effectiveness of this approach proposed in this chapter.

8.4 Remarks and Notes When we control a real plant, it is also desirable to design a control system which not only is stable but also guarantees an adequate level of performance. One approach to this problem is the so-called guaranteed cost control approach which is first introduced by Chang and Peng in 1972 [1]. This approach has the advantage of providing an upper bound on a given performance index, and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound. This chapter has proposed an approach to design state feedback guaranteed cost control for GRNs with interval time-varying delays. This chapter is taken from [2]. We end this chapter by introducing the following items, which are related to this chapter: 1. In [6], Hu et al. proposed an adaptive controller to stabilize GRNs with both finite distributed time delays and discrete time delays. 2. The problem of stabilization design and H∞ control for a class of GRNs with both intrinsic perturbation and extrinsic perturbation is investigated in [5]. 3. By constructing an LKF that contains some triple summation terms, Mathiyalagan and Sakthivel [11] addressed the problems of robust stabilization and H∞ control for stochastic GRNs with constant delays. A state feedback controller is designed to guarantee that the equilibrium point of resultant closed-loop system is meansquare asymptotically stable for all admissible uncertainties. 4. The stabilization of GRNs with leakage delays is concerned in [7]. The sampleddata controller is designed. 5. By using the mathematical induction method and the analysis techniques, Liu and Jiang [9] proposed a periodically intermittent controller to stabilize a class of GRNs with time-varying delays and finite distributed delays. 6. By constructing an appropriate LKF, the problem of finite-time H∞ control for discrete-time GRNs with random delays and partly unknown transition probabilities is addressed in [8]. 7. Pan et al. [13] investigated the problem of robust H∞ feedback control for uncertain stochastic delayed GRNs with additive and multiplicative noise. 8. In [12], a redesigned robust integral sliding mode controller is developed for a class of nonlinear dissipative switched GRNs with affine subsystems. 9. By invoking the input-delay approach and employing the LKF method, Liu et al. [10] investigated the problems of passivity and passification for Markovian jumping GRNs with time-varying delays by a non-uniform sampled-data approach.

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8 Guaranteed Cost Control for Delayed GRNs

References 1. Chang, S., Peng, T.: Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Autom. Control 17(4), 474–483 (1972) 2. Chen, L.L., Zhou, Y., Zhang, X.: Guaranteed cost control for uncertain genetic regulatory networks with interval time-varying delays. Neurocomputing 131, 105–112 (2014) 3. Ghaoui, L.E., Oustry, F., AitRami, M.: A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans. Autom. Control 42(8), 1171–1176 (1997) 4. He, Y., Wu, M., Liu, G.P., She, J.H.: Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Trans. Autom. Control 53(10), 2372–2377 (2008) 5. He, Y., Zeng, J., Wu, M., Zhang, C.K.: Robust stabilization and H∞ controllers design for stochastic genetic regulatory networks with time-varying delays and structured uncertainties. Math. Biosci. 236(1), 53–63 (2012) 6. Hu, J.Q., Liang, J.L., Cao, J.D.: Stabilization of genetic regulatory networks with mixed timedelays: an adaptive control approach. IMA J. Math. Control Inf. 32(2), 343–358 (2015) 7. Li, L., Yang, Y.Q.: On sampled-data control for stabilization of genetic regulatory networks with leakage delays. Neurocomputing 149, 1225–1231 (2015) 8. Liu, A.D., Yu, L., Zhang, D., Zhang, W.A.: Finite-time H∞ control for discrete-time genetic regulatory networks with random delays and partly unknown transition probabilities. J. Franklin Inst. 350(7), 1944–1961 (2013) 9. Liu, Z., Jiang, H.: Exponential stability of genetic regulatory networks with mixed delays by periodically intermittent control. Neural Comput. Appl. 21(6), 1263–1269 (2012) 10. Lu, L., Xing, Z., He, B.: Non-uniform sampled-data control for stochastic passivity and passification of Markov jump genetic regulatory networks with time-varying delays. Neurocomputing 171, 434–443 (2016) 11. Mathiyalagan, K., Sakthivel, R.: Robust stabilization and H∞ control for discrete-time stochastic genetic regulatory networks with time delays. Can. J. Phys. 90(10), 939–953 (2012) 12. Moradi, H., Majd, V.J.: Robust control of uncertain nonlinear switched genetic regulatory networks with time delays: a redesign approach. Math. Biosci. 275, 10–17 (2016) 13. Pan, W., Wang, Z., Gao, H., Li, Y., Du, M.: Robust H∞ feedback control for uncertain stochastic delayed genetic regulatory networks with additive and multiplicative noise. Int. J. Robust Nonlinear Control 20(18), 2093–2107 (2010) 14. Shao, H.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009)

Chapter 9

State Estimation for Delayed Reaction-Diffusion GRNs

This chapter addresses the problem of state estimation for delayed reaction-diffusion GRNs under Dirichlet boundary conditions.

9.1 Problem Formulation Consider the delayed reaction-diffusions GRN model (6.3), that is,   l  ∂ m(t, ˜ x) ∂ ∂ m(t, ˜ x) Dk − Am(t, ˜ x) = ∂t ∂xk ∂xk k=1 +W f ( p(t ˜ − σ(t), x)), t  0,

(9.1a)

  l  ∂ p(t, ˜ x) ∂ ˜ x) ∗ ∂ p(t, Dk − C p(t, ˜ x) = ∂t ∂xk ∂xk k=1 +D m(t ˜ − τ (t), x), t  0,

(9.1b)

where x = col(x1 , x2 , . . . , xl ) ∈ R ⊂ Rl ,  R = {x |xk |  L k , k ∈ l}, A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ), D = diag(d1 , d2 , . . . , dn ), Dk = diag(D1k , D2k , . . . , Dnk ), ∗ ∗ ∗ Dk∗ = diag(D1k , D2k , . . . , Dnk ),

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_9

197

198

9 State Estimation for Delayed Reaction-Diffusion GRNs

m(t, ˜ x) = col(m˜ 1 (t, x), m˜ 2 (t, x), . . . , m˜ n (t, x)), p(t, ˜ x) = col( p˜ 1 (t, x), p˜ 2 (t, x), . . . , p˜ n (t, x)), f ( p(t ˜ − σ(t), x)) = col( f 1 ( p˜ 1 (t − σ(t), x)), f 2 ( p˜ 2 (t − σ(t), x)), . . . , f n ( p˜ n (t − σ(t), x))), f i ( p˜ i (t − σ(t), x)) = gi ( p˜ i (t − σ(t), x) + pi∗ ) − gi ( pi∗ ), i ∈ n, ∗ > 0 denote the diffusion rate matrices. and L k is a constant, Dik > 0 and Dik The time-varying delays are assumed to satisfy

¯ σ(t) ˙  σ¯ d , 0  τ (t)  τ¯ , τ˙ (t)  τ¯d , 0  σ(t)  σ,

(9.2)

where τ¯ , σ, ¯ τ¯d and σ¯ d are known nonnegative real numbers. The Dirichlet boundary conditions and initial conditions associated with GRN (9.1) are given as follows: m(t, ˜ x) = 0, p(t, ˜ x) = 0, x ∈ ∂R, t ∈ [−d, +∞), m(s, ˜ x) = φ(s, x), p(s, ˜ x) = φ∗ (s, x), x ∈ R, s ∈ [−d, 0], where d = max{σ, ¯ τ¯ }, and φ(s, x), φ∗ (s, x) ∈ C 1 ([−d, 0] × R, Rn ). Because of the complexity of delayed reaction-diffusion GRNs, one can obtain only partial state information in the network outputs. To satisfy practical requirement, it is necessary to estimate the states based on network outputs. In the following we assume that the network outputs are ˜ x), z p (t, x) = N p(t, ˜ x), z m (t, x) = M m(t,

(9.3)

where M and N are known constant matrices of appropriate sizes, and z m (t, x) and z p (t, x) represent the expression levels of mRNAs and proteins at (t, x), respectively. To estimate the states of delayed reaction-diffusion GRN (9.1) through available measurement outputs in (9.3), we construct the following state observer:   l  ∂ m(t, ¯ x) ∂ ∂ m(t, ¯ x) Dk − Am(t, ¯ x) = ∂t ∂xk ∂xk k=1 +W f ( p(t ¯ − σ(t), x)) +Km [z m (t, x) − M m(t, ¯ x)], t ≥ 0

(9.4a)

  l  ∂ ¯ x) ∂ p(t, ¯ x) ∗ ∂ p(t, Dk − C p(t, ¯ x) + D m(t ¯ − τ (t), x) = ∂t ∂xk ∂xk k=1 +K p [z p (t, x) − N p(t, ¯ x)], t  0,

(9.4b)

9.1 Problem Formulation

199

where m(t, ¯ x) and p(t, ¯ x) are the estimations of m(t, ˜ x) and p(t, ˜ x), respectively, and Km and K p are the observer gains to be designed later. The Dirichlet boundary conditions and initial conditions associated with the state observer (9.4) are given as follows: m(t, ¯ x) = 0, p(t, ¯ x) = 0, x ∈ ∂R, t ∈ [−d, +∞), ¯ x), p(s, m(s, ¯ x) = φ(s, ¯ x) = φ¯ ∗ (s, x), x ∈ R, s ∈ [−d, 0], ¯ x), φ¯ ∗ (s, x) ∈ C 1 ([−d, 0] × R, Rn ). where d = max{σ, ¯ τ¯ }, and φ(s, ˜ x) − m(t, ¯ x) and e p (t, x) = p(t, ˜ x) − p(t, ¯ x) be the error Let em (t, x) = m(t, state vectors. Then, from (9.1), (9.3) and (9.4), the error system can be easily obtained as follows:  ∂ ∂em (t, x) = ∂t ∂xk k=1 l



∂em (t, x) Dk ∂xk

 − (A + Km M)em (t, x)

+W fˆ(e p (t − σ(t), x)), x ∈ R, t ∈ [−d, +∞),  ∂ ∂e p (t, x) = ∂t ∂xk k=1 l



Dk∗

∂e p (t, x) ∂xk

(9.5a)

 − (C + K p N )e p (t, x)

+Dem (t − τ (t), x), x ∈ R, t ∈ [−d, +∞),

(9.5b)

¯ x), e p (s, x) = φ∗ (s, x) − φ¯ ∗ (s, x), em (s, x) = φ(s, x) − φ(s, x ∈ R, s ∈ [−d, 0], em (t, x) = 0, e p (t, x) = 0, x ∈ ∂R, t ∈ [−d, +∞),

(9.5c) (9.5d)

where ˜ − σ(t), x)) − f ( p(t ¯ − σ(t), x)). fˆ(e p (t − σ(t), x)) = f ( p(t The relationship among fˆi , f i and gi implies that fˆi (s)  ki , ∀0 = s ∈ R, i ∈ n, fˆi (0) = 0, 0  s and hence fˆ(0) = 0, fˆT (z)( fˆ(z) − K z)  0, ∀z ∈ Rn , where K = diag(k1 , k2 , . . . , kn ) > 0.

(9.6)

200

9 State Estimation for Delayed Reaction-Diffusion GRNs

The aim of this chapter is to design state observer (9.4) for the delayed reactiondiffusion GRN (9.1), that is, find suitable observer gains Km and K p such that the error system (9.5) is asymptotically stable (respectively, finite-time stable).

9.2 Infinite-Time State Estimation In this section, by introducing novel integral terms into the LKF and employing the Wirtinger-type integral inequality, the convex technique, Green’s identity, the reciprocally convex technique and Wirtinger’s inequality, we will establish an LMIsbased asymptotic stability criterion of the error system (9.5). The stability criterion depends upon the bounds of delays and their derivatives. Thereby, a state observer was designed, and the observed gains were represented based on a feasible solution to the set of LMIs. Two numerical examples are presented to illustrate the availability and applicability of the proposed design scheme.

9.2.1 Observer Design In this subsection we will design a state observer in the form of (9.4) for the delayed reaction-diffusion GRN (9.1), that is, find a pair of observer gains Km and K p so that the trivial solution of (9.5) is asymptotically stable. For this end, we define e0 = 014n×n , ei = col(0(i−1)n×n , In , 0(14−i)n×n ), i ∈ 14,    t ϕ(t, s, x) = col em (s, x), em (s, x)ds ,   t−t τ¯ ψ(t, s, x) = col e p (s, x), e p (s, x)ds , t−σ¯

ς(t, x) = col(em (t, x), em (t − τ¯ , x), em (t − τ (t), x), e p (t, x), ¯ x), e p (t − σ(t), x), fˆ(e p (t, x)), e p (t − σ, ∂em (t, x) ∂e p (t, x) , , fˆ(e p (t − σ(t), x)), ∂t ∂t  t  t−τ (t) 1 1 em (s, x)ds, em (s, x)ds, τ (t) t−τ (t) τ¯ − τ (t) t−τ¯  t  t−σ(t) 1 1 e p (s, x)ds, e p (s, x)ds). σ(t) t−σ(t) σ¯ − σ(t) t−σ¯

9.2 Infinite-Time State Estimation

201

Theorem 9.1 Let scalars τ¯ , σ, ¯ τ¯d and σ¯ d be known and satisfy (9.2). Then the trivial solution of error system (9.5) is asymptotically stable if there exist matrices Q iT = Q i > 0 (i ∈ 5), RkT = Rk > 0 (k ∈ 4), M Tj = M j > 0 ( j ∈ 2) and G 1 , G 2 , W1 and W2 of appropriate sizes, and diagonal matrices P j > 0 and Λ j > 0 ( j ∈ 2) so that the following LMIs hold for τ ∈ {0, τ¯ } and σ ∈ {0, σ}: ¯   R˜ j G j ˆ  0, j ∈ 2, R j := G Tj R˜ j

(9.7)

Φ(τ , σ) = Φ0 + Φ1 + Φ2 (τ , σ) + Φ3 + Φ4 (τ , σ) + Φ5 (τ , σ) < 0,

(9.8)

where Φ0 = −2e7 Λ1 e7T + e4 Λ1 K eT7 + e7 K Λ1 e4T − 2e8 Λ2 e8T +e6 K Λ2 e8T + e8 Λ2 K e6T , Φ1 = sym(−e9 (P1 A + W1 M)e1T + e9 P1 W eT8 −e9 P1 e9T − e10 (P2 C + W2 N )e4T + e10 P2 DeT3 T −e10 P2 e10 − 0.25π 2 e1 P1 D L e1T − e1 (P1 A + W1 M)e1T +e1 P1 W eT8 − 0.25π 2 e4 P2 D L∗ e4T − e4 (P2 C + W2 N )e4T +e4 P2 DeT3 ), Φ2 (τ , σ) = e1 Q 1 e1T − (1 − τ¯d )e3 Q 1 e3T +e4 Q 3 e4T − (1 − σ¯ d )e6 Q 3 e6T +Δ1 Q 2 ΔT1 + τ (Δ1 Q 2 ΔT2 + Δ2 Q 2 ΔT1 ) −Δ3 Q 2 ΔT3 − τ (Δ3 Q 2 ΔT2 + Δ2 Q 2 ΔT3 ) +Δ4 Q 2 ΔT6 + Δ6 Q 2 ΔT4 +τ (Δ5 Q 2 ΔT6 + Δ6 Q 2 ΔT5 ) + Θ1 Q 4 Θ1T +σ(Θ1 Q 4 Θ2T + Θ2 Q 4 Θ1T ) − Θ3 Q 4 Θ3T −σ(Θ3 Q 4 Θ2T + Θ2 Q 4 Θ3T ) + Θ4 Q 4 Θ6T +Θ6 Q 4 Θ4T + σ(Θ5 Q 4 Θ6T + Θ6 Q 4 Θ5T ), Φ3 = e7 Q 5 e7T − (1 − σ¯ d )e8 Q 5 e8T , Φ4 (τ , σ) = Φ41 − Φ42 (τ ) − Φ43 (σ) −[Δ7 Δ8 ] Rˆ 1 [Δ7 Δ8 ]T − [Θ7 Θ8 ] Rˆ 2 [Θ7 Θ8 ]T , T Φ41 = τ¯ 2 e9 R1 e9T + σ¯ 2 e10 R2 e10 + τ¯ 2 e1 R3 e1T + σ¯ 2 e4 R4 e4T , T T Φ42 (τ ) = τ¯ (τ¯ − τ )e12 R3 e12 + τ¯ τ e11 R3 e11 , T T Φ43 (τ ) = σ( ¯ σ¯ − σ)e14 R4 e14 + σσe ¯ 13 R4 e13 ,

202

9 State Estimation for Delayed Reaction-Diffusion GRNs

Φ5 (τ , σ) = Φ51 − Φ52 − Φ53 τ¯ − τ σ¯ − σ Δ8 M˜ 1 ΔT8 − Θ8 M˜ 2 Θ8T , − τ¯ σ¯ τ¯ 2 σ¯ 2 T Φ51 = e9 M1 e9T + e10 M2 e10 , 2 2 Φ52 = (e1 − e11 )M1 (e1 − e11 )T + (e3 − e12 )M1 (e3 − e12 )T , Φ53 = (e4 − e13 )M2 (e4 − e13 )T + (e6 − e14 )M2 (e6 − e14 )T , Δ1 = [e1 τ¯ e12 ], Δ2 = [e0 e11 − e12 ], Δ3 = [e2 τ¯ e12 ], Δ4 = [τ¯ e12 τ¯ 2 e12 ], Δ5 = [e11 − e12 τ¯ (e11 − e12 )], Δ6 = [e0 e1 − e2 ], Δ7 = [e3 − e2 e3 + e2 − 2e12 ], Δ8 = [e1 − e3 e1 + e3 − 2e11 ], Θ1 = [e4 σe ¯ 14 ], Θ2 = [e0 e13 − e14 ], Θ3 = [e5 σe ¯ 14 ], Θ4 = [σe ¯ 14 σ¯ 2 e14 ], Θ5 = [e13 − e14 σ(e ¯ 13 − e14 )], Θ6 = [e0 e4 − e5 ], Θ7 = [e6 − e5 e6 + e5 − 2e14 ], Θ8 = [e4 − e6 e4 + e6 − 2e13 ], R˜ 1 = diag(R1 , 3R1 ), R˜ 2 = diag(R2 , 3R2 ), 1 1 M˜ 1 = diag(M1 , 3M1 ), M˜ 2 = diag(M2 , 3M2 ), τ¯ σ¯ l l   1 1 ∗ ∗ D , D = D , DL = L 2 k 2 k L L k k k=1 k=1 and A, B, C, L k , Dk , Dk∗ , W and K are defined as previously. Furthermore, when the inequalities (9.7) and (9.8) are feasible, the desired observer is given by (9.4) with Km = P1−1 W1 and K p = P2−1 W2 . Proof Construct the following LKF for the error system (9.5): V (t, em , e p ) =

