1 Capítulo 7 Controle Ótimo: O princípio do máximo O cálculo das variações, o método clássico para atacar problemas de otimização dinâmica, assim como o cálculo comum, requer para sua aplicabilidade a diferenciabilidade das funções que entram no problema. Mais importante que isso é que apenas soluções interiores podem ser manipuladas. Um desenvolvimento mais moderno que pode trabalhar com características não clássicas tais como solução de canto, é encontrado na teoria do controle ótimo. Como seu nome indica, a formulação de controle ótimo do problema de otimização dinâmica foca uma ou mais variáveis que servem como instrumentos de otimização. Diferente, entretanto, do cálculo das variações, onde nosso objetivo é encontrar o caminho temporal ótimo para uma variável estado y, a teoria do controle ótimo tem como sua principal meta a determinação do caminho ca minho ótimo para a variável de controle u. Certamente, logo que o caminho do controle ótimo, u*(t), seja encontrado, nós podemos também encontrar o caminho do estado ótimo, y*(t), que corresponde a ele. De fato, os caminhos u*(t) e y*(t) são usualmente encontrados no mesmo processo. Mas a presença de uma variável de controle como estágio central muda a orientação básica do problema de otimização dinâmica. Duas questões são propostas imediatamente. O que é que uma variável de “controle” faz? E como é seu ajuste dentro do problema da otimização dinâmica? Para responder essas questões, vamos considerar uma economia ilustrativa simples. Suponha que exista numa economia um estoque finito de recursos exauríveis S (tal como carvão ou óleo), como no modelo de Hotelling, com S(0) = S0. Como esse recurso está sendo extraído (e usado), o estoque de recurso será reduzido de acordo com a relação dS (t ) dt
=
− E (t )
onde E(t) denota a taxa de extração do recurso no tempo t. A variável E(t) é qualificada como variável de controle porque possui as duas propriedades seguintes. Primeiro, ela é algo que esta sujeito a nossa escolha arbitrária. Segundo, nossa escolha de E(t) age sobre a variável S(t) que indica o estado do recurso a todo instante do tempo. Conseqüentemente, a variável E(t) é como um mecanismo de pilotagem em que nós podemos manobrar de forma a “dirigir” a variável de estado S(t) para várias posições em qualquer tempo t por meio da equação diferencial dS(t)/dt = - E(t). Por uma pilotagem correta de tal variável de controle, nós poderíamos, consequentemente, visar a otimização de algum critério de performance expresso pelo funcional objetivo. Para o presente exemplo, nós podemos postular que a sociedade quer maximizar a utilidade total derivada do uso do recurso exaurível sobre um dado período de tempo [0,T]. Se não há restrição no estoque final, o problema de otimização dinâmica toma a seguinte forma:
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2 Maximize sujeito a e
T
∫ U ( E )e
− ρ t
0
dS =
dt S (0)
dt
− E (t )
=
S 0
S (T )
livres
( S 0 , T
dados)
Nessa formulação, apenas a variável de controle E entra no funcional objetivo. Mas, de uma maneira geral, espera-se que o funcional objetivo dependa tanto da(s) variável(eis) de estado quanto da(s) variável(eis) de controle. Similarmente, é apenas um caso especial que nesse exemplo o movimento da variável de estado S dependa apenas da variável de controle E . Em geral, o curso do movimento da variável de estado sobre o tempo pode ser afetado tanto por variável (variáveis) de estado quanto por variável (variáveis) de controle, e ainda pela própria variável t. Com esse conhecimento, nós agora continuamos a discussão do método de controle ótimo.
7.1 – O PROBLEMA BÁSICO DE CONTROLE ÓTIMO Para manter uma estrutura introdutória simples, primeiro vamos considerar um problema com uma única variável de estado y e uma única variável de controle u. Como sugerido anteriormente, a variável de controle é o instrumento de política que nos habilita a influenciar a variável de estado. Assim, qualquer escolha do caminho de controle u(t) irá implicar num caminho de estado associado y(t). Nossa tarefa é escolher um caminho ótimo admissível u*(t) no qual, ao longo do caminho de estado ótimo admissível y*(t), iremos otimizar o funcional objetivo sobre o intervalo de tempo [0,T].
