Elementos de optimización dinámica

Table of contents :
Cover
Title page
Copyright page
Contenido
Introducción
Capítulo 1 Cálculo de variaciones
1.1. Condiciones de transversalidad
Capítulo 2 Control óptimo
2.1. Condiciones de transversalidad
2.2. Control óptimo con valor descontado
2.3. Horizonte infinito
Bibliografía

Citation preview

Documento N.° 87

Elementos de optimización dinámica Daniel Ricardo Casas Hernández

Facultad de Ciencias Económicas y Sociales Programa de Economía

Facultad de Ciencias Económicas y Sociales Programa de Economía Bogotá, D. C. 2013

ISSN: 1900-6187 © 2013 Oficina de Publicaciones Cra. 5 N.° 59A-44 Teléfono: 3 48 80 00 exts.: 1224-1227 Fax: 2 17 08 85 [email protected] Dirección editorial Guillermo Alberto González Triana Coordinación editorial Marcela Garzón Gualteros Corrección de estilo Pablo Emilio Daza Velásquez Diagramación Édgar Andrade Carátula Andrea Julieth Castellanos Leal Impresión Xpress Estudio Gráfico y Digital Mayo del 2013

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 j

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 9

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 8

!# "$

*OUSPEVDDJØO $BQÓUVMP  $ÈMDVMP EF WBSJBDJPOFT

$BQÓUVMP  $POUSPM ØQUJNP !# "$

!# "$

#JCMJPHSBGÓB

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 e

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 d

!# "$

* /530%6$$*»/

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 3

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 N

!# "$

$ "1¶56-0  $ «-$6-0 %& 7"3*"$*0/&4

!# "$

!# "$

t0 T V (to , T ) =

!

T

f (t) dt,

to

y = f (t) f (ti )

to T [to , T ]

!ti

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 Ry

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

f

t

to

T U (c(t)) U! > 0

t U !! < 0 ert

to

"T

T

to U (c(t))dt

VP

VF

r

VF = VP · ert !

VP = VF · e−rt T

e−rt U (c(t)) dt

to

to T π(t)

!# "$

!

T

e o

π(t) dt. ˙ c(t) = F (k)−k(t)

e−rt U (c(t))dt T

Q Q(t) = c(t)+I(t)

˙ c(t) = F (k)−k(t)

Q = F (k(t))

˙ k(t) !

T

˙ = I(t) k(t)

˙ e−rt U (F (k)−k(t)) dt

o

k(0) = ko K(T )

to T

to

"T

k(0) = ko , K(T )

−rt

VP

T ,

(1 + r)t

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 RR

"QVOUFT EF $MBTF o %PDVNFOUP / 

s(t) !

T

f (t, s(t), s(t)) ˙ dt

s.a.

s(to ) = so ; s(T ) = sT

to

s1 (t), . . . , sn (t) !

T

f (t, s1 (t), . . . , sn (t), s˙1 (t), . . . , s˙n (t)) dt

s.a.

to

s1 (to ) = s1o , . . . , sn (to ) = sno , s1 (T ) = s1T , . . . , sn (T ) = snT

!# "$

"b

[a, b]

a

Γ

[a, b] f (x)dx

Γ f (x) ∈ Γ

[a, b] [a, b] ϕ

!# "$

ϕ a b

F (x, y, z) M [y] =

!

b

F (x, y(x), y(x)) ˙ dx, a

y(x) [a, b] f si (t) [to , T ]

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 Rk

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

s∗ (t)

[to , T ]

fs! −

d ! f =0 dt s˙

ds ds˙ !! + fs!!˙s˙ + fst. ˙ dt dt ! !! !! !! fs = fss ¨ + fst. ˙ s˙ + fs˙ s˙ s ˙ !! fs! = fss ˙

2

s˙ = ds dt

s¨ = ddts˙ = ddt2s g(t, x(t), x(t)) ˙

∂g ∂ x˙

!# "$

d ! dt gx˙

= gx!˙ s∗ (t)

"T

to

F (t, s∗ , s˙∗ )dt ≥

"T

t ∈ [to , T ]

F (t, s, s)dt ˙

to

v(t) =

x(t)

!5

√

3t + x˙ dt

s.a.

x(1) = 3, x(5) = 7.