5  i=1

Vi (t, em , e p ),

9.2 Infinite-Time State Estimation

203

where  V1 (t, em , e p ) =



R

+

emT (t, x)P1 em (t, x)dx

l  

+

l  

∂eTp (t, x) ∂xk

k=1  R 

V2 (t, em , e p ) =

R



+



eTp (t, x)P2 e p (t, x)dx

P2 Dk∗

∂e p (t, x) dx, ∂xk

emT (s, x)Q 1 em (s, x)dsdx ϕT (t, s, x)Q 2 ϕ(t, s, x)dsdx

R

t−σ(t)  t

R t

t−σ¯



R

t−τ (t)  t t−τ¯  t



V3 (t, em , e p ) =

t

R

+ +  

R

∂emT (t, x) ∂em (t, x) P1 Dk dx ∂xk ∂xk

R

k=1

+

eTp (s, x)Q 3 e p (s, x)dsdx

ψ T (t, s, x)Q 4 ψ(t, s, x)dsdx,

fˆT (e p (s, x))Q 5 fˆ(e p (s, x))dsdx,

t−σ(t) 0  t

∂emT (s, x) ∂em (s, x) R1 dsdθdx ∂s ∂s R −τ¯ t+θ   0 t ∂eTp (s, x) ∂e p (s, x) +σ¯ R2 dsdθdx ∂s ∂s R −σ¯ t+θ   0 t +τ¯ emT (s, x)R3 em (s, x)dsdθdx

V4 (t, em , e p ) = τ¯

R

−τ¯ 0

 

V5 (t, em , e p ) =

+σ¯   0 R0 −σ¯t

t+θ t



t+θ

eTp (s, x)R4 e p (s, x)dsdθdx,

∂emT (s, x) ∂em (s, x) M1 dsdλdθdx ∂s ∂s R −τ¯ θ t+λ   0  0 t ∂eTp (s, x) ∂e p (s, x) + M2 dsdλdθdx. ∂s ∂s R −σ¯ θ t+λ

Similar to the proof of Theorem 6.21, we can obtain  ς T (t, x)Φ0 ς(t, x)dx  0, R  ∂ Vk (t, em , e p )  ς T (t, x)Φk ς(t, x)dx, k = 1, 3, ∂t R ∂ V2 (t, em , e p ) ∂t

(9.9) (9.10)

204

9 State Estimation for Delayed Reaction-Diffusion GRNs

 =

emT (t, x)Q 1 em (t, x)dx  − (1 − τ˙ (t)) emT (t − τ (t), x)Q 1 em (t − τ (t), x)dx R  + eTp (t, x)Q 3 e p (t, x)dx R  − (1 − σ(t)) ˙ eTp (t − σ(t), x)Q 3 e p (t − σ(t), x)dx R  + ϕT (t, t, x)Q 2 ϕ(t, t, x)dx R  − ϕT (t, t − τ¯ , x)Q 2 ϕ(t, t − τ¯ , x)dx R   t ∂ϕ(t, s, x) dsdx ϕT (t, s, x)Q 2 +2 ∂t  R t−τ¯ + ψ T (t, t, x)Q 4 ψ(t, t, x)dx R  − ψ T (t, t − σ, ¯ x)Q 4 ψ(t, t − σ, ¯ x)dx R   t ∂ψ(t, s, x) dsdx ψ T (t, s, x)Q 4 +2 ∂t R t−σ¯   ς T (t, x)Φ2 (τ (t), σ(t))ς(t, x)dx, R

(9.11)

R

∂ V4 (t, em , e p ) = τ¯ 2 ∂t



∂emT (t, x) ∂em (t, x) R1 dx ∂t ∂t R   t ∂emT (s, x) ∂em (s, x) −τ¯ R1 dsdx ∂s ∂s R t−τ¯  ∂eTp (t, x) ∂e p (t, x) +σ¯ 2 R2 dx ∂t ∂t R   t ∂eTp (s, x) ∂e p (s, x) −σ¯ R2 dsdx ∂s ∂s R t−σ¯ +τ¯ 2 emT (t, x)R3 em (t, x)dx R   t emT (s, x)R3 em (s, x)dsdx −τ¯ R t−τ¯ 2 +σ¯ eTp (t, x)R4 e p (t, x)dx R   t eTp (s, x)R4 e p (s, x)dsdx, −σ¯ R

t−σ¯

(9.12)

9.2 Infinite-Time State Estimation

205

 ∂ ∂emT (t, x) ∂em (t, x) τ¯ 2 V5 (t, em , e p ) = M1 dx ∂t 2 R ∂t ∂t   0 t ∂emT (s, x) ∂em (s, x) − M1 dsdθdx ∂s ∂s R −τ¯ t+θ  ∂eTp (t, x) ∂e p (t, x) σ¯ 2 + M2 dx 2 R ∂t ∂t   0 t ∂eTp (s, x) ∂e p (s, x) M2 dsdθdx. − ∂s ∂s R −σ¯ t+θ

(9.13)

Note that   −τ¯

R

  = −τ¯

∂emT (s, x) ∂em (s, x) R1 dsdx ∂s ∂s

t−τ¯ t−τ (t)

R

t−τ¯ t

R

t−τ (t)

  −τ¯

t

∂emT (s, x) ∂em (s, x) R1 dsdx ∂s ∂s ∂emT (s, x) ∂em (s, x) R1 dsdx. ∂s ∂s

(9.14)

Applying Lemma 1.16, one can obtain that   −τ¯

R

t−τ (t)

∂emT (s, x) ∂em (s, x) R1 dsdx ∂s ∂s

t−τ¯  τ¯ ς T (t, x)Δ7 R˜ 1 ΔT7 ς(t, x)dx, − τ¯ − τ (t) R   t ∂emT (s, x) ∂em (s, x) R1 dsdx −τ¯ ∂s ∂s R t−τ (t)  τ¯ − ς T (t, x)Δ8 R˜ 1 Δ8 ς(t, x)dx. τ (t) R

This, together with (9.7), (9.14) and Lemma 1.9, implies that   t ∂emT (s, x) ∂em (s, x) R1 dsdx −τ¯ ∂s ∂s R t−τ¯  − ς T (t, x)[Δ7 Δ8 ] Rˆ 1 [Δ7 Δ8 ]T ς(t, x)dx.

(9.15)

R

Similarly,

 

t ∂eT (s, x) ∂e p (s, x) p R2 dsdx −σ¯ ∂s ∂s  R t−σ¯ − ς T (t, x)[Θ7 Θ8 ] Rˆ 2 [Θ7 Θ8 ]T ς(t, x)dx. R

(9.16)

206

9 State Estimation for Delayed Reaction-Diffusion GRNs

On the other hand, by Lemma 1.14, it yields   −τ¯ =



t

emT (s, x)R3 em (s, x)dsdx t−τ¯   t−τ (t) −τ¯ emT (s, x)R3 em (s, x)dsdx R t−τ¯   t emT (s, x)R3 em (s, x)dsdx −τ¯ R t−τ (t)  t−τ (t)   t−τ (t) τ¯ emT (s, x)ds R3 em (s, x)dsdx − τ¯ − τ (t) R t−τ¯ t−τ¯  t   t τ¯ T − e (s, x)ds R3 emT (s, x)dsdx τ (t) R t−τ (t) m t−τ (t) R



−

R

ς T (t, x)Φ42 (τ (t))ς(t, x)dx.

(9.17)

In the same way,   t −σ¯ eTp (s, x)R4 e p (s, x)dsdx R t−σ¯  − ς T (t, x)Φ43 (σ(t))ς(t, x)dx.

(9.18)

R

Combining (9.12) and, (9.15)–(9.18) yields ∂ V4 (t, em , e p )  ∂t

 R

ς T (t, x)Φ4 (τ (t), σ(t))ς(t, x)dx.

(9.19)

The second term on the right of (9.13) can be divided into three parts:  



∂emT (s, x) ∂em (s, x) M1 dsdθdx ∂s ∂s R −τ¯ t+θ   0  t ∂emT (s, x) ∂em (s, x) M1 dsdθdx =− ∂s ∂s R −τ (t) t+θ   −τ (t)  t−τ (t) T ∂em (s, x) ∂em (s, x) M1 dsdθdx − ∂s ∂s R −τ¯ t+θ   t ∂emT (s, x) ∂em (s, x) M1 dsdx. −(τ¯ − τ (t)) ∂s ∂s R t−τ (t) −

0

t

By using Lemmas 1.14 and 1.16, we can estimate the following inequalities

(9.20)

9.2 Infinite-Time State Estimation

  −

R

0

207



∂emT (s, x) ∂em (s, x) M1 dsdθdx ∂s ∂s

t

−τ (t) t+θ −τ (t)  t−τ (t)

 

∂emT (s, x) ∂em (s, x) M1 dsdθdx ∂s ∂s R −τ¯ t+θ   0  t ∂emT (s, x) 2 dsdθM1 − 2 τ (t) R −τ (t) t+θ ∂s  0  t ∂em (s, x) × dsdθdx ∂s −τ (t) t+θ   −τ (t)  t−τ (t) T ∂em (s, x) 2 dsdθM1 − 2 (τ¯ − τ (t)) R −τ¯ ∂s t+θ  −τ (t)  t−τ (t) ∂em (s, x) dsdθdx × ∂s −τ¯ t+θ  = −2 ς T (t, x)Φ52 ς(t, x)dx. −

(9.21)

R

  −(τ¯ − τ (t))

R

t−τ (t)



(τ¯ − τ (t)) − τ¯

t

∂emT (s, x) ∂em (s, x) M1 dsdx ∂s ∂s

ς T (t, x)Δ8 M˜ 1 ΔT8 ς(t, x)dx.

R

(9.22)

In a similar manner,   −

R

 −2

R 



−σ(t)

  −

0

−σ(t)

−σ¯

∂eTp (s, x)

t

∂s

t+θ



t−σ(t)

M2

∂e p (s, x) dsdθdx ∂s

∂eTp (s, x) ∂s

t+θ

M2

∂e p (s, x) dsdθdx ∂s

ς (t, x)Φ53 ς(t, x)dx T

R

(9.23)

and   −(σ¯ − σ(t)) (σ¯ − σ(t)) − σ¯

R

t t−σ(t)



R

∂eTp (s, x) ∂s

M2

∂e p (s, x) dsdx ∂s

ς T (t, x)Θ8 M˜ 2 Θ8T ς(t, x)dx.

(9.24)

Combining (9.13) and (9.20)–(9.24), we can obtain ∂ V5 (t, em , e p )  ∂t

 R

ς T (t, x)Φ5 (τ (t), σ(t))ς(t, x)dx.

(9.25)

208

9 State Estimation for Delayed Reaction-Diffusion GRNs

From (9.9)–(9.11), (9.19) and (9.25), one can obtain 5  ∂ ∂ V (t, em , e p ) = Vi (t, em , e p ) ∂t ∂t i=1   ς T (t, x)Φ(τ (t), σ(t))ς(t, x)dx. R

Since Φ(τ (t), σ(t)) depends affinely on τ (t) and σ(t), respectively, it follows from (9.8) that ∂t∂ V (t, em , e p ) < 0 for all τ (t) and σ(t) satisfying (9.2). Therefore, the trivial solution of error system (9.5) is asymptotically stable. This completes the proof. We end the subsection by the following remarks on Theorem 9.1. Remark 9.2 Compared with Theorems 6.5, 6.13 and 6.14, the advantages of Theorem 9.1 are as follows: 1. We introduce new integral item like  

0

R

−τ¯



0 θ



∂emT (s, x) ∂em (s, x) M1 dsdλdθdx ∂s ∂s t+λ t

into the LKF, and use the Wirtinger-type integral inequality (instead of Jensen’s inequality) to estimate its derivative, which will achieve more accurate results. 2. The so-called convex technique and the reciprocally convex technique are used simultaneously, which will improve the precision of estimating concentrations of mRNAs and proteins. 1 and τ¯ −τ1 (t) , play a very important 3. The coefficients of some items in ς(t, x), like τ (t) role in simplifying the LMI condition (9.8). 4. The items em (s, x) and e p (s, x) in V2 (t, em , e p ) are revised as ϕ(t, s, x) and ψ(t, s, x), respectively, which will be consistent with V3 (t, em , e p ). Remark 9.3 The approach proposed in this section can easily be applied to establish a delay-dependent and delay-rate-dependent asymptotic stability criterion for GRN (9.1). Due to Remark 9.2 above, the criterion is certainly less conservative than Theorems 6.5, 6.13 and 6.14. In addition, for example (6.19), we test the conservativeness of the stability criterion and ones in Theorems 6.5, 6.13 and 6.14. The computed results, including the maximum values τ¯ = σ¯ and the numbers of decision variables, are listed in Tables 9.1 and 9.2, respectively. Here, the notation “–” means that there is no solution in this case. Table 9.1 illustrates the theoretical results presented in this section, whereas Table 9.2 demonstrates that the computational complexity of our approach is slightly larger than the others. Remark 9.4 Note that Lemma 1.13 (i.e., Wirtinger’s Inequality) is only available for Dirichlet boundary conditions. However, based on the proof of Theorem 9.1, we can still consider the state estimation problem under Neumann boundary conditions

9.2 Infinite-Time State Estimation

209

Table 9.1 Maximum values of τ¯ = σ¯ for different τ¯d = σ¯ d = μ (Remark 9.3) μ 0.77 0.78 0.80 0.81 0.82 0.93 Theorem 6.14 Theorem 6.13 Theorem 6.5 Remark 9.3

>104

– > 104 >104 >104

>104 >104 >104

Table 9.2 Numbers of decision variables of different methods (Remark 9.3)

– >104 >104 >104

– >104 >104 >104

– – 1.5457 1.9033

– – 0.5654 0.7331

1 – – 0.5496 0.7284

Methods

Numbers of decision variables

Theorem 6.14 Theorem 6.13 Theorem 6.5 Remark 9.3

71.5 (4.5n 2 + 10.5n) 30 (1.5n 2 + 5.5n) 78 (5.5n 2 + 9.5n) 105 (8.5n 2 + 9.5n)

and Robin boundary conditions by using the approaches proposed in Theorem 6.9 and [2], respectively. As a result, more conservative LMI conditions compared to Theorem 9.1 will be derived. Remark 9.5 The approach proposed in Theorem 9.1 can be extended to the cases without parameter x in model (9.1) by removing the last two items in V1 (t, em , e p ).

9.2.2 Numerical Examples In this subsection, two numerical examples are provided to demonstrate the availability and applicability of the proposed state observer. Example 9.6 Consider GRN (9.1) with measurements (9.3), the parameters are given as follows: A = diag(2, 2), D = diag(1, 1), C = diag(1.3, 1.3), L 1 = L 2 = 1, K = diag(0.65, 0.65),   0 −0.5 W = , −0.5 0 D1 = D2 = diag(0.3, 0.3), D ∗ 1 = D2∗ = diag(0.6, 0.6),



M = 0.1 −0.1 , N = 0.2 0.3 .

210

9 State Estimation for Delayed Reaction-Diffusion GRNs

When τ¯d = σ¯ d = 0.5 and τ¯ = σ¯ = 0.2, by using the YALMIP Toolbox of MATLAB, one can see that the LMIs in Theorem 9.1 are feasible. The feasible solutions are as follows:     26.5059 3.4568 27.6226 −1.7108 Q1 = , Q3 = , 3.4568 27.2015 −1.7108 27.1431 ⎡ ⎤ 20.5541 1.5757 1.3812 0.2583 ⎢ 1.5757 20.5063 0.2602 1.3493 ⎥ ⎥ Q2 = ⎢ ⎣ 1.3812 0.2602 25.7445 0.0692 ⎦ , 0.2583 1.3493 0.0692 25.7313 ⎡ ⎤ 22.5632 −0.9237 1.7642 −0.1109 ⎢−0.9237 22.2883 −0.1114 1.7162 ⎥ ⎥ Q4 = ⎢ ⎣ 1.7642 −0.1114 25.8597 −0.0199⎦ , −0.1109 1.7162 −0.0199 25.8438     18.3324 −0.3638 3.9195 −0.0156 Q5 = , R1 = , −0.3638 18.2339 −0.0156 3.9230     3.8838 0.0153 25.3457 0.1029 , R3 = , R2 = 0.0153 3.8898 0.1029 25.3572     25.4017 −0.0574 1.3707 −0.0078 , M1 = , R4 = −0.0574 25.3860 −0.0078 1.3747       1.3417 0.0103 −103.0329 −12.1781 M2 = , W1 = , W2 = , 0.0103 1.3459 104.0141 −18.3132 P1 = diag(14.8041, 14.9291), P2 = diag(10.9725, 11.3663), Λ1 = diag(21.9276, 21.9346), Λ2 = diag(14.0881, 14.0768), ⎡

⎤ 0.0485 −0.0011 −1.3295 −0.0067 ⎢−0.0024 0.0312 −0.0042 −1.3332⎥ ⎥ G1 = ⎢ ⎣ 1.3372 0.0077 −3.0676 0.0331 ⎦ , 0.0099 1.3376 0.0253 −3.0583 ⎡ ⎤ 0.1148 0.0001 −1.2980 −0.0112 ⎢0.0003 0.1167 −0.0075 −1.3011⎥ ⎥ G2 = ⎢ ⎣1.3005 −0.0014 −3.0139 −0.0108⎦ . 0.0019 1.3000 −0.0081 −3.0233 Furthermore, we can get the corresponding observer gains as follows: Km = P1−1 W1 =

    −6.9598 −1.1099 , K p = P2−1 W2 = . 6.9672 −1.6112

9.2 Infinite-Time State Estimation

211

Example 9.7 In the special case l = n = 1, GRN (9.1) is simplified into   ∂ ∂ m(t, ˜ x) ∂ m(t, ˜ x) = D1 − Am(t, ˜ x) ∂t ∂x ∂x +W f ( p(t ˜ − σ(t), x)),   ∂ p(t, ˜ x) ∂ ˜ x) ∗ ∂ p(t, = D1 − C p(t, ˜ x) ∂t ∂x ∂x +D m(t ˜ − τ (t), x).

(9.26a)

(9.26b)

Take A = 0.3, D = 1.2, C = 0.2, L 1 = 1, W = −0.1, K = 0.65, D1 = 0.2, D1∗ = 0.1, M = 1, N = 0.7. When τ¯d = 0.5, σ¯ d = 0.3, τ¯ = 1 and σ¯ = 6, by applying the Toolbox YALMIP of MATLAB to solve the LMIs in Theorem 9.1, we can obtain a feasible solution. To save space, only partial solution matrices are listed here:  Q 1 = 0.5484, Q 2 =

 0.6301 −0.3579 , −0.3579 0.7148

R1 = 0.1004, R2 = 0.1025, M1 = 0.0291, M2 = 0.0255, P1 = 0.7815, P2 = 2.9444, W1 = 0.7582, W2 = 0.2572. Moreover, we can get the corresponding observer gains as follows: Km = P1−1 W1 = 0.9702, K p = P2−1 W2 = 0.0874. Furthermore, for σ(t) ≡ 6 and τ (t) ≡ 1, the state responses of GRN (9.26), observer (9.4) and the corresponding error system are given in Figs. 9.1, 9.2 and 9.3.