Características Especiais dos Problemas de Controle ótimo
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3 u
y C
A
c d
b
Trajetória de controle
Trajetória estado
B
a
0
t 1 (a)
t 2
T
t
0
t t 1
t 2 2
T
(b)
Figura 7.1
Uma característica notável da teoria do controle ótimo é a de que um caminho de controle não precisa ser contínuo para se transformar em admissível; ele apenas precisa ser contínuo por partes. Isso significa que ele pode conter saltos descontínuos, como ilustrado na figura 7.1a, apesar de não podermos permitir descontinuidades que envolvam um valor infinito de u. Uma boa ilustração de controle contínuo por partes na vida diária é o liga e desliga da chave do computador ou da ignição. Quando giramos a chave para ligar (u = 1) e desligar (u = 0), a trajetória do controle experimenta um salto. A trajetória de estado y(t), por outro lado, deve ser contínua no período de tempo [0,T]. Mas, como ilustrado na Fig. 7.1b, permite-se que tenha um número finito de pontos agudos, ou quinas. Isto é, para ser admissível, uma trajetória de estado necessita apenas ser diferenciável por partes1. Note que cada ponto agudo sobre a trajetória trajetória do estado aparece no tempo em que o caminho do controle dá um salto. A razão para esse ritmo coincidente está no processo de obtenção da solução do problema. Uma vez que tenhamos encontrado que o segmento do controle ótimo para o intervalo de tempo [0,t1) é, digamos, a curva ab na Fig 7.1a, nós tentamos então determinar o segmento correspondente da trajetória ótima de estado. Ela pode ser, digamos, a curva AB, na Fig. 7.1b, cujos pontos iniciais satisfazem uma dada condição inicial. Para o próximo intervalo, [t1,t2), determinamos novamente o segmento da trajetória de estado ótimo sobre a base do controle ótimo pré encontrado, curva cd, mas dessa vez devemos tomar o ponto B como “ponto inicial” do segmento de estado ótimo. Daí, o ponto B serve como ponto terminal para o primeiro segmento e como ponto inicial para o segundo segmento da trajetória de estado ótimo. Por essa razão, razã o, não há descontinuidade no ponto B, apesar de aparecer como um ponto agudo. Como trajetória de controle admissível, a trajetória admissível deve ter um valor finito y para todo t no intervalo de tempo [0,T]. Outra característica importante é que a teoria de controle ótimo é capaz de manipular diretamente uma restrição na variável de controle tal como a limitação ( ) ∈U
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4 todo t ∈ [0, T ], onde U denota algum conjunto de controle limitado. O conjunto controle pode ser de fato fechado, conjunto convexo, tal como u (t ) ∈ [0,1] . O fato de que U possa ser um conjunto fechado significa que soluções de canto (soluções de fronteiras) podem ser admitidas, o que insere uma importante característica não clássica na estrutura do problema. Quando essa característica é combinada com a possibilidade de saltos descontínuos na trajetória de controle, um fenômeno interessante chamado de solução bang-bang pode ocorrer. Assumindo que o conjunto controle seja U = [0,1], por exemplo, a trajetória do controle ótimo irá saltar como segue: u*(t) = 1 para
t ∈[o, t 1 )
u*(t) = 0 para
t ∈ [t 1 , t 2 )
t 1
<
t 2
u*(t) = 1 para
t ∈ [t 2 , T ]
t 2
<
T