1

√ F (t, x, x) ˙ = 3t + x˙

Fx! = 0

1 ˙ −1/2 , Fx˙! = (3t + x) 2 −3 !! (3t + x) ˙ −3/2 , Fxt ˙ = 4 !! Fxx ˙ = 0, −1 (3t + x) ˙ −3/2 . Fx˙!!x˙ = 4

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 Rj

!# "$ "QVOUFT EF $MBTF o %PDVNFOUP / 

x ¨ −3 (3t + x) ˙ −3/2 − (3t + x) ˙ −3/2 = 0; 4 4 x ¨ = −3

3t + x˙ &= 0 x(t) ˙ = − 3t + c1 , −3 2 t + c1 t + c2 . x(t) = 2 x(1) = 3

x(5) = 7

c1 c2 −3 + c1 + c2 , 2 −75 7= + 5c1 + c2 . 2 3=

!# "$

!# "$

c1 = 10 c2 = −11 2

x(t) =

−3 2 11 t + 10t − . 2 2

Fx˙!!x˙ < 0

V (t) t Fx˙!!x˙

[t0 , T ]

>0 F (t, x(t), x(t)) ˙

Fx˙!!x˙ (x∗ ) < 0 x∗ (t) t

(t, x∗ (t), x∗ (t)) Fx˙!!x˙ > 0 x∗ (t)

V (t)

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 R9

!# "$

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

Fx˙!!x˙ = −1/4(3t + x) ˙ −3/2 −1 10 Fx˙!!x˙ = 4(10) −3/2



!# "$

$POEJDJPOFT EF USBOTWFSTBMJEBE

!

•

XT

T

F (t, x, x) ˙ dt

x(t0 ) = x0

T

•

XT Fx˙! (T ) = 0

T Fx˙! (T ) = 0 F (T ) = 0

T

• •

x(T ) = XT !# "$

XT T F (T ) = x(T ˙ )Fx˙! (T )

XT

s.a.

to

•

•

x(t) = (−3/2)t2 + 10t − 11/2; x∗ (t) x(t) ˙ = −3t + < 0; x∗ (t)

x(t0 ) = x0

•

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 R8

"QVOUFT EF $MBTF o %PDVNFOUP / 

V=

!

T

(2xet + x2 + x˙ 2 ) dt x(1) = 2 x(T ) = xT .

1

xT = e2

T =2

T =2

F (t, x, x) ˙ = 2xet + x2 + x˙ 2 , Fx! = 2et + 2x, Fx˙! = 2x, ˙ !# "$

!! Fxt ˙ = 0, !! Fxx ˙ = 0,

Fx˙!!x˙ = 2, et + x = x ¨. x(t) = c1 et + c2 e−t + 12 tet x(1) = 2 T = 2 x(T ) = e2 1 2 = c1 e + c2 e−1 + e, 2 2 2 −2 e = c1 e + c2 e + e2 .

4−e . 2e(1 − e2 ) e2 (e2 − 4e) c2 = . 2(1 − e2 )

c1 =

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 Re

!# "$

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

T =2

XT Fx˙! (T ) = 0

Fx˙! = 2x; ˙ 1 1 x(t) ˙ = c1 et − c2 e−t + et + tet . 2 2 Fx˙! (T ) = Fx˙! (2) = 2c1 e2 − 2c2 e−2 + e2 + 2e2 = 0. 2c1 e2 + 3e2 − 2c2 e−2 = 0, 1 c1 e + c2 e−1 + e = 2. 2 !# "$

!# "$

−3 − e−2 + 4e−3 , 2(e−2 + 1) 2e + e2 c2 = . 1 + e−2

c1 =

"1 0

− 12 (y 2 + (y˙ + y)2 )dt F (t, y, y) ˙ =

s.a.

y(0) = 1,

y(1) = y1

−1 2 (y + (y˙ + y)2 ). 2

Fy! = − y − (y˙ + y) = − y˙ − 2y,

Fy!˙ = − (y˙ + y),

!! Fyt ˙ = 0,

!! Fyy ˙ = − 1,

Fy!!˙ y˙ = − 1.

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 Rd

"QVOUFT EF $MBTF o %PDVNFOUP / 

y¨ − 2y = 0. √

√

y(t) = C1 e 2t + C2 e− 2t , √ √ √ √ y˙ = 2C1 e 2t − C2 2e− 2t . T

XT

Fy!˙ (t = T ) = 0. y(t)

Fy!˙ (t = T ) = 0

√ √ √ √ √ √ − 2C1 e 2 + 2C2 e− 2 − 2C1 e 2 − 2C2 e− 2 = 0.