9.3 Finite-Time State Estimation This section addresses the problem of finite-time state estimation for delayed reaction-diffusion GRNs under Dirichlet boundary conditions. The purpose is to design a finite-time state observer which is used to estimate the concentrations of mRNAs and proteins via available measurement outputs. By constructing an appropriate LKF and employing the Wirtinger-type integral inequality, Gronwall inequality, convex technique and reciprocally convex technique, we establish an LMIs-

212

9 State Estimation for Delayed Reaction-Diffusion GRNs

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05 10 5 0 0

0.5

1

1.5

2

-0.05 10 5 0 0

0.5

1

1.5

2

Fig. 9.1 The mRNA concentration and its estimation (Example 9.7)

0.5 0.4 0.3 0.2 0.1 0 -0.1 10 5 0 0

0.5

1

1.5

2

0.5 0.4 0.3 0.2 0.1 0 -0.1 10 5 0 0

0.5

1

1.5

2

Fig. 9.2 The protein concentration and its estimation (Example 9.7)

0.2

0.2

0.15

0.15 0.1

0.1

0.05

0.05

0

0 10 5 0 0

0.5

1

1.5

2

-0.05 10 5 0 0

0.5

1

1.5

2

Fig. 9.3 Estimation errors (Example 9.7)

based finite-time stability criterion for the error system. The stability criterion is reaction-diffusion-dependent and delay-dependent. Thereby, we design a finite-time state observer whose gains are represented via a feasible solution to these LMIs. In addition, two numerical examples are presented to illustrate the theoretical results obtained in this section.

9.3 Finite-Time State Estimation

213

9.3.1 Observer Design In this subsection, we will design a finite-time state observer of the form (9.4) for delayed reaction-diffusion GRN (9.1), that is, find a pair of observer gains Km and K p so that the trivial solution of (9.5) is finite-time stable. For convenience, we define ei = col(0(i−1)n×n , In , 0(18−i)n×n ), i ∈ 18, ¯ x), ς(t, x) = col(em (t, x), em (t − τ¯ , x), em (t − τ (t), x), e p (t, x), e p (t − σ, e p (t − σ(t), x), fˆ(e p (t, x)), fˆ(e p (t − σ(t), x)),  t−τ (t) ∂em (t, x) ∂e p (t, x) 1 , , em (s, x)ds, ∂t ∂t τ¯ − τ (t) t−τ¯  t−τ (t)  t−τ (t)  t 1 1 e (s, x)dsdα, em (s, x)ds, m (τ¯ − τ (t))2 t−τ¯ τ (t) t−τ (t) α  t  t  t−σ(t) 1 1 em (s, x)dsdα, e p (s, x)ds, τ 2 (t) t−τ (t) α σ¯ − σ(t) t−σ¯  t−σ(t)  t−σ(t)  t 1 1 e (s, x)dsdα, e p (s, x)ds, p (σ¯ − σ(t))2 t−σ¯ σ(t) t−σ(t) α  t  t 1 e p (s, x)dsdα). σ 2 (t) t−σ(t) α Theorem 9.8 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (9.2), and positive constants α, c1 , c2 and T , the trivial solution of error system (9.5) under Dirichlet boundary conditions is finite-time stable with respect to c1 , c2 and T if there exist diagonal matrices Pk > 0 and Λk > 0, and matrices Q iT = Q i > 0 (i ∈ 5), R Tj = R j > 0, ( j ∈ 4), WkT = Wk > 0, HkT = Hk > 0, Gˆ k and Nk (k ∈ 2) of appropriate sizes, such that the following inequalities hold for τ ∈ {0, τ¯ } and σ ∈ {0, σ}: ¯ 

 Rˆ k Gˆ k  0, k ∈ 2, Gˆ Tk Rˆ k 3 6   Ξ (τ , σ) := Ξi + Ξi (τ , σ) − αe1 P1 e1T − αe4 P2 e4T < 0, i=0

(9.27) (9.28)

i=4

c1 eαT (λ11 + λ12 ) − c2 λmin (P)  0,

(9.29)

214

9 State Estimation for Delayed Reaction-Diffusion GRNs

where π2 π2 P1 D L − P1 A − N1 M)e1T + e1 (− P1 D L − P1 A − N1 M)T e1T 4 4 π2 π2 +e4 (− P2 D L∗ − P2 C − N2 N )e4T + e4 (− P2 D L∗ − P2 C − N2 N )T e4T 4 4 T −2e9 P1 e9T − 2e10 P2 e10 + sym(−e9 (P1 A + N1 M)e1T + e9 P1 W e8T

Ξ1 = e1 (−

+e4 P2 De3T + e10 P2 De3T + e1 P1 W e8T − e10 (P2 C + N2 N )e4T ), Ξ2 = e1 (Q 1 + Q 2 )e1T − e2 Q 2 e2T + (τ¯d − 1)e3 Q 1 e3T + e4 (Q 3 + Q 4 )e4T −e5 Q 4 e5T + (σ¯ d − 1)e6 Q 3 e6T , Ξ3 = (σ¯ d − 1)e8 Q 5 e8T + e7 Q 5 e7T , Ξ4 (τ , σ) = Ξ40 + Ξ41 + Ξ42 + Ξ43 (τ ) + Ξ44 (σ), T Ξ40 = τ¯ 2 (e9 R1 e9T + e1 R3 e1T ) + σ¯ 2 (e10 R2 e10 + e4 R4 e4T ),   Rˆ Gˆ Ξ41 = −[Δ1 Δ2 ] ˆ 1T ˆ 1 [Δ1 Δ2 ]T , G R1   1 Rˆ 2 Gˆ 2 Ξ42 = −[Δ3 Δ4 ] ˆ T ˆ [Δ3 Δ4 ]T , G 2 R2 Ξ43 (τ ) = −τ¯ (τ¯ − τ )Δ5 R˜ 3 ΔT5 − τ¯ τ Δ6 R˜ 3 ΔT6 ,

˜ T Ξ44 (σ) = −σ( ¯ σ¯ − σ)Δ7 R˜ 4 ΔT7 − σσΔ ¯ 8 R 4 Δ8 , Δ1 = [e3 − e2 e3 + e2 − 2e11 e3 − e2 + 6e11 − 12e12 ], Δ2 = [e1 − e3 e1 + e3 − 2e13 e1 − e3 + 6e13 − 12e14 ], Δ3 = [e6 − e5 e6 + e5 − 2e15 e6 − e5 + 6e15 − 12e16 ], Δ4 = [e4 − e6 e4 + e6 − 2e17 e4 − e6 + 6e17 − 12e18 ], Δ5 = [e11 e11 − 2e12 ], Δ6 = [e13 e13 − 2e14 ], Δ7 = [e15 e15 − 2e16 ], Δ8 = [e17 e17 − 2e18 ], Ξ5 (τ , σ) = Ξ50 + Ξ51 (τ ) + Ξ52 (σ), τ¯ 2 σ¯ 2 T e9 W1 e9T + e10 W2 e10 , 2 2 τ¯ − τ Δ2 Wˆ 1 ΔT2 , Ξ51 (τ ) = −Θ1 W˜ 1 Θ1T − Θ2 W˜ 1 Θ2T − τ¯ Ξ50 =

9.3 Finite-Time State Estimation

215

σ¯ − σ Ξ52 (σ) = −Θ3 W˜ 2 Θ3T − Θ4 W˜ 2 Θ4T − Δ4 Wˆ 2 ΔT4 , σ¯ Ξ6 (τ , σ) = Ξ60 + Ξ61 (τ ) + Ξ62 (σ) − (τ¯ − τ )Θ1 H˜ 1 Θ1T − (σ¯ − σ)Θ3 H˜ 2 Θ3T , Ξ60 =

τ¯ 3 σ¯ 3 T , e9 H1 e9T + e10 H2 e10 6 6

Θ1 = [e1 − e13 e1 + 2e13 − 6e14 ], Θ2 = [e3 − e11 e3 + 2e11 − 6e12 ], Θ3 = [e4 − e17 e4 + 2e17 − 6e18 ], Θ4 = [e6 − e15 e6 + 2e15 − 6e16 ], 3 Ξ61 (τ ) = − τ (e1 − 2e14 )H1 (e1 − 2e14 )T 2 3 − (τ¯ − τ )(e3 − 2e12 )H1 (e3 − 2e12 )T , 2 3 Ξ62 (σ) = − σ(e4 − 2e18 )H2 (e4 − 2e18 )T 2 3 − (σ¯ − σ)(e6 − 2e16 )H2 (e6 − 2e16 )T , 2 Ξ0 = −2e7 Λ1 e7T + e4 K Λ1 e7T + e7 Λ1 K eT4 −2e8 Λ2 e8T + e6 K Λ2 e8T + e8 Λ2 K eT6 , W˜ i = diag(2Wi , 4Wi ), Rˆ i = diag(Ri , 3Ri , 5Ri ), Wˆ i = diag(Wi , 3Wi , 5Wi ), H˜ i = diag(2Hi , 4Hi ) i = 1, 2, R˜ j = diag(R j , 3R j ), W¯ j = diag(2W j , 16W j ), j = 3, 4, P = diag(P1 , P2 ), 1 λ11 = λmax (P1 ) + τ¯ λmax (Q 1 ) + τ¯ λmax (Q 2 ) + τ¯ 3 λmax (W1 ) 6 l  1 1 + λmax (P1 )λmax (Dk ) + τ¯ 3 λmax ((R1 ) + (R3 )) + τ¯ 4 λmax (H1 ), 2 24 k=1 λ12 = λmax (P2 ) + σλ ¯ max (Q 3 ) + σλ ¯ max (Q 4 ) + σλ ¯ max (Q 5 )λmax (K T K ) l  1 + σ¯ 3 λmax (W2 ) + λmax (P2 )λmax (Dk∗ ) 6 k=1 1 1 + σ¯ 3 λmax ((R2 ) + (R4 )) + σ¯ 4 λmax (H2 ), 2 24 and A, D, C, W , K , D L and D L∗ are the same with previous ones. Furthermore, when the inequalities (9.27)–(9.29) are feasible, the desired finitetime state observer is given by (9.4) with Km = P1−1 N1 and K p = P2−1 N2 .

216

9 State Estimation for Delayed Reaction-Diffusion GRNs

Proof It is similar to those of Theorems 6.21 and 9.1. Since the inequality (9.29) is not an LMI, it can not be solved by the Toolbox YALMIP of MATLAB. The following theorem offers a method to transform the nonlinear problem into a linear one. Theorem 9.9 For given scalars τ¯ , σ, ¯ τ¯d and σ¯ d satisfying (9.2), and positive constants α, c1 , c2 and T , the trivial solution of error system (9.5) under Dirichlet boundary conditions is finite-time stable with respect to c1 , c2 and T if there exist real numbers λqi > 0 (i ∈ 5), λwj > 0, λ pj > 0, λh j > 0 ( j ∈ 2), λr j > 0 ( j ∈ 4), λ p > 0, diagonal matrices Pk > 0, Λk > 0 and matrices Q iT = Q i > 0 (i ∈ 5), R Tj = R j > 0 ( j ∈ 4), WkT = Wk > 0, HkT = Hk > 0, Gˆ k and Nk (k ∈ 2) of appropriate sizes, such that (9.27), (9.28) and the following LMIs hold for τ ∈ {0, τ¯ } and σ ∈ {0, σ}: ¯ 0  Q i  λqi I, i ∈ 5,

(9.30)

0  P j  λ pj I, j ∈ 2,

(9.31)

0  R j  λr j I, j ∈ 4,

(9.32)

0  W j  λwj I, j ∈ 2,

(9.33)

0  H j  λh j I, j ∈ 2,

(9.34)

λ p I  P,

(9.35)

1 1 c1 eαT (λ p1 + τ¯ λq1 + τ¯ λq2 + τ¯ 3 λw1 + τ¯ 3 (λr 1 + λr 3 ) 6 2 l  1 + λ p1 λmax (Dk ) + λ p2 + σλ ¯ q3 + σλ ¯ q4 + τ¯ 4 λh1 24 k=1 1 1 1 + σλ ¯ q5 λmax (K T K ) + σ¯ 3 λw2 + σ¯ 3 (λr 2 + λr 4 ) + σ¯ 4 λh2 6 2 24 l  + λ p2 λmax (Dk∗ ))  c2 λ p ,

(9.36)

k=1

where Dk , Dk∗ and K are the same with previous ones. Furthermore, when LMIs (9.27), (9.28) and (9.30)–(9.36) are feasible, the desired finite-time state observer is given by (9.4) with Km = P1−1 N1 and K p = P2−1 N2 . Proof The proof is omitted, since it is similar to one of Theorem 6.25. Remark 9.10 The approach proposed in this section can easily be applied to establish a delay-dependent asymptotic stability criterion (i.e., (9.27) and (9.28) with α = 0) for delayed reaction-diffusion GRN (9.1). Next we test the conservativeness of

9.3 Finite-Time State Estimation

217

Table 9.3 Maximum values of τ¯ = σ¯ with different τ¯d = σ¯ d = μ (Remark 9.10) 0.77 0.78 0.80 0.81 0.82 0.93 μ Theorem 6.14 Theorem 6.13 Theorem 6.5 Remark 9.3 Remark 9.10

>104 >104 >104 >104 >104

– >104 >104 >104 >104

– >104 >104 >104 >104

– >104 >104 >104 >104

– – 1.5457 1.9033 2.0968

– – 0.5654 0.7331 0.8750

1 – – 0.5496 0.7284 0.8749

the stability criterion, Remark 9.3 and Theorems 6.5, 6.13 and 6.14. For the same numerical example, the compute results are listed in Table 9.3, which shows that the approach proposed in this section is less conservative than ones in Remark 9.3 and Theorems 6.5, 6.13 and 6.14.

9.3.2 Numerical Examples In this section, we will give two numerical examples to demonstrate the availability of the state observer proposed previously. Example 9.11 Consider delayed reaction-diffusion GRN (9.37) and the measurement output (9.3), where A = diag(1.1, 1.2), D = diag(0.4, 0.7), C = diag(0.7, 1.3), L 1 = L 2 = L 3 = 1, K = diag(0.65, 0.65),   0 0 W = , −0.5 0 D1 = D2 = D3 = diag(0.1, 0.1), D1∗ = D2∗ = D3∗ = diag(0.2, 0.2),



M = 0.8 −0.2 , N = 0.2 −0.3 . When τ¯d = σ¯ d = 1.1, τ¯ = σ¯ = 1.5, c1 = 1, c2 = 5, T = 10 and α = 0.002, by using the Toolbox YALMIP of MATLAB, we find that the LMIs given in Theorem 9.9 are feasible. The solution matrices are listed as follows:     0.7959 −0.0210 0.0023 0.0000 Q 1 = 10−4 × , Q2 = , −0.0210 0.4004 0.0000 0.0025

218

9 State Estimation for Delayed Reaction-Diffusion GRNs

   0.0764 −0.0007 0.0017 −0.0000 Q 3 = 10 × , Q4 = , −0.0007 0.1026 −0.0000 0.0017       0.2326 −0.0007 −0.0022 0.0106 Q 5 = 10−3 × , N1 = , N2 = , −0.0007 0.3722 −0.0009 −0.0110     0.1203 0.0095 0.1163 0.0001 , R2 = 10−3 × , R1 = 10−3 × 0.0095 0.3044 0.0001 0.0915     0.0042 0.0001 0.0025 −0.0000 R3 = , R4 = , 0.0001 0.0058 −0.0000 0.0024     0.2237 −0.0031 0.2615 0.0021 , W2 = 10−4 × , W1 = 10−4 × −0.0031 0.1566 0.0021 0.2833     0.4289 0.0011 0.3348 −0.0027 , H2 = 10−3 × , H1 = 10−3 × 0.0011 0.4408 −0.0027 0.3022 −3



P1 = 10−3 ∗ diag(0.0088, 0.0088), P2 = 10−3 ∗ diag(0.0086, 0.0086), Λ1 = diag(0.0147, 0.0155), Λ2 = diag(0.0022, 0.0018), λq1 = 4.7916 × 10−4 , λq2 = 0.0027, λq3 = 5.0061 × 10−4 , λq4 = 0.0020, λq5 = 0.0012, λw1 = 0.0021, λw2 = 0.0021, λ p1 = 0.0090, λ p2 = 0.0088, λh1 = 0.0067, λh2 = 0.0066, λr 1 = 8.7912 × 10−4 , λr 2 = 7.8762 × 10−4 , λr 3 = 0.0061, λr 4 = 0.0030, λ p = 0.0085. Moreover, the corresponding observer gains can be obtained as follows: Km =

P1−1 N1

    −0.2511 1.2297 −1 = , K p = P2 N2 = . −0.0998 −1.2865

Example 9.12 In the special case l = n = 1, delayed reaction-diffusion GRN (9.1) is simplified into ∂ ∂ m(t, ˜ x) = ∂t ∂x



∂ p(t, ˜ x) ∂ = ∂t ∂x

∂ m(t, ˜ x) D1 ∂x 

D1∗



∂ p(t, ˜ x) ∂x

− Am(t, ˜ x) + W f ( p(t ˜ − σ(t), x)), (9.37a)  − C p(t, ˜ x) + D m(t ˜ − τ (t), x).

(9.37b)

9.3 Finite-Time State Estimation

219

Take A = 0.5, D = 1.2, C = 0.3, L 1 = 1, W = −1.5, K = 0.65, D1 = 0.1, D1∗ = 0.2, M = 0.1, N = 0.2. When τ¯d = σ¯ d = τ¯ = σ¯ = 1, c1 = 1.2, c2 = 5, T = 10 and α = 0.002, by using the Toolbox YALMIP of MATLAB to solve the LMIs in Theorem 9.9, we can obtain a feasible solution. To save space, only partial solution matrices are listed here: P1 = 0.0033, P2 = 0.0033, N1 = 0.0413, N2 = 0.0253. Furthermore, the corresponding observer gains are Km = P1−1 N1 = 12.3357, K p = P2−1 N2 = 7.5780.

1.5 1 0.5 0 -0.5 -1 -1.5 8

¯ x) m(t,

m(t, ˜ x)

The state responses of delayed reaction-diffusion GRN (9.37), observer (9.4) and the corresponding error system (9.5) are given in Figs. 9.4, 9.5 and 9.6.