então estaremos “ricocheteando”* (“banging ”) ”) sucessivamente entre um e outro limite do conjunto de controle U ; daí, o nome “bang-bang ”. ”. Finalmente, chamamos a atenção de que o problema básico da teoria do controle ótimo, diferente do cálculo das variações, tem um estado terminal livre (linha terminal vertical) ao invés de um ponto terminal fixo. A primeira razão para isso é que: No desenvolvimento das condições fundamentais de primeira ordem conhecido como princípio máximo, invocamos a noção de um Δu arbitrário. Qualquer Δu arbitrário deve, portanto, implicar num Δ y associado. Se o problema tem um estado terminal fixo, precisamos prestar atenção se o Δ y associado irá para o estado terminal designado. Assim, a escolha de Δu pode não ser inteiramente e verdadeiramente arbitrária. Se o problema tem um estado terminal livre (linha terminal vertical), por outro lado, então podemos arbitrar um Δu sem qualquer preocupação com o destino final de y. E isto simplifica o problema. O problema básico Baseado na discussão precedente, podemos colocar o problema básico do controle ótimo como
(7.1)
Maximize Maximize
V
sujeito sujeito a
y
T
∫ F (t , y, u )dt
=
0
=
y (0) e
f (t , y, u ) =
A
u (t ) ∈ U
y(T ) livre
( A, T
para todo t ∈ [0, T ]
dados )
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5 estabelecidas com mais especificidade e menos confusão. Quando for encontrado um problema de minimização, podemos sempre reformulá-lo como um problema de maximização simplesmente colocando o sinal de menos no funcional objetivo. Por T T exemplo, minimizar ∫ F (t , y, u )dt é equivalente a maximizar ∫ − F (t , y, u )dt . 0
0
Em (7.1), o funcional objetivo ainda toma a forma de uma integral definida, mas a função integrando F não inclui o argumento y como no cálculo das variações. Ao contrário, existe um novo argumento u. A presença da variável de controle u necessita de uma ligação entre u e y, para nos dizer como u afeta especificamente o curso tomado pela variável de estado y. Essa informação é fornecida pela equação y f (t , y, u ) , onde o símbolo com ponto y , denotando a derivada dy/dt, é uma notação alternativa para o símbolo y´ usado antes2. No tempo inicial, os dois primeiros argumentos na função f devem tomar valores dados t = 0 e y(0) = A, então apenas o terceiro argumento está sob nossa escolha. Para alguma política escolhida em t = 0, digamos, u1 (0) , essa equação produzirá um valor específico para y , digamos, y1 (0) , que impõe uma direção específica para a qual a variável y move-se. Uma política diferente u 2 (0) , geralmente nos dará um valor diferente, y 2 (0) , via a função f . E um argumento similar aplicar-se-á a outros pontos do tempo. O que essa equação faz, todavia, é fornecer um mecanismo pelo qual nossa escolha do controle u poderá ser transformada num padrão específico de movimento da variável de estado y. Por essa razão, essa equação é conhecida como a equação de movimento para a variável de estado (ou, para simplificar, a equação de estado). Normalmente, a ligação entre u e y pode ser adequadamente descrita pela equação diferencial de primeira ordem y f (t , y, u ) . Entretanto, se existir um padrão de mudança da variável de estado y que não possa ser capturado pela primeira derivada y mas que requer o uso da segunda derivada y d y / dt , então a equação de estado tomará a forma de uma equação diferencial de segunda ordem, que nós deveremos transformar num par de equações diferenciais de primeira ordem. A complicação é que, no processo de transformação, uma variável de estado adicional deverá ser introduzida no problema. Um exemplo de tal situação pode ser encontrado na seção 8.4. Nós usaremos consistentemente a letra f minúscula como símbolo da função na equação de movimento, e reservaremos a letra maiúscula F para a função integrando no funcional objetivo. Assume-se que as funções F e f são contínuas em todos os seus argumentos, e possuem derivadas parciais de primeira ordem contínuas com respeito a t e y, mas não necessariamente com respeito a u. O resto do problema (7.1) consiste de especificações com relação aos conjuntos de fronteira e de controle. Da mesma forma que o caso da linha-terminal-vertical é básico e foi implementado, outras especificações de ponto-terminal também podem ser acomodadas. Igualmente para o conjunto controle, o caso básico é de U ser um conjunto aberto U = ( + ) . Se entretanto, a escolha de U de fato é não restritiva, em tal caso poderemos, ʹ