!# "$

y(0) = 1 C1 = √ ( 2+ C2 = √ ( 2+

1 = C 1 + C2

√ √ e− 2 ( 2 − 1) √ √ √ 1)e 2 + e− 2 ( 2 √ √ e 2 ( 2 + 1) √ √ √ 1)e 2 + e− 2 ( 2

y(t) = 0,01e

√ 2

+ 0,99e−

!# "$

√

− 1) − 1)

2

.

, .

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 R3

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 RN

!# "$

$ "1¶56-0  $ 0/530- »15*.0

V (t) =

!

T

F (t, x(t), u(t)) dt

s.a.

t0

x(t) ˙ = g(t, x(t), u(t)) !# "$

x(t0 ) = x0 ; x(T ) = xT . !# "$

H = F (t, x, u) + λ(t)g(t, x, u). x, u, λ

x(t), u(t), λ(t) t

x(t)

u(t) x(t), u(t) V (t)

Hu! = 0 Hx! = − λ˙

Hλ! = x˙

!# "$

λ(t)

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 ky

!# "$

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

!

T

f (t, x, u) dt

s.a.

x ¨ = g(t, x, u);

x(t0 ) = x0 , x(T ) = xT

t0

x˙ = z !

x ¨ = z˙ = g(t, x, u)

T

f (t, u, x, z) dt

s.a.

t0

x˙ = z; z˙ = g(t, u, x, z); x(t0 ) = x0 ; x(T ) = xT ; z(t0 ) = x(t ˙ 0 ); z(T ) = x(T ˙ ).

H = f (t, x, u) + λ1 (z) + λ2 g(t, x, u) !# "$

Hu! = 0 Hz! = − λ˙ 1 H ! = − λ˙ 2 x

x˙ = z

z˙ = g(t, u, x)



$POEJDJPOFT EF USBOTWFSTBMJEBE

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 kR

!# "$ "QVOUFT EF $MBTF o %PDVNFOUP / 

T

XT

T

XT

T

λ(T ) = 0

XT

H(t = T ) = 0

T XT H(t = T ) = 0 !

1 0

λ(T ) = 0

1 2 (x + u2 ) dt 4

s.a.

x˙ = x + u

x(0) = 2

x(1) = 0

H = 1/4(x2 + u2 ) + λ(x + u)

!# "$

1 Hu! = u + λ = 0 2 1 Hx! = x + λ = − λ˙ 2 x˙ = x + u

x˙ = x − 2λ −1 x−λ λ˙ = 2

# $ # x˙ 1 = −1 λ˙ 2

$# $ −2 x −1 λ

% % %1 − r −2 %% % −1 =0 % −r − 1% 2

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 kk

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

(1 − r)(−1 − r) − 1 = 0. r= ± r &

r=

1−

√

2

−1 2

√

√

2

2

' & ' & ' 0 v √−2 · 1 = 0 v2 − 2−1

√ (1 − 2)v1 − 2v2 = 0 √ 1 − v1 − ( 2 + 1)v2 = 0 2 !# "$

√ 2

v1 = 1 r= −

v1 = 1

√

v2 = 1−2

2 √ & 1+ 2

'& ' & ' 0 v1 −2√ = −1 0 v −1 + 2 2 2 √ (1 + 2)v1 − 2v2 = 04 √ 1 − v1 + (−1 + 2)v2 = 0 2 v2 =

√ 2+1 2

( ) ( ) & ' √ √ 1√ 1 x = c1 1− 2 e 2t + c2 1 + √2 e− 2t . λ 2 2 x(t) λ(t) u = − 2λ u(t) = − 2(c1 e

u(t) √ 2t

√

+ c 2 e−

!# "$

2t

)

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 kj

"QVOUFT EF $MBTF o %PDVNFOUP / 

x(0) = 2 x(1) = 0

c1 = c2 =

2e−

√

2

√ e 2

e

− e− √ 2e 2

√

2

− e−

√

2

√ 2

u˙ = 0 x = u x˙ = 0 x = −u x −

u˙ = 0

+

+

−

!# "$

u

+

!

1√ 0

−

u dt

−

s.a.

x˙ = − u

H=

√

x˙ = 0

x(0) = 1,

u + λ(−u)

!# "$

+

x(1) = 0.