6

t

4

2

0 0

0.5

1

1.5

2

1 0.5 0 -0.5 -1 -1.5 -2 8

6

4

t

x

2

0 0

0.5

1

1.5

2

x

Fig. 9.4 The mRNA concentration and its estimation (Example 9.12)

1

p¯(t, x)

p˜(t, x)

2

0 -1 -2 8

6

t

4

2

0 0

0.5

1

x

1.5

2

3 2 1 0 -1 -2 -3 8

6

t

4

2

Fig. 9.5 The protein concentration and its estimation (Example 9.12)

0 0

0.5

1

x

1.5

2

220

9 State Estimation for Delayed Reaction-Diffusion GRNs

1

ep (t, x)

em (t, x)

1.5 0.5 0 -0.5 -1 8

6

t

4

2

0 0

0.5

1

1.5

x

2

1.5 1 0.5 0 -0.5 -1 -1.5 8

6

t

4

2

0 0

0.5

1

1.5

2

x

Fig. 9.6 Estimation errors (Example 9.12)

9.4 Remarks and Notes This chapter investigated the problem of state estimation for delayed reaction- diffusion GRNs. Infinite-time and finite-time state observers have been designed to estimate the concentrations of mRNAs and proteins based on available network outputs, which guarantees that the error system is asymptotically and finite-time stable, respectively. The results related to infinite-time and finite-time state observers are taken from [3] and [1], respectively.

References 1. Fan, X., Xue, Y., Zhang, X., Ma, J.: Finite-time state observer for delayed reaction-diffusion genetic regulatory networks. Neurocomputing 227, 18–28 (2017) 2. Han, Y.Y., Zhang, X.: Stability analysis for delayed regulatory networks with reaction-diffusion terms (in Chinese). J. Nat. Sci. Heilongjiang Univ. 31(1), 32–40 (2014) 3. Zhang, X., Han, Y., Wu, L., Wang, Y.: State estimation for delayed genetic regulatory networks with reaction-diffusion terms. IEEE Trans. Neural Netw. Learn. Syst. 29(2), 299–309 (2018)

Chapter 10

H∞ State Estimation for Delayed Stochastic GRNs

This chapter addresses the problem of robust H∞ filter for a class of uncertain stochastic GRNs with mixed delays. The uncertain stochastic GRNs under consideration are extended to involve Itô-type stochastic disturbance, norm-bounded uncertainties, time-varying discrete delays and distributed delays. By constructing an appropriate LKF and using reciprocal convex technique, LMIs-based sufficient conditions were presented to guarantee that the filtering error systems are robustly asymptotically mean square stable with pre-specified disturbance attenuation level. Furthermore, two numerical examples are given to illustrate the effectiveness of the proposed approach.

10.1 Model Description Let (Ω, F, P) be a probability space with the sample space Ω, the σ-algebra F of subsets of the sample space, and the probability measure P. E {·} denotes the mathematical expectation operator with respect to probability measure P. Consider the delayed GRN model (1.8), that is, ˙˜ m(t) = −Am(t) ˜ + W f ( p(t ˜ − σ(t))), t  0,

(10.1a)

˙˜ = −C p(t) p(t) ˜ + D m(t ˜ − τ (t)), t  0.

(10.1b)

When we take the Itô-type stochastic disturbance, external disturbance, normbounded uncertainties and constant distributed delays into model (10.1), a class of uncertain stochastic GRN model with mixed delays can be described as follows: dm(t) ˜ = A(t)dt + B(t)dω1 (t), © Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_10

(10.2a) 221

222

10 H∞ State Estimation for Delayed Stochastic GRNs

d p(t) ˜ = C(t)dt + D(t)dω2 (t),

(10.2b)

dym (t) = A y (t)dt + B y (t)dω1 (t),

(10.2c)

dy p (t) = C y (t)dt + D y (t)dω2 (t),

(10.2d)

˜ = φ p (t), ∀t ∈ [−d, 0] m(t) ˜ = φm (t), p(t)

(10.2e)

with ˜ − σ(t))) A(t) = (A + ΔA(t))m(t) ˜ + (A1 + ΔA1 (t)) f ( p(t  t f ( p(s))ds ˜ + (A0 + ΔA0 (t))ν(t), +(A2 + ΔA2 (t)) t−h 1

˜ − σ(t))) B(t) = (B + ΔB(t))m(t) ˜ + (B1 + ΔB1 (t)) f ( p(t  t f ( p(s))ds, ˜ +(B2 + ΔB2 (t)) t−h 1

˜ − τ (t)) C(t) = (C + ΔC(t)) p(t) ˜ + (C1 + ΔC1 (t))m(t  t m(s)ds ˜ + (C0 + ΔC0 (t))ν(t), +(C2 + ΔC2 (t)) t−h 2

˜ − τ (t)) D(t) = (D + ΔD(t)) p(t) ˜ + (D1 + ΔD1 (t))m(t  t m(s)ds, ˜ +(D2 + ΔD2 (t)) t−h 2

˜ + (A4 + ΔA4 (t)) f ( p(t ˜ − σ(t))) A y (t) = (A3 + ΔA3 (t))m(t)  t +(A5 + ΔA5 (t)) f ( p(s))ds ˜ + (A y + ΔA y (t))ν(t), t−h 1

˜ + (B4 + ΔB4 (t)) f ( p(t ˜ − σ(t))) B y (t) = (B3 + ΔB3 (t))m(t)  t +(B5 + ΔB5 (t)) f ( p(s))ds, ˜ t−h 1

˜ + (C4 + ΔC4 (t))m(t ˜ − τ (t)) C y (t) = (C3 + ΔC3 (t)) p(t)  t +(C5 + ΔC5 (t)) m(s)ds ˜ + (C y + ΔC y (t))ν(t), t−h 2

˜ + (D4 + ΔD4 (t))m(t ˜ − τ (t)) D y (t) = (D3 + ΔD3 (t)) p(t)  t +(D5 + ΔD5 (t)) m(s)ds, ˜ t−h 2

where m(t) ˜ := col(m˜ 1 (t), . . . , m˜ n (t)) and p(t) ˜ := col( p˜ 1 (t), . . . , p˜ n (t)) are concentrations of mRNAs and proteins for GRNs with n nodes at time t, respectively; ym (t) and y p (t) represent the expression levels of mRNAs and proteins at time t, respectively; ν(t) is the external disturbance which belongs to L 2 [0, +∞);

10.1 Model Description

223

ωi (t)(i = 1, 2) are two one-dimensional Wiener processes on (Ω, F, P) satisfying E{ωi (t)} = 0 and E{ωi2 (t)} = dt; φm (t) and φ p (t) are the initial functions on [−d, 0] with d = 2 max{h 1 , h 2 , τ2 , σ2 }; f ( p(t)) ˜ := col( f 1 ( p˜ 1 (t)), f 2 ( p˜ 2 (t)), . . . , f n ( p˜ n (t))) represents the feedback regulation of the protein in the process of transcription; matrices A = diag(−a1 , . . . , −an ), C = diag(−c1 , . . . , −cn ) with ai > 0 and ci > 0 (i ∈ n) represent the rate of degradation of mRNAs and proteins, respectively; C1 = diag(−c11 , . . . , −c1n ) with c1i > 0(i ∈ n) represents the translation rate of the i-th node; A, B, C, D, A0 , A y , C0 , C y , Ai , Ci , Bi and Di (i ∈ 5) are the constant matrices of appropriate sizes; ΔA(t), ΔB(t), ΔC(t), ΔD(t), ΔA0 (t), ΔA y (t), ΔC0 (t), ΔC y (t), ΔAi (t), ΔBi (t), ΔCi (t) and ΔDi (t)(i ∈ 5) are the corresponding norm-bounded uncertainties; τ (t) and σ(t) are the time-varying discrete delays; and h i (i = 1, 2) are the constant distributed delays. We introduce the following assumptions throughout the chapter. Assumption 10.1 The time-varying delays τ (t) and σ(t) satisfy 0  τ1  τ (t)  τ2 , τ˙ (t)  τd < ∞,

(10.3a)

˙  σd < ∞, 0  σ1  σ(t)  σ2 , σ(t)

(10.3b)

where τ1 , τ2 , σ1 , σ2 , σd and τd are known scalars. Assumption 10.2 The function f i (s) satisfies the following condition: f i (0) = 0, 0 

f i (s)  ki , ∀0 = s ∈ R, i ∈ n, s

(10.4)

where k1 , . . . , kn are constants. Assumption 10.3 The norm-bounded uncertainties ΔA(t), ΔB(t), ΔC(t), ΔD(t), ΔA0 (t), ΔA y (t), ΔC0 (t), ΔC y (t), ΔAi (t), ΔBi (t), ΔCi (t) and ΔDi (t) (i ∈ 5) are assumed to be of the form ⎡ ⎤ ΔA(t) ΔB(t) ΔC(t) ΔD(t) ⎢ΔA1 (t) ΔB1 (t) ΔC1 (t) ΔD1 (t)⎥ ⎢ ⎥ ⎢ΔA2 (t) ΔB2 (t) ΔC2 (t) ΔD2 (t)⎥ ⎢ ⎥ ⎢ΔA3 (t) ΔB3 (t) ΔC3 (t) ΔD3 (t)⎥ ⎢ ⎥ ⎣ΔA4 (t) ΔB4 (t) ΔC4 (t) ΔD4 (t)⎦ ΔA5 (t) ΔB5 (t) ΔC5 (t) ΔD5 (t)

224

10 H∞ State Estimation for Delayed Stochastic GRNs

⎤ E0 ⎢ E1⎥ ⎢ ⎥ ⎢ E2 ⎥

⎥ =⎢ ⎢ E 3 ⎥ F(t) Ha Hb Hc Hd , ⎢ ⎥ ⎣ E4 ⎦ E ⎡ 5 ⎤ ⎤ ⎡ ΔA0 (t) E a0 ⎢ ΔC0 (t) ⎥ ⎢ E c0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎣ΔA y (t)⎦ = ⎣ E ay ⎦ F(t) He , E cy ΔC y (t) ⎡

(10.5a)

(10.5b)

where Ha , Hb , Hc , Hd , He , E a0 , E c0 , E ay , E cy and E j ( j = 0, 1, . . . , 5) are known matrices of appropriate sizes, and F(t) is a time-varying uncertain matrix satisfying F T (t)F(t)  I, ∀t  0.

(10.6)

For convenience, we denote K = diag(k1 , . . . , kn ), τ12 = τ2 − τ1 and σ12 = σ2 − σ1 . In this chapter, we will design the following full-order filter (10.7) to estimate the concentrations of mRNAs and proteins in (10.2). ˆ + B f dym (t), dm(t) ˆ = A f m(t)dt

(10.7a)

ˆ + D f dy p (t), d p(t) ˆ = C f p(t)dt

(10.7b)

m(t) ˆ = 0, p(t) ˆ = 0, ∀t ∈ [−d, 0],

(10.7c)

where m(t) ˆ and p(t) ˆ are the filter state vectors, A f , B f , C f and D f are appropriately dimensioned filter gains to be designed. Define ˜ m(t)), ˆ e p (t) = col( p(t), ˜ p(t)), ˆ em (t) = col(m(t), φ˜ m (t) = col(φm (t), 0), φ˜ p (t) = col(φ p (t), 0), then the filtering error system is as follows:

with

˜ ˜ + B(t)dω dem (t) = A(t)dt 1 (t), t  0,

(10.8a)

˜ ˜ + D(t)dω de p (t) = C(t)dt 2 (t), t  0,

(10.8b)

em (t) = φ˜ m (t), e p (t) = φ˜ p (t), ∀t ∈ [−d, 0]

(10.8c)

10.1 Model Description

225

˜ ˜ ˜ ˜ A(t) = ( A˜ + Δ A(t))e m (t) + ( A1 + Δ A1 (t))I1 f (e p (t − σ(t)))  t +( A˜ 2 + Δ A˜ 2 (t))I1 f (e p (s))ds t−h 1

+( A˜ 3 + Δ A˜ 3 (t))ν(t), ˜ ˜ ˜ ˜ B(t) = ( B˜ + Δ B(t))e m (t) + ( B1 + Δ B1 (t))I1 f (e p (t − σ(t)))  t +( B˜ 2 + Δ B˜ 2 (t))I1 f (e p (s))ds, t−h 1

˜ ˜ ˜ ˜ C(t) = (C˜ + ΔC(t))e p (t) + (C 1 + ΔC 1 (t))I1 em (t − τ (t))  t +(C˜ 2 + ΔC˜ 2 (t))I1 em (s)ds t−h 2

+(C˜ 3 + ΔC˜ 3 (t))ν(t), ˜ ˜ ˜ ˜ D(t) = ( D˜ + Δ D(t))e p (t) + ( D1 + Δ D1 (t))I1 em (t − τ (t))  t +( D˜ 2 + Δ D˜ 2 (t))I1 em (s)ds, t−h 2

where    A1 A2 A0 A 0 ˜ ˜ ˜ , A1 = , A2 = , A3 = , B f A4 B f A5 B f Ay B f A3 A f    B 0 B1 B2 , B˜ 1 = , B˜ 2 = , B˜ = B f B4 B f B5 B f B3 0     C 0 C1 C2 C0 , C˜ 1 = , C˜ 2 = , C˜ 3 = , C˜ = D f C4 D f C5 D f Cy D f C3 C f    D 0 D1 D2 , D˜ 2 = , D˜ = , D˜ 1 = D f D4 D f D5 D f D3 0   ΔA(t) 0 ΔA1 (t) ˜ = Δ A(t) , Δ A˜ 1 (t) = , B f ΔA4 (t) B f ΔA3 (t) 0   ΔA2 (t) ΔA0 (t) , Δ A˜ 3 (t) = , Δ A˜ 2 (t) = B f ΔA5 (t) B f ΔAy(t)   ΔB1 (t) ΔB(t) 0 ˜ , Δ B˜ 1 (t) = , Δ B(t) = B f ΔB4 (t) B f ΔB3 (t) 0   ΔC(t) 0 ΔB2 (t) ˜ , ΔC(t) = , Δ B˜ 2 (t) = B f ΔB5 (t) D f ΔC3 (t) 0   ΔC1 (t) ΔC2 (t) , ΔC˜ 2 (t) = , ΔC˜ 1 (t) = D f ΔC4 (t) D f ΔC5 (t)   ΔD(t) 0 ΔC0 (t) ˜ , Δ D(t) = , ΔC˜ 3 (t) = D f ΔC y(t) D f ΔD3 (t) 0 A˜ =



226

10 H∞ State Estimation for Delayed Stochastic GRNs

 ΔD1 (t) ΔD2 (t) , Δ D˜ 2 (t) = , D f ΔD4 (t) D f ΔD5 (t) 

f ( p(t)) ˜ I1 = I 0 , f (e p (t)) = . f ( p(t)) ˆ

Δ D˜ 1 (t) =



For the sake of convenience, let

A = A˜ 02n×9n A˜ 1 02n×4n A˜ 2 02n×n A˜ 3 ,

B = B˜ 02n×9n B˜ 1 02n×4n B˜ 2 02n×2n ,

C = 02n×2n C˜ 02n×n C˜ 1 02n×11n C˜ 2 C˜ 3 ,

D = 02n×2n D˜ 02n×n D˜ 1 02n×11n D˜ 2 02n×n ,

˜ 02n×9n Δ A˜ 1 (t) 02n×4n Δ A˜ 2 (t) 02n×n Δ A˜ 3 (t) , ΔA(t) = Δ A(t)

˜ ΔB(t) = Δ B(t) 02n×9n Δ B˜ 1 (t) 02n×4n Δ B˜ 2 (t) 02n×2n ,

˜ ΔC(t) = 02n×2n ΔC(t) 02n×n ΔC˜ 1 (t) 02n×11n ΔC˜ 2 (t) ΔC˜ 3 (t) ,

˜ ΔD(t) = 02n×2n Δ D(t) 02n×n Δ D˜ 1 (t) 02n×11n Δ D˜ 2 (t) 02n×n . Then system (10.8) becomes dem (t) = f 1 (t)dt + f 2 (t)dω1 (t),

(10.9a)

de p (t) = f 3 (t)dt + f 4 (t)dω2 (t)

(10.9b)

f 1 (t) = (A + ΔA(t))ξ(t), f 2 (t) = (B + ΔB(t))ξ(t),

(10.10a)

f 3 (t) = (C + ΔC(t))ξ(t), f 4 (t) = (D + ΔD(t))ξ(t),

(10.10b)

with

where ξ(t) = col(m(t), ˜ m(t), ˆ p(t), ˜ p(t), ˆ I1 em (t − τ1 ), I1 em (t − τ (t)), I1 em (t − τ2 ), I1 e p (t − σ1 ), I1 e p (t − σ(t)), I1 e p (t − σ2 ), I1 f (e p (t)), I1 f (e p (t − σ(t))), I1 f 1 (t), I1 f 2 (t), I1 f 3 (t), I1 f 4 (t),  t  t f (e p (s))ds, I1 em (s)ds, ν(t)). I1 t−h 1

t−h 2

10.1 Model Description

227

Definition 10.4 For a given positive constant γ, the filtering error system (10.8) is said to be mean square robustly asymptotically stable with disturbance attenuation level γ, if system (10.8) with ν(t) = 0 is mean square robustly asymptotically stable, and under the zero initial condition the following inequality



∞

E 0

    2 2 2 em (t)2 + e p (t)2 dt  γ

∞

0

ν(t)22 dt

holds for any nonzero ν(t) ∈ L 2 [0, +∞).