=
=
2
≡
−∞
∞
2
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6
Um caso especial Como um caso especial, considere o problema onde a escolha de u não é restringida, e onde a equação de movimento toma uma forma particularmente simples y
=
u
Então o problema de controle ótimo fica
(7.2)
T
∫ F (t , y, u )dt
Maximize Maximize
V
sujeito a
y
=
0
=
y (0)
u =
y (T )
A
livres
( A, T
dados )
Substituindo a equação de movimento na função integrando, entretanto, podemos eliminar e reescrever o problema como y
(7.2’)
Maximize Maximize sujeito a
V
T
∫ F (t , y, y )dt
=
0
y (0)
=
A
y (T )
livres
( A, T
dados )
Este é precisamente o problema de cálculo das variações com linha terminal vertical. A ligação fundamental entre o cálculo das variações e a teoria do controle ótimo é, então, evidente. Mas, as equações de movimento encontradas nos problemas de controle ótimo são geralmente mais complicadas que em (7.2).
7.2 O PRINCÍPIO DO MÁXIMO O resultado mais importante na teoria do controle ótimo – uma condição necessária de primeira ordem – é conhecida como o princípio do máximo. Esse termo foi cunhado pelo matemático russo L S Pontryagin e seus associados3. Como mencionado na seção 1.4, entretanto, a mesma técnica foi independentemente descoberta por Maguns Hestenes, um matemático da Universidade da Califórnia, Los Angeles, que depois também expandiu os resultados de Pontryagin. O enunciado do princípio do máximo envolve o conceito da função Hamiltoniana e da variável co-estado. Devemos, entretanto, primeiro explicar esses conceitos.
A variável de co-estado e a função Hamiltoniana
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7
Três tipos de variáveis foram apresentadas no problema (7.1): t ( tempo), tempo), y (estado) e u (controle). No processo de solução, outro tipo de variável emerge. Ela é chamada de variável de co-estado (ou variável auxiliar) e será designada por λ. Como veremos, a variável de co-estado é similar ao multiplicador de Lagrange e, como tal, tem caráter de uma variável de valoração, medindo o preço sombra de uma variável de estado associada. Como y e u, a variável λ pode tomar diferentes valores em diferentes pontos do tempo. Assim, o símbolo λ é na verdade uma notação simplificada para λ (t). (t). O veículo pelo qual a variável de co-estado entra no problema do controle ótimo é a função Hamiltoniana, ou simplesmente, o Hamiltoniano, que figura com muito destaque no processo de solução. Denotando por po r H H , o Hamiltoniano é definido como (7.3)
H (t , y, u, λ ) F (t , y, u ) + λ (t ) f (t , y, u ) ≡
Desde que H consiste da função integrando F mais o produto da variável de co-estado pela função f , ele é naturalmente uma função com quatro argumentos: t , y, u bem como λ. Note que, em (7.3), nós designamos um coeficiente unitário para F , o que entra em contraste com o coeficiente ainda indeterminado λ (t) (t) de f . Rigorosamente falando, o Hamiltoniano deveria ser escrito como (7.4)
H (t , y, u, λ )
λ 0 F (t , y, u) + λ (t ) f (t , y, u )
≡