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 k9

!# "$

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

1 Hu! = u−1/2 − λ = 0. 2 ! ˙ Hx = 0 = − λ. x˙ = − u.

λ˙ = 0

λ(t) = c1 1 −1/2 = c1 2u

c2 = ac12

u(t) = c2

1

x˙ = − c2 x(t) = x(0) = 1 x(1) = 0

−c2 t + c3 c3 = 1, !# "$

x(t) = − t + 1

c2 = 1

u(t) = 1

λ(t) =

1 2

u(t) !! ≤ 0 Huu

V (t) =

!T 0

x(t)

e−rt [py(t) − wN (t) − Pk I(t)] dt s.a. y(t) = F (K(t), N (t)),

p·y I

˙ I(t) = K(t) + δK(t).

wN + Pk I K

N

r V (t)

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 k8

!# "$ "QVOUFT EF $MBTF o %PDVNFOUP / 

!∞ ˙ e−rt [P F (K(t), N (t)) − wN (t) − PK (K(t) + δK(t))] dt 0

N (t) K(t) G = e−rt [P F (K, N ) − wN − Pk (K˙ + δK)]. N (t) K(t) d ! G = 0; dt K˙ D G!N − G!N˙ = 0; dt

G!K −

! e−rt [P FK − PK δ − PK r] = 0;

e−rt [P FN − w] = 0.

! P FK = PK (δ + R),

P FN! = w

!# "$

PK (δ + r) P w FN! = P

! FK =

FN! = wp δ>r

! > 0; δ < r F ! < 0; FK K ! P FK

PK (δ + r)

!∞ e−rt [P Y (t), wN (t) − PK I(t)] dt

s.a.

0

K˙ = I − δK

K N

I H = e−rt [P F (K, N ) − wN − PK I] + λ(I − δK)

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 ke

!# "$

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

! HN = 0:

e−rt [P FN! − w] = 0

HI! = 0:

e−rt [−PK ] + λ = 0

! ˙ HK = − λ: H ! = K˙

! e−rt [−P FK ] − λδ = − λ˙

λ

P FN! = w:

FN =

w . P

λ = e−rt PK . ! λ˙ = λδ − e−rt P FK . ˙ K = I − δK. !# "$

λ = − e−rt PK . ! −re−rt PK = − e−rt PK δ − e−rt P FK .

! −rPK = PK δ − P FK . ! P FK = PK (δ + r).



$POUSPM ØQUJNP DPO WBMPS EFTDPOUBEP e−δt VF = eδt VP

VP

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 kd

!# "$ "QVOUFT EF $MBTF o %PDVNFOUP / 

VF

δ !

T

e−δt F (t, u, x) dt s.a.

t0

x˙ = g(t, u, x);

x(t0 ) = x0 ;

x(T ) = xT .

H = e−δt F (t, u, x) + λg(t, u, x).

ˆ = F (t, u, x) + µg(t, u, x). H ˆ = eδt H, µ = λeδt H !# "$

!# "$

ˆ −δt = H, λ = µe−δt . He

 )PSJ[POUF JOåOJUP T

!∞ F (t, u(t), x(t)) dt

s.a.

t0

x(t) ˙ = g(t, u(t), x(t));

x(t0 ) = x0 .

U ∗ (t), X ∗ (t) λ∗ (t) U ∗ (t) X ∗ (t) t→∞

λ(t)[x(t) − x∗ (t)] ≥ 0.

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 k3

%BOJFM 3JDBSEP $BTBT )FSOÈOEF[

c(t) !∞ V (b, t0 ) = e−δt

c(t) dt

s.a.

t0

s˙ = rs − c

S(t0 ) = S0 > 0,

H = e−δt

t→∞

S(t) = 0.

C + λ(rs − c).

ˆ = eδt H, H ˆ= H

!# "$

µ(t) = λ(t)eδt .

C + µ(rs − c), ˆ c! = 0; H 1 = µ. c

ˆ ! = − µ˙ = µr; H s

µ(t) = µ0 e−δt .

λ(t) = µ0 e−(r + δ)t , S = rs − c. 1 rt e , µ0 1 S − rs = − ert , µ0 tert S(t) = − + αert . µ0 C(t) =

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

!# "$

S;2 kN

"QVOUFT EF $MBTF o %PDVNFOUP / 

S(b) = S0 & ' −tert t0 −rt S(t) = + S0 e + ert . µ0 µ0

t→∞

µ0 e(−r + δ)t [S(t) − S ∗ (t)] ≥ 0.

t→∞ S(t) = 0

t→∞

µ0 e

(−r + δ)t

&

tert + µo

&

S0 e

−rt

t0 + µ0

'

e

rt

'

≥ 0.

µ0

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 jy

!# "$

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 jR

!# "$

#*#-*0(3"'¶"

(2a ed.).

!# "$

!# "$

!# "$

KBM

RN /2 DmMBQ /2 kyRj

kR,Rj

S;2 jk

!# "$

!# "$

!# "$

!# "$