10.2 Main Results In this section we will derive a delay-dependent sufficient condition of mean square robust asymptotic stability with disturbance attenuation level γ for the filtering error system (10.8). This requires the following proposition which is related to the stochastic differential equation dx(t) = f (t)dt + h(t)dω(t), where ω(t) is a onedimensional Wiener process.  Q Q˜ ˜  0, scalars τ1 Proposition 10.5 For given matrices Q and Q satisfying ˜ T Q Q and τ2 subject to τ12 := τ2 − τ1 > 0, and functions τ : R → [τ1 , τ2 ] and f : R → Rn such that the integrations concerned are well defined, the following inequality holds: 

 −τ1  t f T (s)Q f (s)dsdθ E L −τ2

t+θ

 E τ12 f T (t)Q f (t) −

  1 T Q Q˜ η(t) , ∀t ∈ R, η (t) ˜ T Q Q τ12

where η(t) = col(x(t − τ1 ) − x(t − τ (t)), x(t − τ (t)) − x(t − τ2 )). Proof Based on the definition of the weak infinitesimal operator L[2], we get from Lemma 1.14 that  −τ1  t f T (s)Q f (s)dsdθ L −τ2

t+θ

= τ12 f T (t)Q f (t) − = τ12 f (t)Q f (t) − T



t−τ1

t−τ2  t−τ1 t−τ (t)

f T (s)Q f (s)ds  f (s)Q f (s)ds − T

t−τ (t) t−τ2

f T (s)Q f (s)ds

228

10 H∞ State Estimation for Delayed Stochastic GRNs

 τ12 f T (t)Q f (t) − 1 − τ2 − τ (t)



1 τ (t) − τ1

t−τ (t)





t−τ1

f T (s)ds Q

t−τ (t)  t−τ (t)

f T (s)ds Q

t−τ2

t−τ1 t−τ (t)

f (s)ds

f (s)ds.

t−τ2

Using Newton-Leibniz formula, we have that 

t−τ1

t−τ (t)

 f (s)ds = x(t − τ1 ) − x(t − τ (t)) −

t−τ1

h(s)dω(s) t−τ (t)

and 

t−τ (t)

 f (s)ds = x(t − τ (t)) − x(t − τ2 ) −

t−τ2

t−τ (t)

h(s)dω(s). t−τ2

Based on the above equalities, we obtain that  −

t−τ1 t−τ (t)

 f T (s)ds Q

t−τ1 t−τ (t)

f (s)ds

 −[x(t − τ1 ) − x(t − τ (t))]T Q[x(t − τ1 ) − x(t − τ (t))]  t−τ1 +2[x(t − τ1 ) − x(t − τ (t))]T Q h(s)dω(s) t−τ (t)

and  −

t−τ (t) t−τ2

 f (s)ds Q T

t−τ (t)

f (s)ds

t−τ2

 −[x(t − τ (t)) − x(t − τ2 )]T Q[x(t − τ (t)) − x(t − τ2 )]  t−τ (t) T +2[x(t − τ (t)) − x(t − τ2 )] Q h(s)dω(s). t−τ2

So, from the above inequalities and Lemma 1.9, we have that 

 −τ1  t T f (s)Q f (s)dsdθ E L t+θ −τ2   E τ12 f T (t)Q f (t) τ12 1 [x(t − τ1 ) − x(t − τ (t))]T Q[x(t − τ1 ) − x(t − τ (t))] − τ12 τ (t) − τ1  1 τ12 − [x(t − τ (t)) − x(t − τ2 )]T Q[x(t − τ (t)) − x(t − τ2 )] τ12 τ2 − τ (t)  

1 T Q Q˜ T η(t) . η (t) ˜ T  E τ12 f (t)Q f (t) − Q Q τ12

10.2 Main Results

229

This completes the proof. Now we can investigate the following result which gives a sufficient condition guaranteeing the existence of H∞ filters Theorem 10.6 For given scalars γ > 0, τ1 , τ2 , σ1 , σ2 , τd , σd , h 1 and h 2 , the filtering error system (10.8) subject to (10.3)–(10.6) is mean square robustly asymptotically stable with disturbance attenuation level γ if there exist matrices Ti := diag(ti1 , . . . , tin ) > 0, PiT = Pi > 0, WiT = Wi > 0 (i = 1, 2), Q Tj = Q j > 0 ( j ∈ 5), RlT = Rl > 0 (l ∈ 6), G, Q˜ 5 , R˜ 5 , J , M, N and S, and scalars εm (m ∈ 4) such that    Q 5 Q˜ 5 R5 R˜ 5 R3 G  0,  0,  0, (10.11) G T R6 Q˜ T5 Q 5 R˜ 5T R5 ⎡ ⎤ Υ˜ Γ1 Γ2 Γ3 Γ4 ⎢Γ T −ε1 I 0 0 0 ⎥ ⎢ 1 ⎥ ⎢Γ T 0 −ε2 I 0 (10.12) 0 ⎥ ⎢ 2 ⎥ < 0, ⎣Γ T 0 ⎦ 0 −ε I 0 3 3 0 0 −ε4 I Γ4T 0 where Υ˜ = Φ˜ + ε1 ΓaT Γa + ε2 ΓbT Γb + ε3 ΓcT Γc + ε4 ΓdT Γd , ⎡ ⎤ ⎡ T T ⎤ T Φ B T P1 DT P2 [e1 e2 ]P1 Γ˜1 + e13 J Θ1 ⎦, Φ˜ = ⎣ P1 B −P1 0 ⎦ , Γ1 = ⎣ 0 P2 D 0 −P2 0 ⎡ T ⎡ T T ⎤ ⎤ T e14 MΘ2 [e3 e4 ]P2 Γ˜3 + e15 N Θ3 ⎦, Γ2 = ⎣ P1 Γ˜2 ⎦ , Γ3 = ⎣ 0 0 0 ⎡ T ⎤ e16 SΘ4

Γ4 = ⎣ 0 ⎦ , Γi = Γ˜i 0 0 , i = a, b, c, d, P2 Γ˜4 Φ = Φ1 + Φ 2 + Φ 3 , T e19 , Φ1 = (e1 − e2 )T (e1 − e2 ) + (e3 − e4 )T (e3 − e4 ) − γ 2 e19

 



Φ2 = sym e1T e2T P1 A + e3T e4T P2 C , Φ3 =

6 

Φ3 j + sym((I1 A − e13 )T J T e13 + (I1 B − e14 )T M T e14

j=1

+(I1 C − e15 )T N T e15 + (I1 D − e16 )T S T e16 ),

230

10 H∞ State Estimation for Delayed Stochastic GRNs

Φ31 = e1T Q 1 e1 + e5T (Q 2 − Q 1 )e5 − e7T Q 2 e7 T +e3T R1 e3 + e8T (R2 − R1 )e8 − e10 R2 e10 ,

Φ32 = e1T Q 3 e1 − (1 − τd )e6T Q 3 e6    T   T  R3 G e3 R3 G e9 e3 e9 − (1 − σd ) , + e11 e12 G T R6 e11 G T R6 e12 T Φ33 = e13 (τ1 Q 4 + τ12 Q 5 )e13 − τ1−1 (e1 − e5 )T Q 4 (e1 − e5 )  T   Q 5 Q˜ 5 e5 − e6 −1 e5 − e6 −τ12 , e6 − e7 Q˜ T5 Q 5 e6 − e7 T (σ1 R4 + σ12 R5 )e15 − σ1−1 (e3 − e8 )T R4 (e3 − e8 ) Φ34 = e15  T   R5 R˜ 5 e8 − e9 −1 e8 − e9 , −σ12 e9 − e10 R˜ 5T R5 e9 − e10 T T T Φ35 = h 21 e11 W1 e11 + h 22 e1T W2 e1 − e17 W1 e17 − e18 W2 e18 ,

Φ36 = sym(−(e11 − K e3 )T T1 e11 − (e12 − K e9 )T T2 e12 ),

diag(Ξ j , He ), i f j = a, c, ˜ Γj = diag(Ξ j , 0), i f j = b, d,

diag(H j , 010n×10n , H j , 04n×4n , H j , 0n×n ), i f j = a, b, Ξj = diag(02n×2n , H j , 02n×2n , H j , 011n×11n , H j ), i f j = c, d,  

E a0 ˜ Γ1 = Ξ 1 , Γ˜2 = Ξ1 0 , B f E ay  

E c0 ˜ Γ 3 = Ξ2 , Γ˜4 = Ξ2 0 , D f E cy



Θ1 = I1 Ξ1 E a0 , Θ2 = I1 Ξ1 0 ,



Θ3 = I1 Ξ2 E c0 , Θ4 = I1 Ξ2 0 ,    E1 E2 E0 0 02n×9n 02n×4n 02n×n , Ξ1 = B f E4 B f E5 B f E3 0     E0 0 E1 E2 02n×11n , Ξ2 = 02n×2n 02n×n D f E4 D f E5 D f E3 0  0, . . . , 0 I 0, . . . , 0 ei =       i ∈ 19. i−1

19−i

Proof Due to the Schur complement lemma and Lemma 1.11, it follows from (10.12) that ¯ ¯ ¯ ¯ Φ˜ + sym(Γ1 F(t)Γ a + Γ2 F(t)Γb + Γ3 F(t)Γc + Γ4 F(t)Γd ) < 0

10.2 Main Results

231

for any F(t) satisfying (10.6), where ¯ F(t) = diag(F(t), . . . , F(t)).    19

By direct computation, we obtain ⎡

⎤ Φ + ΔΦ(t) (B + ΔB(t))T P1 (D + ΔD(t))T P2 ⎣ P1 (B + ΔB(t)) ⎦ 0, implies that ρ1 E{em (t)22 + e p (t)22 }  E{V (t)}  E{V (0)}   2 2  ρ2 sup E{φ˜ m (t) } + sup E{φ˜ p (t) } 2

t∈[−d, 0]

t∈[−d, 0]

2

for some positive constants ρ1 and ρ2 . On the other hand, we can derive E{LV (t)}  −ρ3 E{em (t)22 + e p (t)22 } for some positive constant ρ3 , which implies that

 E 0

 ρ−1 3 ρ2

t

 (em (s)22

+

e p (s)22 )ds 

 sup E{φ˜ m (t)22 } +

t∈[−d, 0]

sup E{φ˜ p (t)22 } .

t∈[−d, 0]

By Lemmas 1.19 and 1.20 , it is easy to see that the filtering error system (10.8) with ν(t) = 0 is mean square robustly asymptotically stable. For any nonzero ν(t), under the zero initial condition we introduce the following index 

 t   2 2 2 T em (s)2 + e p (s)2 − γ ν (s)ν(s) ds . J (t) = E 0

Clearly,



   em (s)22 + e p (s)22 − γ 2 ν T (s)ν(s) + LV (s) ds − E{V (t)} 0 

 t   em (s)22 + e p (s)22 − γ 2 ν T (s)ν(s) + LV (s) ds E 0   t ξ T (s) Φ + ΔΦ(s) + (B + ΔB(s))T P1 (B + ΔB(s)) E 0   +(D + ΔD(s))T P2 (D + ΔD(s)) ξ(s)ds , ∀t > 0.

J (t) = E

t

234

10 H∞ State Estimation for Delayed Stochastic GRNs

Hence, if (10.12) holds, then J (t) < 0 for any t > 0 from (10.14). Therefore, the filtering error system (10.8) is mean square robustly asymptotically stable with disturbance attenuation level γ. This completes the proof. The following theorem offers an approach to design an H∞ filter in terms of LMIs for the filtering error system (10.8) by using Theorem 10.6. Theorem 10.7 For given scalars γ > 0, τ1 , τ2 , σ1 , σ2 , τd , σd , h 1 and h 2 , the filtering error system (10.8) subject to (10.3)–(10.6) is mean square robustly asymptotically stable with disturbance attenuation level γ if there exist matrices Ti :=  Pi1 Pi2 T > 0, WiT = Wi > 0 (i = 1, 2), Q Tj = diag(ti1 , . . . , tin ) > 0, Pi = Pi := Pi2 Pi2 Q j > 0 ( j ∈ 5), RlT = Rl > 0 (l ∈ 6), A f , B f , C f , D f , G, Q˜ 5 , R˜ 5 , J , M, N and S, and scalars εm (m ∈ 4) such that (10.11) and ⎡

⎤ Υˆ Γˆ1 Γˆ2 Γˆ3 Γˆ4 ⎢Γˆ T −ε I 0 0 0 ⎥ ⎢ 1 ⎥ 1 ⎢ ˆT ⎥ 0 ⎥ < 0, ⎢Γ2 0 −ε2 I 0 ⎢ T ⎥ ⎣Γˆ3 0 0 −ε3 I 0 ⎦ Γˆ4T 0 0 0 −ε4 I where Υˆ = Φˆ + ε1 ΓaT Γa + ε2 ΓbT Γb + ε3 ΓcT Γc + ε4 ΓdT Γd , ⎡ ⎤ ⎡ T T ⎤ T Φ¯ Bˆ T Dˆ T [e1 e2 ]Γ¯1 + e13 J Θ1 ⎢ ⎥ ⎦, Φˆ = ⎣ Bˆ −P1 0 ⎦ , Γˆ1 = ⎣ 0 ˆ 0 D 0 −P2 ⎤ ⎡ T ⎡ T T ⎡ T ⎤ ⎤ T e14 MΘ2 e16 SΘ4 [e3 e4 ]Γ¯3 + e15 N Θ3 ⎦ , Γˆ4 = ⎣ 0 ⎦ , Γˆ2 = ⎣ Γ¯2 ⎦ , Γˆ3 = ⎣ 0 Γ¯4 0 0 Φ¯ = Φ1 + Φ¯ 2 + Φ3 , 



 Φ¯ 2 = sym e1T e2T Aˆ + e3T e4T Cˆ ,  

P11 E a0 + B f E ay Γ¯1 = Ξ¯ 1 , Γ¯2 = Ξ¯ 1 0 , P12 E a0 + B f E ay  

P21 E c0 + D f E cy Γ¯3 = Ξ¯ 2 , Γ¯4 = Ξ¯ 2 0 , P22 E c0 + D f E cy  E P 11 0 + B f E 3 0 0n×9n P11 E 1 + B f E 4 Ξ¯ 1 = P12 E 0 + B f E 3 0 0n×9n P12 E 1 + B f E 4 0n×4n P11 E 2 + B f E 5 0n×n , 0n×4n P12 E 2 + B f E 5 0n×n

(10.28)

10.2 Main Results

235

 0 P E + D f E 3 0 0n×n P21 E 1 + D f E 4 Ξ¯ 2 = n×2n 21 0 0n×2n P22 E 0 + D f E 3 0 0n×n P22 E 1 + D f E 4 0n×11n P21 E 2 + D f E 5 , 0n×11n P22 E 2 + D f E 5

Aˆ = Aˆ 02n×9n Aˆ 1 02n×4n Aˆ 2 02n×n Aˆ 3 ,

Bˆ = Bˆ 02n×9n Bˆ 1 02n×4n Bˆ 2 02n×2n ,

Cˆ = 02n×2n Cˆ 02n×n Cˆ 1 02n×11n Cˆ 2 Cˆ 3 ,

Dˆ = 02n×2n Dˆ 02n×n Dˆ 1 02n×11n Dˆ 2 02n×n ,   P11 A1 + B f A4 P11 A + B f A3 A f ˆ ˆ , A1 = , A= P12 A + B f A3 A f P12 A1 + B f A4   P A + B f A5 P A + B f Ay , Aˆ 3 = 11 0 , Aˆ 2 = 11 2 P12 A2 + B f A5 P12 A0 + B f A y   P B + B f B3 0 P B + B f B4 , Bˆ = 11 , Bˆ 1 = 11 1 P12 B1 + B f B4 P12 B + B f B3 0   P C + D f C3 C f P B + B f B5 , Cˆ = 21 , Bˆ 2 = 11 2 P12 B2 + B f B5 P22 C + D f C3 C f   P C + D f C4 P C + D f C5 Cˆ 1 = 21 1 , Cˆ 2 = 21 2 , P22 C1 + D f C4 P22 C2 + D f C5   P D + D f D3 0 P C + D f Cy , Dˆ = 21 , Cˆ 3 = 21 0 P22 C0 + D f C y P22 D + D f D3 0   P D + D f D4 P D + D f D5 , Dˆ 2 = 21 2 , Dˆ 1 = 21 1 P22 D1 + D f D4 P22 D2 + D f D5 and Θm (m ∈ 4), Γi (i = a, b, c, d), e j ( j ∈ 19), Φ1 and Φ3 are defined as in Theorem 10.6. In this case, the parameters of the desired filter can be given by −1 −1 −1 −1 A f , B f = P12 B f , C f = P22 C f , D f = P22 Df. A f = P12

When

t t−h 1

f ( p(s))ds ˜ and

t t−h 2

m(s)ds ˜ vanish, system (10.2) reduces to

˜ − σ(t))) dm(t) ˜ = [(A + ΔA(t))m(t) ˜ + (A1 + ΔA1 (t)) f ( p(t +(A0 + ΔA0 (t))ν(t)]dt + [(B + ΔB(t))m(t) ˜ +(B1 + ΔB1 (t)) f ( p(t ˜ − σ(t)))]dω1 (t),

(10.29a)

˜ − τ (t)) d p(t) ˜ = [(C + ΔC(t)) p(t) ˜ + (C1 + ΔC1 (t))m(t +(C0 + ΔC0 (t))ν(t)]dt + [(D + ΔD(t)) p(t) ˜ +(D1 + ΔD1 (t))m(t ˜ − τ (t))]dω2 (t),

(10.29b)

236

10 H∞ State Estimation for Delayed Stochastic GRNs

dym (t) = [(A3 + ΔA3 (t))m(t) ˜ + (A4 + ΔA4 (t)) f ( p(t ˜ − σ(t))) +(A y + ΔA y (t))ν(t)]dt + [(B3 + ΔB3 (t))m(t) ˜ +(B4 + ΔB4 (t)) f ( p(t ˜ − σ(t)))]dω1 (t),

(10.29c)

˜ + (C4 + ΔC4 (t))m(t ˜ − τ (t)) dy p (t) = [(C3 + ΔC3 (t)) p(t) +(C y + ΔC y (t))ν(t)]dt + [(D3 + ΔD3 (t)) p(t) ˜ +(D4 + ΔD4 (t))m(t ˜ − τ (t))]dω2 (t),

(10.29d)

˜ = φ p (t), ∀t ∈ [−d, 0]. m(t) ˜ = φm (t), p(t)

(10.29e)

We finish this section by the following theorem which gives an approach to design H∞ filter (10.7) for system (10.29). Theorem 10.8 For given scalars γ > 0, τ1 , τ2 , σ1 , σ2 , τd and σd , the filtering error system of system (10.29) subject to (10.3)–(10.6) is mean square robustly asymptotically stable with H∞ disturbance attenuation level γ if there exist matri P P i1 i2 > 0 (i = 1, 2), Q Tj = Q j > ces Ti := diag(ti1 , . . . , tin ) > 0, PiT = Pi := Pi2 Pi2 0 ( j ∈ 5), RlT = Rl > 0 (l ∈ 6), A f , B f , C f , D f , G, Q˜ 5 , R˜ 5 , J , M, N and S, and scalars εm (m ∈ 4) such that (10.11) and ⎡ ˘ ⎤ Υ Γ˘1 Γ˘2 Γ˘3 Γ˘4 ⎢Γ˘1T −ε1 I 0 0 0 ⎥ ⎢ ⎥ ⎢Γ˘ T 0 −ε2 I 0 0 ⎥ ⎢ 2 ⎥ < 0, ⎣Γ˘ T 0 ⎦ 0 −ε I 0 3 3 T Γ˘4 0 0 0 −ε4 I