onde λ é uma constante não negativa, também ainda indeterminada. Para o problema (7.1) da linha-terminal-vertical, a constante λ torna-se sempre não nula (estritamente positiva); assim, ela pode normalizada para o valor unitário, reduzindo (7.4) a (7.3). O fato de ser λ ≠ 0 no problema básico é devido a duas condições do princípio do máximo. Primeiro, os multiplicadores λ e λ (t) (t) não podem desaparecer simultaneamente em nenhum ponto do tempo. Segundo, a solução do problema da linha-terminal-vertical deve satisfazer a condição de transversalidade λ (T) (T) = 0, que será explicada na discussão que se segue. A condição λ (T) (T) = 0 requer um valor não nulo para λ em t = T . Mas, desde que λ é uma constante não negativa concluímos que λ é uma constante positiva, que pode ser 0
0
0
0
0
0
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8 seja, onde a função F não tem importância no processo da solução5. É exatamente esse o motivo pelo qual se pode pôr o coeficiente λ igual a zero, para fazer a função F sair do Hamiltoniano. Como muitos dos problemas encontrados em economia são do tipo onde a função F tem importância, a prática prevalecente entre os economistas é simplesmente assumir λ > 0 , normalizando-o então para a unidade e usando o Hamiltoniano (7.3), sempre que o problema não for daquele com uma linha terminal vertical. Essa é a prática que seguiremos. 0
0
O princípio do máximo Em contraste com a equação de Euler que é uma simples equação diferencial de segunda ordem na variável de estado y, o princípio do máximo envolve duas equações diferenciais de primeira ordem na variável de estado y e na variável de co-estado λ . Junto com essas equações, é exigido também que o Hamiltoniano seja maximizado com respeito as variáveis de controle u em todo ponto do tempo. Para uma eficiência pedagógica, primeiro explicamos e discutimos as condições envolvidas, antes de fornecer a racionalidade do princípio do máximo. Para o problema em (7.1), e com o Hamiltoniano definido em (7.3), as condições para o princípio do máximo são H (t , y , u, λ )
Max u
(7.5)
y
λ
∂ H =
=
∂λ −
∂ H ∂ y
λ (T )
O símbolo
=
0
Max H
para todo t ∈ [0, T ]
[equação de movimento para y ] [equação de movimento para
λ ]
[condição de transver salidade]
significa que o Hamiltoniano deve ser maximizado
u
exclusivamente com respeito a u como variável de escolha. Um modo equivalente de expressar essa condição é
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9 Max H
é uma exigência muito mais extensa desse requerimento. Na fig. 7.2, desenhamos
u
três curvas, cada uma indicando um possível gráfico do Hamiltoniano H contra a variável de controle u em um ponto específico do tempo, para valores específicos de y e λ. Assumese que a região de controle é o intervalo fechado [a,c]. Para a curva 1, que é diferenciável com respeito a u, o máximo de H ocorre em u = b, um ponto interior da região de controle U ; nesse caso, a equação ∂ H / ∂u 0 pode de fato servir para identificar o controle ótimo em cada ponto do tempo. Mas, se a curva 2 é a curva relevante, então o controle que maximiza H em U , é u = c, um ponto limite de U . Assim a condição ∂ H / ∂u 0 não se aplica ainda que a curva seja diferenciável. E no caso da curva 3, com o Hamiltoniano =
=
H
Curva 1
Curva 2
Curva 3
u a
0
b
c
U Figura 1 7.2
linear em u, o máximo de H ocorre em u = a, outro ponto limite e a condição ∂ H / ∂u 0 é novamente inaplicável porque a derivada não é igual a zero em nenhum lugar. Em resumo, enquanto a condição ∂ H / ∂u 0 pode servir ao propósito quando o Hamiltoniano é diferenciável com respeito a u e produz uma solução interior, o fato de que a região de =
=