(10.30)

where Υ˘ = Φ˘ + ε1 Γ˘aT Γ˘a + ε2 Γ˘bT Γ˘b + ε3 Γ˘cT Γ˘c + ε4 Γ˘dT Γ˘d , ⎡ ⎤ ⎡ T T ⎤ T Ψ B˘ T D˘ T [e˘1 e˘2 ]Γ´1 + e˘13 J Θ´ 1 ⎦, Φ˘ = ⎣ B˘ −P1 0 ⎦ , Γ˘1 = ⎣ 0 0 D˘ 0 −P2 ⎡ T ⎡ T T ⎡ T ⎤ ⎤ ⎤ T e˘14 M Θ´ 2 e˘16 S Θ´ 4 [e˘3 e˘4 ]Γ´3 + e15 N Θ´ 3 ⎦ , Γ˘4 = ⎣ 0 ⎦ , Γ˘2 = ⎣ Γ´2 ⎦ , Γ˘3 = ⎣ 0 0 Γ´4 0 Ψ = Ψ1 + Ψ2 + Ψ3 , T e˘17 , Ψ1 = (e˘1 − e˘2 )T (e˘1 − e˘2 ) + (e˘3 − e˘4 )T (e˘3 − e˘4 ) − γ 2 e˘17 



 Ψ2 = sym e˘1T e˘2T A˘ + e˘3T e˘4T C˘ ,

10.2 Main Results

Ψ3 =

237

4 

Ψ3 j + sym((I1 A´ − e˘13 )T J T e˘13 − (e˘11 − K e˘3 )T T1 e˘11

j=1

−(e˘12 − K e˘9 )T T2 e˘12 + (I1 B´ − e˘14 )T M T e˘14 +(I1 C´ − e˘15 )T N T e˘15 + (I1 D´ − e˘16 )T S T e˘16 ), Ψ31 = e˘1T Q 1 e˘1 + e˘5T (Q 2 − Q 1 )e˘5 − e˘7T Q 2 e˘7 T +e˘3T R1 e˘3 + e˘8T (R2 − R1 )e˘8 − e˘10 R2 e˘10 , Ψ32 = e˘1T Q 3 e˘1 − (1 − τd )e˘6T Q 3 e˘6    T   T  R3 G e˘3 R3 G e˘9 e˘ e˘9 − (1 − σ , + 3 ) d e˘11 e˘12 G T R6 e˘11 G T R6 e˘12 T Ψ33 = +e˘13 (τ1 Q 4 + τ12 Q 5 )e˘13 − τ1−1 (e˘1 − e˘5 )T Q 4 (e˘1 − e˘5 )  T   Q 5 Q˜ 5 e˘5 − e˘6 −1 e˘5 − e˘6 , −τ12 e˘6 − e˘7 Q˜ T5 Q 5 e˘6 − e˘7 T Ψ34 = +e˘15 (σ1 R4 + σ12 R5 )e˘15 − σ1−1 (e˘3 − e˘8 )T R4 (e˘3 − e˘8 )  T   R5 R˜ 5 e˘8 − e˘9 −1 e˘8 − e˘9 , −σ12 e˘9 − e˘10 R˜ 5T R5 e˘9 − e˘10

A´ = A˜ 02n×9n A˜ 1 02n×4n A˜ 3 ,



B´ = B˜ 02n×9n B˜ 1 02n×5n ,

C´ = 02n×2n C˜ 02n×n C˜ 1 02n×10n C˜ 3 ,

D´ = 02n×2n D˜ 02n×n D˜ 1 02n×11n ,  

P11 E a0 + B f E ay ´ ˘ Γ 1 = Ξ1 , Γ´2 = Ξ˘ 1 0 , P12 E a0 + B f E ay  

P21 E c0 + D f E cy Γ´3 = Ξ˘ 2 , Γ´4 = Ξ˘ 2 0 , P22 E c0 + D f E cy   P11 E 1 + B f E 4 P11 E 0 + B f E 3 0 02n×9n Ξ˘ 1 = 02n×4n , P12 E 1 + B f E 4 P12 E 0 + B f E 3 0    P E + D f E3 0 P E + D f E4 Ξ˘ 2 = 02n×2n 21 0 02n×n 21 1 02n×10n , P22 E 1 + D f E 4 P22 E 0 + D f E 3 0

˘ A = Aˆ 02n×9n Aˆ 1 02n×4n Aˆ 3 ,

B˘ = Bˆ 02n×9n Bˆ 1 02n×5n ,

C˘ = 02n×2n Cˆ 02n×n Cˆ 1 02n×10n Cˆ 3 ,

238

10 H∞ State Estimation for Delayed Stochastic GRNs



D˘ = 02n×2n Dˆ 02n×n Dˆ 1 02n×11n ,

diag(Ξ˘ j , He ), i f j = a, c, Γ˘ j = diag(Ξ˘ j , 0), i f j = b, d,

i f j = a, b, diag(H j , 010n×10n , H j , 04n×4n ), Ξ˘ j = diag(02n×2n , H j , 02n×2n , H j , 010n×10n ), i f j = c, d,



Θ´ 1 = I1 Ξ˘ 1 E a0 , Θ´ 2 = I1 Ξ˘ 1 0 ,



Θ´ 3 = I1 Ξ˘ 3 E c0 , Θ´ 4 = I1 Ξ˘ 3 0 ,  0, . . . , 0 I 0, . . . , 0 e˘i =       , i ∈ 17, i−1

17−i

ˆ C, ˆ B, ˆ D, ˆ Aˆ j , Cˆ j ( j = 1, 3), Bˆ 1 and Dˆ 1 are defined as in Theorem 10.7. In and A, this case, the parameters of the desired filer can be given by −1 −1 −1 −1 A f , B f = P12 B f , C f = P22 C f , D f = P22 Df. A f = P12

Proof By removing V6 (t) in the proof of Theorem 10.6, Theorem 10.8 can be proven in view of the proof process of Theorems 10.6 and 10.7. Remark 10.9 In [11], the deigned filter only depends on τ˙ (t) and σ(t) ˙ with τ˙ (t)  τd < 1 and σ(t) ˙  σd < 1. But the obtained results in Theorem 10.8 depend on not only τ (t) and σ(t) with 0  τ1  τ (t)  τ2 and 0  σ1  σ(t)  σ2 but also τ˙ (t) ˙  σd < ∞. So, the applied range of Theorem and σ(t) ˙ with τ˙ (t)  τd < ∞ and σ(t) 10.8 is more extensive than ones in [11]. Remark 10.10 Compared with Theorem 10.8 and [17, Theorem 4.1], we not only remove some free-weighting matrices but also reduce complexity of the LKF used in [17] (i.e., the terms involving f 2 (s) and f 4 (s) in the LKF were removed). This reduces the amount of calculation.

10.3 Numerical Examples In this section, we present the following numerical examples to show the validity of the proposed main results above. Example 10.11 Consider the uncertain stochastic GRNs (10.29) with the following parameters: A = diag(−3, −3), A0 = diag(−0.3, −0.3), A3 = diag(0.1, 1.3), A y = diag(−0.6, 0.3),

10.3 Numerical Examples

239

B = D = diag(0.41, 0.42), C = diag(−3.5, −3.5), C1 = diag(2.12, 0.22), C0 = diag(0.7, −0.2), C3 = diag(3.1, 0.03), C y = diag(−0.9, 0.5), E 0 = E 1 = E 3 = diag(0.1, 0.3),

E ay

Ha = Hb = Hc = Hd = diag(0.55, 0.21),   0 0 0.2 0.11 , E a0 = E c0 = , A1 = −0.2 0 0.17 0.3    0.3 0.2 −0.3 0.7 0.55 0.67 = , E cy = , He = , 0.2 0.3 0.2 −0.7 0.3 −1

A4 = B1 = B3 = B4 = C4 = D1 = D3 = D4 = E 4 = 0. Assume that K = diag(0.65, 0.65), τ1 = 0.5, τ2 = 0.7, τd = 0.93, σ1 = 0.1, σ2 = 0.3 and σd = 0.31. Firstly, by using the toolbox YALMIP of MATLAB to solve LMIs (10.11) and (10.30), we obtain that the minimal disturbance attenuation level γ = 0.5380, and the corresponding filter gains are given by 

 −2.1525 −0.4247 −0.9509 −212.8030 , Bf = , −0.2530 −0.0666 −0.1080 −30.6348   −2.9542 0.0026 2.6854 6.6927 , Df = . C f = 104 × −0.0019 −0.0004 −0.0003 0.6085

A f = 103 ×

Example 10.12 Consider the uncertain stochastic GRN (10.2) with the following parameters: A = diag(−3, −3), A0 = diag(−0.3, −0.3), A3 = diag(0.1, 1.3), B = D = diag(0.41, 0.42), C = diag(−3.5, −3.5), C1 = diag(2.12, 0.22), C0 = diag(0.7, −0.2), C3 = diag(3.1, 0.03), C2 = C5 = C y = diag(−0.9, 0.5), A y = A2 = A5 = diag(−0.6, 0.3), E a0 = E ay = E c0 = E cy = diag(0.01, 0.02), E i = H j = diag(0.01, 0.02), i = 0, . . . , 5, j = a, b, c, d, e,

240

10 H∞ State Estimation for Delayed Stochastic GRNs 5

2

4

1.5 1

w2 (t)

w1 (t)

3 2 1

0 -0.5

0 -1

0.5

-1 0

1

2

3

4

5

6

7

t

-1.5

0

1

2

3

4

5

6

7

t

Fig. 10.1 Wiener processes (Example 10.12)



 0 0 0 0 A1 = , A4 = B1 = B2 = B4 = B5 = , −0.2 0 −0.1 0 B3 = C4 = D1 = D2 = D3 = D4 = D5 = diag(0.1, 0.2), τ1 = σ1 = σd = 0.1, τ2 = σ2 = 2, τd = 1.2, h 1 = 2, h 2 = 3. Assume that K = diag(0.65, 0.65). By taking γ = 1.7855 and using the toolbox YALMIP of MATLAB, we found that the feasible solution of LMIs (10.11) and (10.28) in Theorem 10.7 exists, and the filter gains are given by   −278.7503 −52.6429 −0.8962 −0.0910 Af = , Bf = , −52.3806 −56.8680 −0.0754 −1.0494   −156.8648 −4.2084 −2.7113 0.0670 , Df = . Cf = −0.6558 −3.6896 0.0100 −1.1497 Furthermore, let σ(t) = 1.05 + 0.95 sin(2t/19), τ (t) = 1.05 + 0.95 sin(24t/19), s2 h 1 = 2, h 2 = 3, F(t) ≡ diag(1, 1), ν(t) ≡ 0 and f i (s) = 1+s 2 , i = 1, 2. When the Wiener processes ω1 (t) and ω2 (t) are shown as in Fig. 10.1, the simulation results of the trajectories, estimations and estimation errors of the variables m(t) ˜ and p(t) ˜ with initial conditions col(1, −0.3) and col(2, 0.4) are given in Figs. 10.2, 10.3 and 10.4, respectively. The simulation results illustrate that the uncertain stochastic GRN (10.2) is mean square asymptotically stable. Remark 10.13 In Example 10.12, the true values and estimations of the concentrations of mRNAs and proteins are listed in Figs. 10.2 and 10.3. From which, we can see that the filter has tracked the original system in the presence of Itô-type stochastic disturbance, norm-bounded parameter uncertainties, time-varying discrete delays, and constant distributed delays.

10.3 Numerical Examples

241

1.2

m ˜ 1 (t) m ˆ 1 (t)

˜ 2 (t) and m m ˆ 2 (t)

m ˜ 1 (t) and m ˆ 1 (t)

1 0.8 0.6 0.4 0.2 0 -0.2

0

1

2

3

4

5

6

7

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4

m ˜ 2 (t) m ˆ 2 (t)

0

1

2

3

t

t

4

5

6

7

Fig. 10.2 The mRNA concentrations and their estimations (Example 10.12) 2.5

p˜2 (t) pˆ2 (t)

0.3

p˜2 (t) and pˆ2 (t)

2

p˜1 (t) and pˆ1 (t)

0.4

p˜1 (t) pˆ1 (t)

1.5 1 0.5 0

0.2 0.1 0 -0.1

-0.5 0

1

2

3

4

5

6

-0.2 0

7

1

2

3

t

4

5

6

7

t

Fig. 10.3 The protein concentrations and their estimations (Example 10.12) 1.2

pˆ(t)

0.6 0.4 0.2 0 -0.2

1.5 1 0.5 0

-0.4 -0.6

p˜1 (t) − pˆ1 (t) p˜(t)2 − pˆ2 (t)

2

p˜(t)

m(t) ˆ

0.8

m(t) ˜

2.5

m ˜ 1 (t) − m ˆ 1 (t) m(t) ˜ 2−m ˆ 2 (t)

1

0

1

2

3

4

5

6

t

Fig. 10.4 Estimation errors (Example 10.12)

7

-0.5

0

1

2

3

4

t

5

6

7

242

10 H∞ State Estimation for Delayed Stochastic GRNs

10.4 Remarks and Notes This chapter is concerned with the problem of robust H∞ filter for uncertain stochastic GRNs with mixed delays. The GRN model under consideration involves Itô-type stochastic disturbance, norm-bounded parameter uncertainties, time-varying discrete delays and constant distributed delays. By constructing an appropriate LKF and using reciprocal convex technique, we obtain less conservative LMI conditions which guarantee the filtering error systems are mean square robustly asymptotically stable with pre-specified disturbance attenuation level. The chapter is taken from [18]. We end this chapter by introducing the following items, which are related to this chapter: 1. In [19], the linear filtering problem for nonlinear stochastic GRNs with constant delays is investigated. Extending the model in [19] to parameter uncertainties and time-varying delays, the authors of [11] have studied the robust H∞ filtering problem. 2. Lakshmanan et al. [3] designed state estimator for delayed GRNs with randomly occurring uncertainties by employing the LKF method and LMI technique. 3. For a class of delayed stochastic GRNs, the problem of designing robust H∞ observer-based controller has been addressed in [16], the problem of sampled-data H∞ filter has been studied in [15], and the design approaches of state estimators have been developed in [6, 17, 20]. 4. Delay-dependent stochastic stability analysis and filter design for uncertain stochastic GRNs with time-varying delays has been achieved in [5]. 5. The problem of H∞ filtering design for a class of delayed Markovian jumping GRNs with intrinsic fluctuations and extrinsic noises has been discussed in [1, 10]. While the state estimation problem for delayed Markovian jumping GRNs has been investigated in [4, 7–9, 12, 13]. 6. By using the LKF approach and the reciprocally convex technique, Revathi et al. [14] designed an H∞ filter for a class of Markovian switching stochastic delayed GRNs. The exponential H∞ filtering problem is investigated in [21] for a class of switching-type stochastic GRNs with random sensor delays by using the average dwell time approach.

References 1. Chen, B., Yu, L., Zhang, W.A.: H∞ filtering for Markovian switching genetic regulatory networks with time-delays and stochastic disturbances. Circuits Syst. Signal Process. 30(6), 1231– 1252 (2011) 2. Kushur, H.: Stochastic Stability and Control. Accademic, New York (1976) 3. Lakshmanan, S., Park, H., Jung, H.Y., Balasubramaniam, P., Lee, S.M.: Design of state estimator for genetic regulatory networks with time-varying delays and randomly occurring uncertainties. Biosystems 111(1), 51–70 (2013)

References

243

4. Lee, T.H., Lakshmanan, S., Park, J.H., Balasubramaniam, P.: State estimation for genetic regulatory networks with mode-dependent leakage delays, time-varying delays, and Markovian jumping parameters. IEEE Trans. Nanobiosci. 12(4), 363–375 (2013) 5. Li, Y., Liu, G.P.: Delay-dependent robust dissipative filtering of stochastic genetic regulatory networks with time-varying delays. IET Control Theory Appl. 7(11), 1520–1528 (2013) 6. Liang, J.L., Lam, J.: Robust state estimation for stochastic genetic regulatory networks. Int. J. Syst. Sci. 41(1), 47–63 (2010) 7. Liang, J.L., Lam, J., Wang, Z.D.: State estimation for Markov-type genetic regulatory networks with delays and uncertain mode transition rates. Phys. Lett. A 373(47), 4328–4337 (2009) 8. Liu, J., Tian, E., Gu, Z., Zhang, Y.: State estimation for Markovian jumping genetic regulatory networks with random delays. Commun. Nonlinear Sci. Numer. Simul. 19(7), 2479–2492 (2014) 9. Lv, B., Liang, J.L., Cao, J.D.: Robust distributed state estimation for genetic regulatory networks with Markovian jumping parameters. Commun. Nonlinear Sci. Numer. Simul. 16(10), 4060– 4078 (2011) 10. Mohammadian, M., H. Abolmasoumi, A., R. Momeni, H.: H∞ mode-independent filter design for Markovian jump genetic regulatory networks with time-varying delays. Neurocomputing 87(15), 10–18 (2012) 11. Mousavi, S.M., Majd, V.J.: Robust filtering of extended stochastic genetic regulatory networks with parameter uncertainties, disturbances, and time-varying delays. Neurocomputing 74, 2123–2134 (2011) 12. Rakkiyappan, R., Chandrasekar, A., Rihan, F.A., Lakshmanan, S.: Corrigendum to “exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays” [Math. Biosci., 251 (2014) 30–53]. Math. Biosci. 255, 91 (2014) 13. Rakkiyappan, R., Chandrasekar, A., Rihan, F.A., Lakshmanan, S.: Exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic timevarying delays. Math. Biosci. 251, 30–53 (2014) 14. Revathi, V.M., Balasubramaniam, P., Ratnavelu, K.: Mode-dependent H∞ filtering for stochastic Markovian switching genetic regulatory networks with leakage and time-varying delays. Circuits Syst. Signal Process. 33(11), 3349–3388 (2014) 15. Shen, B., Wang, Z.D., Liang, J.L., Liu, X.H.: Sampled-data H∞ filtering for stochastic genetic regulatory networks. Int. J. Robust Nonlinear Control 21, 1759–1777 (2011) 16. Shokouhi-Nejad, H., Rikhtehgar-Ghiasi, A.: Robust H∞ observer-based controller for stochastic genetic regulatory networks. Math. Biosci. 250, 41–53 (2014) 17. Wang, W., Zhong, S., Liu, F.: Robust filtering of uncertain stochastic genetic regulatory networks with time-varying delays. Chaos Solitons Fractals 45(7), 915–929 (2012) 18. Wang, Y.T., Zhang, X., Hu, Z.R.: Delay-dependent robust H∞ filtering of uncertain stochastic genetic regulatory networks with mixed time-varying delays. Neurocomputing 166, 346–356 (2015) 19. Wang, Z., Lam, J., Wei, G., Fraser, K., Liu, X.: Filtering for nonlinear genetic regulatory networks with stochastic disturbances. IEEE Trans. Autom. Control 53(10), 2448–2457 (2008) 20. Wei, G.L., Wang, Z.D., Lam, J., Fraser, K., Rao, G.P., Liu, X.H.: Robust filtering for stochastic genetic regulatory networks with time-varying delays. Math. Biosci. 220, 73–80 (2009) 21. Zhang, D., Yu, L., Wang, Q.G.: Exponential H∞ filtering for switched stochastic genetic regulatory networks with random sensor delays. Asian J. Control 13(5), 749–755 (2011)

Chapter 11

H∞ State Estimation for Delayed Discrete-Time GRNs

This chapter is concerned with the problem of H∞ state estimation for a class of discrete-time GRNs with random delay and external disturbance. The random delay is described by a Markovian chain. The aim is to estimate the concentrations of mRNAs and proteins by designing H∞ filter based on available measurement outputs. By using the LKF method, a sufficient LMI condition is first established to ensure the filtering error system to be stochastically stable with a prescribed H∞ disturbance attenuation level. The condition is dependent on the transition probability matrix of the random delay. Then, the filter gains are represented via a feasible solution of the LMIs. Moreover, an optimization problem with LMIs constraints is established to design an H∞ filter which ensures an optimal H∞ disturbance attenuation level. The effectiveness of the proposed approach is illustrated by a numerical example.

11.1 Problem Formulation Suppose the discretization step-size h is a fixed positive real number. For any t ∈ [0, +∞), set k = t/ h, where · represents the integral function, then k is a non-negative integer for any t ∈ [kh, (k + 1)h). Let σ(t)/ h = ω(k) and τ (t)/ h = κ(k). Then, the continuous-time GRN model (1.5) can be transformed into the following approximated form: m˙ i (t) = −ai m i (t) +

n 

wi j g j (P j (k − ω(k))) + Ji , i ∈ n,

(11.1a)

j=1

p˙ i (t) = −ci pi (t) + di Mi (k − κ(k)), i ∈ n,

© Springer Nature Switzerland AG 2019 X. Zhang et al., Analysis and Design of Delayed Genetic Regulatory Networks, Studies in Systems, Decision and Control 207, https://doi.org/10.1007/978-3-030-17098-1_11

(11.1b)

245

246

11 H∞ State Estimation for Delayed Discrete-Time GRNs

where P j (k − ω(k)) and Mi (k − κ(k)) represent pi (kh − ω(k)h) and m i (kh − κ(k)h), respectively, and the other symbols are the same with previous ones. Multiplying both sides of (11.1a) with eai t ((11.1b) with eci t , respectively), integrating over the interval [kh, t] with t < (k + 1)h, and taking the limit t → (k + 1)h, we obtain Mi (k + 1) = e

−ai h

Mi (k) + φi (h)

n 

wi j g j (P j (k − ω(k)))

j=1

+φi (h)Ji , i ∈ n, Pi (k + 1) = e−ci h Pi (k) + ϕi (h)di Mi (k − κ(k)), i ∈ n,

(11.2a) (11.2b)

where φi (h):=(1 − e−ai h )/ai > 0, ϕi (h):=(1 − e−ci h)/ci > 0. Clearly, GRN (11.2) can rewritten as the following compact matrix form: M(k + 1) = AM(k) + W g(P(k − ω(k))) + J,

(11.3a)

P(k + 1) = C P(k) + D M(k − κ(k)),

(11.3b)

where M(k) = col(M1 (k), M2 (k), . . . , Mn (k)), P(k) = col(P1 (k), P2 (k), . . . , Pn (k)), g(P(k)) = col(g1 (P1 (k)), g2 (P2 (k)), . . . , gn (Pn (k))), J = col(φ1 (h)J1 , φ2 (h)J2 , . . . , φn (h)Jn ),     A = diag e−a1 h , e−a2 h , . . . , e−an h , C = diag e−c1 h , e−c2 h , . . . , e−cn h ,   D = diag (ϕ1 (h)d1 , ϕ2 (h)d2 , . . . , ϕn (h)dn ) , W = φi (h)wi j n×n . In Corollary 2.7, it has been shown that GRN (11.3) has at least one nonnegative equilibrium point. Let (M ∗ , P ∗ ) be the equilibrium point of (11.3), that is, M ∗ = AM ∗ + W g(P ∗ ) + J, P ∗ = C P ∗ + D M ∗ .

11.1 Problem Formulation

247

To simplify the analysis, one can transform the equilibrium point to the origin by the relation xm (k) = M(k) − M ∗ and x p (k) = P(k) − P ∗ . Then the transformed GRN is changed as follows: xm (k + 1) = Axm (k) + W f (x p (k − ω(k))), x p (k + 1) = C x p (k) + Dxm (k − κ(k)), where f (·) = g(· + P ∗ ) − g(P ∗ ). For every i ∈ n, since gi is a monotonic function in Hill form, one can easily obtain that f i is a monotonically increasing function with saturation and satisfies f i (0) = 0, 0  f i (s)/s  ki , ∀0 = s ∈ R,

(11.5)

where ki is defined as previously. Set K = diag(k1 , k2 , . . . , kn ). When we take external perturbations into account, and assume that ω(k) = κ(k) = d(k), where d(k) is random delay (i.e., d(k) is a Markovian chain with state space ¯ and transition probability matrix π, where d¯ is a fixed positive integer), a class of d stochastic discrete-time GRN model with random delays is represented as follows: xm (k + 1) = Axm (k) + W f (x p (k − d(k))) + E 1 w(k),

(11.6a)

x p (k + 1) = C x p (k) + Dxm (k − d(k)) + F1 v(k),

(11.6b)

where A, W , C, D, E 1 and F1 are constant matrices of appropriate sizes, and w(k) and v(k) are external disturbance signals. Assume that the initial conditions of (11.6) are as follows: ¯ −d¯ + 1, . . . , 0. xm (k) = θm (k), x p (k) = θ p (k), k = −d, Let ym (k) = C1 xm (k) + E 2 w(k), y p (k) = C2 x p (k) + F2 v(k)

(11.7)

be the expression levels of mRNA and protein, and z m (k) = G 1 xm (k), z p (k) = G 2 x p (k) be the estimated signals, where C1 , C2 , E 2 , F2 , G 1 and G 2 are constant matrices of appropriate sizes. In complex GRNs, only the partial information of the network states can be usually measured. Therefore, in order to obtain the states of GRNs, it is necessary to estimate them via available measurements (11.7) (see [6]). We will use the following full order linear filter:

248

11 H∞ State Estimation for Delayed Discrete-Time GRNs

xˆm (k + 1) = A f xˆm (k) + B f ym (k),

(11.8a)

xˆ p (k + 1) = C f xˆ p (k) + D f y p (k),

(11.8b)

zˆ m (k) = G 1 f xˆm (k) + H1 f ym (k),

(11.8c)

zˆ p (k) = G 2 f xˆ p (k) + H2 f y p (k),

(11.8d)

where xˆm (k), xˆ p (k), zˆ m (k) and zˆ p (k) are the estimates of xm (k), x p (k), z m (k) and z p (k), respectively, and A f , B f , C f , D f ∈ Rn×n and G 1 f , G 2 f , H1 f , H2 f ∈ Rl×n are the filter gains to be determined. Set x˜m (k) = col(xm (k), xˆm (k)), x˜ p (k) = col(x p (k), xˆ p (k)), em (k) = z m (k) − zˆ m (k), e p (k) = z p (k) − zˆ p (k). Then the filtering error system can be expressed as ¯ x˜m (k + 1) = A¯ x˜m (k) + W¯ f (I1 x˜ p (k − d(k))) + Ew(k),

(11.9a)

¯ 1 x˜m (k − d(k)) + Fv(k), ¯ x˜ p (k + 1) = C¯ x˜ p (k) + DI

(11.9b)

em (k) = G¯ 1 f x˜m (k) + H¯ 1 f w(k),

(11.9c)

e p (k) = G¯ 2 f x˜ p (k) + H¯ 2 f v(k),

(11.9d)

¯ −d¯ + 1, . . . , 0, x˜m (k) = θ˜m (k), x˜ p (k) = θ˜ p (k), k = −d,

(11.9e)

where θ˜m (k) = col(θm (k), 0), θ˜ p (k) = col(θ p (k), 0),      W C 0 A 0 , W¯ = , C¯ = , 0 D f C2 C f B f C1 A f       D F1 E1 D¯ = , F¯ = , , E¯ = B f E2 D f F2 0

A¯ =



    G¯ 1 f = G 1 − H1 f C1 −G 1 f , G¯ 2 f = G 2 − H2 f C2 −G 2 f ,

11.1 Problem Formulation

249

  H¯ 1 f = −H1 f E 2 , H¯ 2 f = −H2 f F2 , I1 = I 0 . For convenience, for a nonnegative integer k we define ¯ x˜ p (k), x˜ p (k − 1), . . . , x˜ p (k − d)}. ¯ Θk = {x˜m (k), x˜m (k − 1), . . . , x˜m (k − d), Definition 11.1 [7] When w(k) = 0 and v(k) = 0, the filtering error system (11.9) is said to be stochastically stable, if ∞ 

E{ x˜m (k) 22 + x˜ p (k) 22 | Θ0 , d(0)} < ∞

k=0

for every initial condition Θ0 and initial mode d(0), where E{·} represents the mathematical expectation operator. Definition 11.2 For a given scalar γ > 0, the filtering error system (11.9) is said to be stochastically stable with H∞ disturbance attenuation level γ if it is stochastically stable when w(k) = 0 and v(k) = 0, and under the zero initial conditions satisfies the following inequality ∞ 

 E x˜m (k) 22 + x˜ p (k) 22 Θ0 = 0, d(0)

k=0

< γ2

∞    w(k) 22 + v(k) 22 k=0

for all non-zero w(k), v(k) and initial mode d(0). The objective of this chapter is to design a filter (11.8) such that the filtering error system (11.9) is stochastically stable with H∞ disturbance attenuation level γ.

11.2

H∞ Filter Design

The stability analysis for the filtering error system (11.9) with w(k) = 0 and v(k) = 0 is presented by the following theorem. Theorem 11.3 The filtering error system (11.9) with w(k) = 0 and v(k) = 0 is stochastically stable, if there exist matrices ς := diag (ς1 , ς2 , . . . , ςn ) > 0, μ := ¯ and P jT = P j > 0 diag (μ1 , μ2 , . . . , μn ) > 0, PiT (r ) = Pi (r ) > 0 i ∈ 6; r ∈ d ( j = 2, 3, 5, 6) such that the following matrix inequalities (11.10)–(11.11) hold for ¯ all r ∈ d. ˜ ) + Ω(r ˆ ) < 0, Ω(r ) := Ω(r

(11.10)

250

11 H∞ State Estimation for Delayed Discrete-Time GRNs

P¯ j (r ) < P j , j = 2, 3, 5, 6,

(11.11)

where

d¯ 2 + d¯ T ¯ T ¯ ¯ ˆ Ω(r ) = Λ1 P1 (r )Λ1 + Λ2 d P3 (r ) + P3 Λ2 + ΛT3 P¯4 (r )Λ3 , 2   Λ1 = A¯ 0 0 W¯ 0 0 ,   Λ2 = A¯ − I 0 0 W¯ 0 0 ,   ¯ 1000 , Λ3 = 0 C¯ DI ⎡ Ω11 (r ) 0 Ω13 (r ) 0 0 T ⎢ 0 (r ) 0 0 −I −P 4 1 ςK ⎢ T ⎢Ω (r ) 0 Ω33 (r ) 0 0 ˜ ) = ⎢ 13 Ω(r ⎢ 0 (r ) Ω (r ) 0 0 Ω 44 45 ⎢ T ⎣ 0 0 Ω45 (r ) Ω55 (r ) −K ςI1 T Ω56 (r ) 0 −K μCI1 −K μDI1 0

⎤ 0 Ω26 (r )⎥ ⎥ Ω36 (r )⎥ ⎥, 0 ⎥ ⎥ Ω56 (r )⎦ Ω66 (r )

Ω11 (r ) = (d¯ − 1)P2 + P¯2 (r ) − P1 (r ) − Ω13 (r ), Ω13 (r ) =

1 1 P3 (r ) + P3 , Ω26 (r ) = −I1T C T μK , r r

Ω33 (r ) = −P2 (r ) − Ω13 (r ), Ω36 (r ) = −I1T D T μK , Ω44 (r ) = −P5 (r ) − Ω45 (r ), Ω45 (r ) =

1 1 P6 (r ) + P6 , r r

Ω55 (r ) = (d¯ − 1)P5 + P¯5 (r ) − Ω56 (r ) − Ω45 (r ) − 2ς, ¯ (d¯ 2 + d) P6 , Ω66 (r ) = −Ω56 (r ) − 2μ, Ω56 (r ) = −d¯ P¯6 (r ) − 2 P¯i (r ) =

d¯ 

πr s Pi (s), i ∈ 6.

s=1

Proof Choose an appropriate LKF for the filtering error system (11.9) with w(k) = 0 and v(k) = 0 as follows:

11.2 H∞ Filter Design

V (Θk , k, d(k)) =

251 3 

[Vm,i (Θk , k, d(k)) + V p,i (Θk , k, d(k))]

(11.12)

i=1

with Vm,1 (Θk , k, d(k)) = x˜mT (k)P1 (d(k))x˜m (k), V p,1 (Θk , k, d(k)) = x˜ Tp (k)P4 (d(k))x˜ p (k), k−1 

Vm,2 (Θk , k, d(k)) =

x˜mT (i)P2 (d(k))x˜m (i)

i=k−d(k)

+

−1 

k−1 

x˜mT (i)P2 x˜m (i),

¯ i=k+ j j=−d+1 k−1 

V p,2 (Θk , k, d(k)) =

f T (I1 x˜ p (i))P5 (d(k)) f (I1 x˜ p (i))

i=k−d(k)

+

−1 

k−1 

f T (I1 x˜ p (i))P5 f (I1 x˜ p (i)),

¯ i=k+ j j=−d+1

Vm,3 (Θk , k, d(k)) =

−1 

k−1 

η T (i)P3 (d(k))η(i)

j=−d(k) i=k+ j

+

−1  −1  k−1 

η T (i)P3 η(i),

j=−d¯ l= j i=k+l

V p,3 (Θk , k, d(k)) =

−1 

k−1 

ζ T (i)P6 (d(k))ζ(i)

j=−d(k) i=k+ j

+

−1  −1  k−1 

ζ T (i)P6 ζ(i),

j=−d¯ l= j i=k+l

where η(k) = x˜m (k + 1) − x˜m (k) and ζ(k) = f (I1 x˜ p (k + 1)) − f (I1 x˜ p (k)). By taking the forward difference of the functions Vm,i (Θk , k, d(k)) (i = 1, 2, 3) along with the solution of system (11.9), one can obtain that E{Vm,1 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − Vm,1 (Θk , k, r ) =

d¯  s=1

πr s x˜mT (k + 1)P1 (s)x˜m (k + 1) − x˜mT (k)P1 (r )x˜m (k)

252

11 H∞ State Estimation for Delayed Discrete-Time GRNs

= x˜mT (k + 1) P¯1 (r )x˜m (k + 1) − x˜mT (k)P1 (r )x˜m (k),

(11.13)

E{Vm,2 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − Vm,2 (Θk , k, r ) =

d¯ 

k 

πr s

s=1

+

i=k+1−s −1 

−1 

x˜mT (i)P2 x˜m (i) −

k−1 

x˜mT (i)P2 x˜m (i)

¯ i=k+ j j=−d+1

x˜mT (k) P¯2 (r )x˜m (k) − x˜mT (k − r )P2 (r )x˜m (k − r ) k−1 k−1   T ¯ + x˜m (i) P2 (r )x˜m (i) − x˜mT (i)P2 (r )x˜m (i) i=k+1−r i=k+1−d¯ +

−1 

x˜mT (k)P2 x˜m (k) −

¯ j=−d+1



x˜mT (i)P2 (r )x˜m (i)

i=k−r

k 

¯ i=k+1+ j j=−d+1

=

k−1 

x˜mT (i)P2 (s)x˜m (i) −

k−1 

x˜mT ( j)P2 x˜m ( j)

j=k+1−d¯

x˜mT (k) P¯2 (r )x˜m (k) − x˜mT (k − r )P2 (r )x˜m (k − r ) + (d¯ k−1 k−1   + x˜mT (i) P¯2 (r )x˜m (i) − x˜mT (i)P2 x˜m (i) i=k+1−d¯ i=k+1−d¯

− 1)x˜mT (k)P2 x˜m (k)

 x˜mT (k)[(d¯ − 1)P2 + P¯2 (r )]x˜m (k) − x˜mT (k − r )P2 (r )x˜m (k − r ), E{Vm,3 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − Vm,3 (Θk , k, r ) =

d¯ 

πr s

s=1

+

−1 



j=−d¯ l= j



η T (i)P3 (s)η(i) −

j=−s i=k+1+ j

−1  −1 

−1 

k 

k 

+

η T (i)P3 (r )η(i)

j=−r i=k+ j

η T (i)P3 η(i) −

i=k+1+l

η T (k) P¯3 (r )η(k) −

j=−d¯

k−1 −1  

k−1 



η T (i)P3 η(i)

i=k+l −1 

η T (k + j)P3 (r )η(k + j)

j=−r

−1 

k−1 

d¯ 2 + d¯ T η (k)P3 η(k) η T (i) P¯3 (r )η(i) + 2 ¯ i=k+1+ j

j=−d

−

−1 

k−1 

η T (i)P3 η(i) −

j=−d¯ i=k+1+ j



−1  j=−d¯

d + d¯  η T (k) d¯ P¯3 (r ) + P3 η(k) 2 ¯2

η T (k + j)P3 η(k + j)

(11.14)

11.2 H∞ Filter Design

− +

253

−1 

−1  1 η T (k + j) P3 (r ) η(k + j) r j=−r j=−r −1 

k−1 

η T (i) P¯3 (r )η(i)

j=−d¯ i=k+1+ j

−

−1 

k−1 

η T (i)P3 η(i)

j=−d¯ i=k+1+ j −1 

−1  1 − η (k + j) P3 η(k + j) r j=−r j=−r

d¯ 2 + d¯ P3 η(k)  η T (k) d¯ P¯3 (r ) + 2

−

T

−1 

−1  1 η T (k + j) (P3 (r ) + P3 ) η(k + j). r j=−r j=−r

(11.15)

Similarly, the following inequalities (11.16)–(11.18) can be derived. E{V p,1 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − V p,1 (Θk , k, r ) = =

d¯ 

πr s x˜ Tp (k + 1)P4 (s)x˜ p (k + 1) − x˜ Tp (k)P4 (r )x˜ p (k)

s=1 x˜ Tp (k

+ 1) P¯4 (r )x˜ p (k + 1) − x˜ Tp (k)P4 (r )x˜ p (k),

(11.16)

E{V p,2 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − V p,2 (Θk , k, r ) =

d¯ 

k 

πr s

s=1

i=k+1−s

k−1 

−

f T (I1 x˜ p (i))P5 (s) f (I1 x˜ p (i))

f T (I1 x˜ p (i))P5 (r ) f (I1 x˜ p (i))

i=k−r −1 

+

k 

f T (I1 x˜ p (i))P5 f (I1 x˜ p (i))