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10 Partindo-se para as outras partes de (7.5), notamos que a condição y ∂ H / ∂λ não é nada mais que uma reafirmação da equação de movimento da variável de estado originalmente especificada em (7.1). O único motivo de re-expressar y como a derivada parcial de H com respeito a variável de co-estado λ é mostrar a simetria entre essa equação de movimento e a variável de co-estado. Note, entretanto, que na última equação de movimento, λ é o negativo da derivada parcial de H com respeito a variável de estado y. Juntas, as duas equações de movimento são referidas coletivamente como o sistema Hamiltoniano, ou sistema canônico (significando o sistema de equações diferenciais padrão) para o dado d ado problema. Contudo nós temos mais que uma equação e quação diferencial para tratar na teoria do controle ótimo – uma para cada variável de estado e uma para cada variável de co-estado – cada equação diferencial é apenas de primeira ordem. Desde que a variável de controle nunca aparece na forma derivada, não existe equação diferencial para u no sistema Hamiltoniano. Mas, da solução básica de (7.5) pode-se, se desejado, obter uma equação diferencial para a variável de controle. E, em alguns modelos, pode ser mais conveniente tratar com um sistema dinâmico nas variáveis ( y, y, u) no lugar do sistema canônico nas variáveis ( y, y, λ). A última condição em (7.5) é a condição de transversalidade para o problema de estado-terminal-livre – aquele com uma linha terminal vertical. Como nós esperaríamos, tal condição diz respeito apenas ao que deveria ocorrer no tempo terminal T. =
EXEMPLO 1: Ache a curva de menor distância de um ponto P dado para uma linha reta L dada. Nós já tínhamos encontrado esse problema no cálculo das variações. Para reformulálo como um problema de controle ótimo, seja o ponto P(0,A), e assuma, sem perda de generalidade, que a linha L é uma linha vertical. (Se a posição da linha não for vertical, pode-se sempre fazer que seja através de uma rotação apropriada nos eixos). A função F previamente usada (1 + y ) pode ser reescrita como (1 +u ) , onde y u ou y u . Para converter o problema de minimização-de-distância para maximização, devemos também, colocar o sinal de menos no integrando. Então, nosso problema é ʹ
2 1/ 2
Maximize Maximize
2 1/ 2
T
(
V = ∫ − 1 + u 2
)
1/ 2
dt
ʹ
=
=
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11 ∂ H ∂u
=
−
1 2
(1
+
u
2
)
−1/ 2
(2u ) + λ = 0
Isso produz a solução6 (7.9)
u (t )
=
(
)
2 −1/ 2
λ 1 − λ
Além disso a diferenciação de
∂ H / ∂u
usando a regra do produto produz
2
∂ H 2
∂ u
=
(
− 1+ u
)
2 −3 / 2
<
0
Assim, o resultado em (7.9) maximiza H . Mesmo que (7.9) expresse u em termos de λ, nós vemos agora encontrar uma solução para λ .
Etapa ii Para fazer isso, nós recorremos a equação de movimento da variável de co-estado λ
=
−
∂ H ∂ y
(7.10)
em (7.5). Mas, como (7.8) mostra que H é independente de y, temos
λ
=
−
∂ H =
∂ y
0
⇒
λ (t )
=
constante
Convenientemente, a condição de transversalidade λ (T ) 0 em (7.5) é suficiente para definir a constante. Pois, se λ é uma constante, então seu valor em t = T também é seu valor para todo t . Assim, =
(7.10’)
λ * (t )
=
0 para todo
t ∈ [0, T ]
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12
(7.12)
y
=
0
y (t )
=
constante
Mais ainda, a condição inicial y(0) = A habilita-nos a definir essa constante e escrever (7.12’)
y * (t )
=
A
y
y* (t ) A =
A
B
Figura 7.3
t 0
T
Esse caminho y*, ilustrado na Fig. 7.3, é uma linha reta horizontal. Alternativamente, ele pode ser visto como um caminho ortogonal para a linha terminal vertical.