¯ i=k+1+ j j=−d+1 −1 

−

k−1 

f T (I1 x˜ p (i))P5 f (I1 x˜ p (i))

¯ i=k+ j j=−d+1

= f (I1 x˜ p (k)) P¯5 (r ) f (I1 x˜ p (k)) − f T (I1 x˜ p (k − r ))P5 (r ) f (I1 x˜ p (k − r )) T

254

11 H∞ State Estimation for Delayed Discrete-Time GRNs

+

d¯ 

k−1 

πr s

s=1

i=k+1−s

k−1 

−

f T (I1 x˜ p (i))P5 (s) f (I1 x˜ p (i))

f T (I1 x˜ p (i))P5 (r ) f (I1 x˜ p (i))

i=k+1−r −1 

+

[ f T (I1 x˜ p (k))P5 f (I1 x˜ p (k)) − f T (I1 x˜ p (k + j))P5 f (I1 x˜ p (k + j))]

¯ j=−d+1

 f (I1 x˜ p (k))[(d − 1)P5 + P¯5 (r )] f (I1 x˜ p (k)) − f T (I1 x˜ p (k − r ))P5 (r ) f (I1 x˜ p (k − r )) T

+

k−1 

f T (I1 x˜ p (i)) P¯5 (r ) f (I1 x˜ p (i))

i=k+1−d¯

− 

k−1 

f T (I1 x˜ p (i))P5 f (I1 x˜ p (i))

i=k+1−d¯ T f (I1 x˜ p (k))[(d¯ − 1)P5 + P¯5 (r )] f (I1 x˜ p (k)) − f T (I1 x˜ p (k − r ))P5 (r ) f (I1 x˜ p (k − r )),

(11.17)

E{V p,3 (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − V p,3 (Θk , k, r ) =

d¯ 

πr s

s=1

+

−1 

k 

j=−s i=k+1+ j

−1  −1 



j=−d¯ l= j



d¯ 

πr s

s=1

+

d¯ 

−1 

k 

πr s

−1 

−1 

k−1 

ζ (i)P6 ζ(i) − T

i=k+1+l

ζ T (i)P6 (r )ζ(i) 

ζ (i)P6 ζ(i) T

i=k+l

ζ T (k)P6 (s)ζ(k) −

−1 

ζ T (k + j)P6 (r )ζ(k + j)

j=−r

−1 

k−1 

ζ T (i)P6 (s)ζ(i)

j=−d¯ i=k+1+ j k−1 

ζ T (i)P6 (r )ζ(i) +

j=−r i=k+1+ j

−

k−1 −1   j=−r i=k+ j

j=−d

s=1

−

ζ T (i)P6 (s)ζ(i) −

k−1 

ζ T (i)P6 ζ(i) −

j=−d¯ i=k+1+ j

 ζ T (k)Ω56 ζ(k) −

d¯ 2 + d¯ T ζ (k)P6 ζ(k) 2

−1 

ζ T (k + j)P6 ζ(k + j)

j=−d −1  j=−r

ζ T (k + j)Ω45

−1  j=−r

ζ(k + j).

(11.18)

11.2 H∞ Filter Design

255

In view of (11.5), we can conclude that − 2 f T (I1 x˜ p (k))ς f (I1 x˜ p (k)) + 2 x˜ Tp (k)I1T ς K f (I1 x˜ p (k))  0, −2 f T (I1 x˜ p (k + 1))μ f (I1 x˜ p (k + 1)) + 2 x˜ Tp (k + 1)I1T μK f (I1 x˜ p (k + 1))  0.

(11.19a)

(11.19b)

Now, combining (11.13)–(11.19) results in E{V (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − V (Θk , k, r )  ξ T (k)Ω(r )ξ(k),

(11.20)

where Ω(r ) is defined as in (11.10), and ξ(k) = col(x˜m (k), x˜ p (k), x˜m (k − r ), f (I1 x˜ p (k − r )), f (I1 x˜ p (k)), f (I1 x˜ p (k + 1))). Due to (11.10), formula (11.20) results in E{V (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r }  V (Θk , k, r ) − min λr { x˜m (k) 22 + x˜ p (k) 22 }, ¯ r ∈d

(11.21)

where λr denotes the minimal eigenvalue of −Ω(r ). Since E{E{V (Θk+1 , k + 1, d(k + 1)) | Θk , d(k)} | Θ0 , d(0)} = E{V (Θk+1 , k + 1, d(k + 1)) | Θ0 , d(0)}, we obtain E{ x˜m (k) 22 + x˜ p (k) 22 |Θ0 , d(0)} 

min λr

¯ r ∈d

−1

V (Θ0 , 0, d(0)) < ∞

by taking the conditional expectation E{·|Θ0 , d(0)} and summing from k = 0 to +∞ on both sides of (11.21). Consequently, by Definition 11.1, one can conclude from the above inequality that the filtering error system (11.9) is stochastically stable, and the proof is thus completed. Remark 11.4 The novel LKF in the proof of Theorem 11.3 is selected to be of (11.12), which is concerned with not only the triple summation term but also the consideration of sufficiently the information of the random delay described by a Markovian chain. Theorem 11.3 does not give a design procedure for the desired filter. However, from which, we can easily give an approach to design an H∞ filter for GRN (11.6)

256

11 H∞ State Estimation for Delayed Discrete-Time GRNs

such that the filtering error system (11.9) is stochastically stable with H∞ disturbance attenuation level γ. ¯ the filtering error Theorem 11.5 For given a scalar γ > 0 and a positive integer d, system (11.9) is stochastically stable with H∞ disturbance attenuation level γ, if ¯ there exist matrices PiT (r ) = Pi (r ) > 0 (i ∈ 6), P jT = P j > 0 for each r ∈ d, ( j = 2, 3, 5, 6), 

Rk1 Rk2 Rk := Rk3 Rk2

T , det Rk2 = 0, k = 1, 2,

ς := diag(ς1 , ς2 , . . . , ςn ) > 0, μ := diag(μ1 , μ2 , . . . , μn ) > 0, A¯ f , B¯ f , C¯ f , D¯ f , G 1 f , ¯ H1 f , G 2 f and H2 f , such that the following LMIs (11.22) and (11.23) hold for r ∈ d: ⎡

Υ11 (r ) 0 0 0 ⎢ 0 Υ22 (r ) 0 0 ⎢ ⎢ 0 0 Υ (r ) 0 33 Υ (r ) := ⎢ ⎢ 0 0 0 −I ⎢ ⎣ 0 0 0 0 T T T T Υ16 Υ26 Υ36 Υ46

0 0 0 0 −I T Υ56

⎤ Υ16 Υ26 ⎥ ⎥ Υ36 ⎥ ⎥ < 0, Υ46 ⎥ ⎥ Υ56 ⎦ Υ66

P¯ j (r ) < P j , j = 2, 3, 5, 6, where Υ11 (r ) = P¯1 (r ) − R1 − R1T , d¯ 2 + d¯ P3 − R1 − R1T , Υ22 (r ) = d¯ P¯3 (r ) + 2 Υ33 (r ) = P¯4 (r ) − R2 − R2T , P¯i (r ) =

d¯ 

πr s Pi (s), i ∈ 6,

s=1

Υ16 = R1T Ψ1 + (I1 + I2 )T ( B¯ f Ψ2 + A¯ f Ψ3 ), Υ26 = R1T Ψ4 + (I1 + I2 )T ( B¯ f Ψ2 + A¯ f Ψ3 ), Υ36 = R2T Ψ5 + (I1 + I2 )T ( D¯ f Ψ6 + C¯ f Ψ7 ),     I1 = I 0 , I2 = 0 I ,

(11.22)

(11.23)

11.2 H∞ Filter Design

257

 Ψ1 =

 AI1 0 0 W 0 0 E 1 0 , 0 00 0 00 0 0

  Ψ2 = CI1 0 0 0 0 0 E 2 0 ,   Ψ3 = I2 0 0 0 0 0 0 0 ,   (A − I )I1 0 0 B 0 0 E 1 0 Ψ4 = , 00 0 00 0 0 −I2   0 CI1 DI1 0 0 0 0 F1 Ψ5 = , 0 0 0 0000 0   Ψ6 = 0 C2 I1 0 0 0 0 0 F2 ,   Ψ7 = 0 I2 0 0 0 0 0 0 ,   Υ46 = G¯ 1 f 0 0 0 0 0 H¯ 1 f 0 ,   Υ56 = 0 G¯ 2 f 0 0 0 0 0 H¯ 2 f , ⎡

Υ66

⎤ ˜ ) 0 Ω(r 2 T  = ⎣ 0 −γ 2 I 0 ⎦ , 2 = 0 0 0 0 0 −F1T μK , 0 −γ 2 I T2

and K , G¯ 1 f , G¯ 2 f , H¯ 1 f and H¯ 2 f are defined as previously. Moreover, when the LMIs in (11.22)–(11.23) are feasible, the desired filter is given by (11.8) with −1 ¯ −1 ¯ −1 ¯ −1 ¯ A f , B f = R12 B f , C f = R22 C f , D f = R22 Df. A f = R12

(11.24)

Proof Let A f , B f , C f and D f be defined as in (1.8). Then it is easy to verify that Υ16 = R1T Λ¯ 1 , Υ26 = R1T Λ¯ 2 and Υ36 = R2T Λ¯ 3 , where       Λ¯ 1 = Λ1 E¯ 0 , Λ¯ 2 = Λ2 E¯ 0 , Λ¯ 3 = Λ3 0 F¯ , and Λ1 , Λ2 , Λ3 , E¯ and F¯ are defined as previously. This, together with (11.22) and Lemma 1.8, implies that

258

11 H∞ State Estimation for Delayed Discrete-Time GRNs

⎡

− P¯1−1 (r )

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0



d¯ 2 +d¯ 2

− d¯ P¯3 (r ) +

0 0 0 Λ¯ T1

0 0 0 Λ¯ T2

P3

−1

0

0

0 0 −1 ¯ − P4 (r ) 0 0 −I 0 0 T Λ¯ T3 Υ46

⎤ 0 Λ¯ 1 ⎥ 0 Λ¯ 2 ⎥ ⎥ 0 Λ¯ 3 ⎥ ⎥ < 0. 0 Υ46 ⎥ ⎥ −I Υ56 ⎦ T Υ56 Υ66

Due to the Schur complement lemma, the above inequality is equal to ¯ < 0, (r ) + 

(11.25)

where ⎡

⎤ 0 0 0 T T ¯ = ⎣0 −γ 2 I 0 ⎦ + Υ46  Υ46 + Υ56 Υ56 , 2 0 0 −γ I ⎡ ⎤ ˜ ) 0 2 Ω(r (r ) = ⎣ 0 0 0 ⎦ + Λ¯ T1 P¯1 (r )Λ¯ 1 T2 0 0

d¯ 2 + d¯ P3 Λ¯ 2 + Λ¯ T3 P¯4 (r )Λ¯ 3 , +Λ¯ T2 d¯ P¯3 (r ) + 2 and hence

d¯ 2 + d¯ P3 Λ¯ 2 + Λ¯ T3 P¯4 (r )Λ¯ 3 < 0. Λ(r ) := Υ66 + Λ¯ T1 P¯1 (r )Λ¯ 1 + Λ¯ T2 d¯ P¯3 (r ) + 2 Noting that Ω(r ) is a submatrix of Λ(r ), we can conclude that Ω(r ) < 0. By Theorem 11.3, the filtering error system (11.9) with w(k) = 0 and v(k) = 0 is stochastically stable. Choose the same LKF as in (11.12) for the filtering error system (11.9) and employ the similar approach in the proof of Theorem 11.3, one has ΔVk := E{V (Θk+1 , k + 1, d(k + 1)) | Θk , d(k) = r } − V (Θk , k, r )  E{δ T (k)(r )δ(k)}, (11.26) where δ(k) = col(ξ(k), w(k), v(k)), and ξ(k) is defined as previously. To deal with the H∞ performance, the following performance function is considered: JK :=

K  k=0

    2

 

 x˜m (k) 2  w(k) 2    E  −γ 

Θ = 0, d(0) . v(k) 2 0 x˜ p (k) 2

11.2 H∞ Filter Design

259

Due to the zero initial condition and E{E{V (Θk+1 , k + 1, d(k + 1)) | Θk , d(k)} | Θ0 = 0, d(0)} = E{V (Θk+1 , k + 1, d(k + 1)) | Θ0 = 0, d(0)}, it is easy to see from (11.25) and (11.26) that JK =

K  k=0

−

      x˜m (k) 2  w(k) 2 2    E   x˜ p (k)  − γ  v(k)  + ΔVk | Θ0 = 0, d(0) 2 2

K 

E{ΔVk | Θ0 = 0, d(0)}

k=0

=

K  k=0

   2     x˜m (k) 2 2  w(k)    E  −γ  + ΔVk | Θ0 = 0, d(0) v(k) 2 x˜ p (k) 2

−E{V (Θ K +1 , K + 1, d(K + 1)) | Θ0 = 0, d(0)} +V (Θ0 = 0, 0, d(0)) K   E{[ x˜m (k) 22 + x˜ p (k) 22 − γ 2 w(k) 22 − γ 2 v(k) 22 k=0

+ΔVk | Θ0 = 0, d(0)} K  ¯ E{δ T (k)((r ) + )δ(k) | Θ0 = 0, d(0)} < 0.  k=0

Let K → ∞, it is concluded from Definition 11.2 that the filtering error system (11.9) is stochastically stable with H∞ disturbance attenuation level γ. The proof is thus completed. Remark 11.6 What can be seen from Theorem 11.5 is that the scalar γ can be calculated as an optimization variable to obtain the minimum H∞ disturbance attenuation level. To be more specific, the minimal H∞ disturbance attenuation level can be obtained by solving the following convex optimization problem: min

s.t. (11.22)−(11.23)

β, β = γ 2 .

(11.27)

∗ Note that if there exists a solution √ β∗ to the problem (11.27), then the minimal H∞ disturbance attenuation level is β .

260

11 H∞ State Estimation for Delayed Discrete-Time GRNs

11.3 A Numerical Example In this section we will illustrate the effectiveness of the proposed approach by testing the following numerical example. Example 11.7 Consider GRN (11.6) with the following parameters: 

 0 0 A = diag(0.3679, 0.3679), W = , −0.126 0 C = diag(0.6065, 0.3679), D = diag(0.3935, 0.6321),       0.5 0.4 0.6 E1 = , E 2 = F1 = , F2 = , 0 0.2 0.3 G 2 = G 1 = C2 = C1 = diag(0.2, 0.3). The regulation function is taken as gi (x) =

x2 ,i 1+x 2

= 1, 2, and hence

K = diag(0.65, 0.65). Suppose the bound of the time delay is d¯ = 3, then d(k) ∈ {1, 2, 3}. Assume that the transition probability matrix π is given by ⎡

⎤ 0.3 0.5 0.2 π = ⎣0.4 0.3 0.3⎦ . 0.2 0.5 0.3 By solving the optimization problem (11.27), it can be obtained that the optimal disturbance attenuation level γ ∗ is 0.2025, and the corresponding filter gains are as follows:     0.1179 −0.0000 −1.2500 −0.0000 , Bf = , Af = −0.0448 0.3029 0.4472 −1.6554     0.3114 0.4332 −1.8405 0.8750 Cf = , Df = , 0.0300 0.0442 −0.0417 −0.4223     −0.2000 0.0000 −0.1068 −0.1655 , G2 f = , G1 f = −0.1000 0.0000 −0.0534 −0.0827     0.0000 −0.0000 0.3635 −0.5397 H1 f = , H2 f = . −0.5000 1.0000 −0.3182 0.7301

11.3 A Numerical Example

261

In the following simulation setup, the noise signal is chosen as  w(k) = v(k) =

sin(0.3k), k  20, 0, k > 20.

Let the filtering error system run by random sequence d(k), the trajectories and their estimations of the mRNAs and proteins are shown in Figs. 11.1 and 11.2, respectively, where the solid line and dotted line describe the state trajectories and estimations of mRNAs and proteins, respectively. From which, it can be seen that the filtering error converges to zero in the absence of disturbances Next, we illustrate the H∞ performance of the filtering error system (11.9). By direct computation, we have 60    w(k) 22 + v(k) 22 = 20.4462. k=0

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0

0

-0.2 -0.4

-0.1

-0.6

-0.2

-0.8 0

10

20

30

40

50

60

-0.3 0

10

20

30

40

50

60

30

40

50

60

Fig. 11.1 The mRNA concentrations and their estimations (Example 11.7) 2

0.8

1.5

0.6

1

0.4

0.5

0.2

0

0

-0.5 -1

-0.2

-1.5

-0.4

-2

0

10

20

30

40

50

60

-0.6

0

10

20

Fig. 11.2 The protein concentrations and their estimations (Example 11.7)

262

11 H∞ State Estimation for Delayed Discrete-Time GRNs

For values of 1000random sequences of d(k), we obtain by MATLAB that the 2 2 maximum of 60 k=0 x˜ m (k) 2 + x˜ p (k) 2 is 0.1186, and hence the maximum disturbance attenuation level is:   60 2 2   k=0 x˜m (k) 2 + x˜ p (k) 2 = 0.1647 = 0.0761 < γ ∗ . 60 2 2 20.9454 k=0 w(k) 2 + v(k) 2 This verifies that the H∞ disturbance attenuation level is below the given upper bound.

11.4 Remarks and Notes In this chapter, we investigate the problem of H∞ filtering on a class of discrete-time GRNs with random delays. The random delay is described as a Markovian chain. The chapter is taken from [12]. We end this chapter by introducing the following items, which are related to this chapter: 1. The H∞ state estimation for discrete-time GRNs with random time delays have been investigated in [1, 7, 8, 15]. Wang et al. [11] extended the GRN model to a class of stochastic systems. 2. Zhang et al. [13] presented the set-values filtering for discrete-time delayed GRNs with time-varying parameters and bounded external noise. 3. Banu and Balasubramaniam [2] addressed the problem of non-fragile observer design for discrete-time delayed GRNs with randomly occurring uncertainties. 4. Wan et al. [9] presented the robust non-fragile H∞ state estimator for discrete-time uncertain delayed GRNs with exogenous disturbances. 5. In [4], the problem of event-triggered H∞ state estimation for discrete-time delayed stochastic GRNs with both Markovian jumping parameters is investigated. 6. In [10], the problem of mean-square asymptotic stability of discrete-time GRNs with random time delays has been addressed by utilizing a class of indicator functions and discrete-time Jensen inequality. 7. Zhao et al. [14] proposed a mean-square asymptotic stability criterion for discretetime delayed stochastic GRNs. 8. Cao and Ren [3] investigated the problem of global exponential stability of the discrete-time delayed GRNs, and Li et al. [5] studied the global robust exponential stability of uncertain discrete-time delayed GRNs.

References

263

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