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13
u * (t )
=
⎧2 ⎨ ⎩0
se
⎧> ⎫ ⎬3 < ⎩ ⎭
λ (t ) ⎨
As soluções u* = 2 e u* = 0 são, certamente, soluções de canto. Note que, pelo fato de H ser linear em u, a condição de primeira ordem usual ∂ H / ∂u 0 é inaplicável na nossa busca por u*. =
(t), dado que ele é necessário em (7.14). Da Etapa ii Nossa próxima tarefa é determinar λ (t), equação de movimento de λ, nós teremos a equação diferencial λ
=
−
∂ H =
∂ y
H Max H
Curva 1 λ > 3
Curva 2 λ < 3
−2 − λ
Max H
Figura 7.4
ou λ + λ = −2
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14 (7.16)
τ
=
2 − ln 2,5 ≅ 1,084
Conseqüentemente, o controle ótimo em (7.14) pode ser reescrito mais especificamente em duas fases: (7.17)
Fase I: u * Fase II: u *
I
≡
II
u *[0,τ )
≡
2 u *[τ ,2] 0 =
=
u*
(a) 2
0
39,339
Fase II
Fase I
1
τ
= 1,083
2
t
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15 y
−y
=
0
com solução geral (7.19) y * II y *[τ ,2] cet (c arbitrária) ≡
=
Note que a constante c não pode ser definida pelas condições iniciais y(0) = 4 dada em (7.13) porque já estamos na fase II, depois de t = 0. Nem pode ser definida por qualquer condição terminal porque o estado terminal é livre. Entretanto, o leitor lembrará do requerimento de que o caminho ótimo y deva ser contínuo, como ilustra a Fig. 7.1b. Conseqüentemente, o valor inicial de y*II deve ser igual ao valor de y*I em τ. Porquanto, y * I
e
=
2 3e
τ
−1
[por (7.18)]
[por (7.19)] encontramos, igualando essas duas expressões e resolvendo para c, que portanto o caminho ótimo y na fase II é y * II
(7.19’)
y* II 2 3 e =
−
τ
−
≈
τ
=
2e
5,324e
c
t
O valor de y* no tempo de troca τ é aproximadamente
2(3e1,096
− 1)
=
15,739 .
2(3 − e − ) , τ
=
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16 Maximize Maximize sujeito a
∫ ( y − u )dt 2
2
0
y
=
y (0)
u =
y (2)
0
livre
u (t ) não restrito
Verifique que o Hamiltoniano é maximizado ao invés de minimizado. 4 Encontre os caminhos ótimos das variáveis de controle, estado e co-estado para Maximize sujeito a
1
∫ −
1
( y
2
+
2 y = u − y 0
y (0) = 1
)
u 2 dt
y (1)
livre u (t ) não restrito
Verifique que o Hamiltoniano é maximizado ao invés de minimizado. [Sugestão: Duas equações de movimento devem ser resolvidas simultaneamente. Revise o material sobre equações diferenciais simultâneas em Alpha C. Chiang, Fundamental Methods of Mathematical Economics, 3ed., McGraw-Hill, New York, 1984, Séc. 18.2]
7.3
A RACIONALIDADE DO PRINCÍPIO DO MÁXIMO
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17
Etapa i Como primeiro passo no desenvolvimento do princípio do máximo, incorporaremos a equação de movimento dentro do funcional objetivo e então reexpressaremos o funcional em termos do Hamiltoniano. O leitor observará que, se a variável y sempre obedece a equação de movimento, então a quantidade [ f (t , y, u ) − y ] irá seguramente tomar um valor zero para todo t no intervalo [0,T]. Assim, usando a noção dos multiplicadores de Lagrange, podemos formar uma expressão λ (t )[ f (t , y, u) − y ] para cada valor va lor de t, e ainda ter um valor zero. Apesar de existir um número infinito de valores de t no intervalo [0,T], somando λ (t )[ f (t , y, u ) − y ] sobre o tempo no período [0,T] ainda iremos produzir um valor geral zero: (7.21)
T
∫ λ (t )[ f (t , y, u) − y ]dt 0
=
0
Por essa razão, podemos aumentar o antigo funcional objetivo com a integral em (7.21) sem
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18
− ∫
T
0
dt = λ (t ) y
−λ (T ) yT
+
λ (0) y 0
T
+
∫ y(t )λ dt 0
Consequentemente, pela substituição desse resultado, o novo funcional objetivo pode ser re-escrito como (7.22’’)
T
H (t , y, u, λ ) y (t )λ ]dt − λ (T ) y ∫ [
ν =
Ω1
(0) y
T + λ
+
0
Ω2
0 Ω3
A expressão υ é composta da soma de três termos, Ω1, Ω2 e Ω3. Note que enquanto o termo Ω1, uma integral, cobre todo período de planejamento [0,T], o termo Ω2 diz respeito exclusivamente ao tempo terminal T, e Ω3 diz respeito apenas ao tempo inicial.
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19
(7.25)
y (t )
=
y * (t )+ ∈ q(t )
Além disso, se T e yT são variáveis, também teremos (7.26)
T
=
T * + ∈ ΔT
e yT = y *T + ∈ ΔyT (implicando
dT d ∈
=
ΔT
e
dyT d ∈
=
Δ yT )
Observando as expressões de u e y em (7.24) e (7.25), podemos expressar υ em termos de ∈ , portanto poderemos aplicar a condição de primeira ordem ∂υ / ∂ε 0 . A nova versão de υé =
(7.27)
T (∈)
∫
υ =
{ H [t , y * + ∈ q(t ), u * + ∈ p(t ), λ ] + λ [ y * + ∈ q(t )]}dt − λ (T ) yT + λ (0) y0
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20 satisfazer (7.30). Colocando o componente integral igual a zero, podemos deduzir duas condições: λ
=
−
∂ H ∂ y
e
∂ H =
0
∂u
A primeira nos dá a equação de movimento para a variável de co-estado λ (ou simplesmente, a equação de co-estado). E a segunda representa uma versão frágil da condição “ Max H ” – frágil no sentido de que é previamente assumido que H seja u
diferenciável com respeito a u e que exista uma solução interior. Desde que o problema básico tem um T fixo e um yT livre, o termo ΔT em (7.30) é automaticamente igual a zero, mas o termo Δ yT não. A fim de que façamos a expressão − λ (T )Δ yT desaparecer, devemos impor a restrição
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21 para demonstrar esse ponto. p onto. Para nossos propósitos, prop ósitos, é suficiente afirmar que, com c om um ponto terminal fixo, a transversalidade é substituída pela condição y(T ) yT =
( T yT dados) ,
Linha Terminal Horizontal (O Problema do Pontofinal-Fixo) Se o problema tem uma linha terminal horizontal (com um tempo terminal livre mas um “ponto final” fixo, significando um estado terminal fixo), então yT é fixo (Δ yT 0), mas T não é ( ΔT é arbitrário). Do segundo e terceiro termos componentes em (7.30), é fácil ver que a condição de transversalidade para esse caso é =
(7.31)
[ H ]
t T =
=
0
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22 No outro resultado, y *T ymin , desde que a restrição terminal é atingida, os min caminhos admissíveis da vizinhança de y consistiram apenas daqueles que tem estado terminal yT ≥ y min . Se avaliamos (7.25) para t = T e permitimos y *T ymin , obtemos min min =
=
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23 No primeiro resultado, a restrição terminal não é atingida, e a condição de transversalidade para o problema com uma linha terminal horizontal regular ainda é válido: (7.36)
[ H ]
t T =
=
0
para T* < Tmax
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24
Etapa ii
Da equação de movimento de co-estado λ
=
−
∂ H =
∂ y
−λ
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25 Agora as condições limites y(0)=1 e y(1)=0 são diretamente aplicáveis, e Etapa iv elas nos dão a seguinte valores definitivos para c e k : c
1 =
e
k
4e =
2
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26 Segue-se então de (7.42) que (7.44)
u*(t) = 0
Etapa iii Da equação de movimento para y, encontramos
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