Optimal Control via Nonsmooth
                    Analysis

Loewen, Philip D.

doi:10.1090/crmp/002

Cited by 105 publications

(56 citation statements)

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“…Tais resultados serão necessários para a obtenção de soluções para inclusões dinâmicas em escalas temporais. Os correspondentes resultados para a obtenção de soluções para inclusões diferenciais, podem ser encontrados em [8].…”

Section: Propriedades De Multifunçõesunclassified

Sobre Inclusões Dinâmicas em Escalas Temporais

Santos¹

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. No presente trabalho consideramos inclusões dinâmicas em escalas temporais cujos campos vetoriais podem ser ilimitados. Então, fazendo uso de propriedades de teoria da medida e de multifunções, provamos um resultado de existência de soluções.Palavras-chave. Inclusões Dinâmicas, Existência de Soluções, Escalas Temporais IntroduçãoO estudo de inclusões dinâmicas em escalas temporais pode ser encontrado, por exemplo, em [1,2,[4][5][6]9]. Uma escala temporalé um subconjunto fechado e não-vazio de números reais. Aqui usamos uma escala temporal limitada T, onde a = min T e b = max T são tais que a < b.Neste trabalho, nós provamos um resultado de existência de soluções para inclusões dinâmicas em escalas temporais cujos campos vetoriais podem ser ilimitados. O resultadó e obtido de modo similar ao teorema [ [7], Theorem 2.2]. Entretanto, diferente de [7], nós supomos que os campos vetoriais das inclusões dinâmicas são Lipschitz mensuráveis. PreliminaresConceitos e resultados básicos da teoria de escalas temporais podem ser encontrados em [9] e [10]. Integração em escalas temporaisDenotamos o conjunto de funções f :Denote por B m a σ-Álgebra de Borel de R m e por ∆ × B m a σ-Álgebra produto entre ∆ e B m .Os teoremas 2.1 e 2.2 dados abaixo são obtidos de modo análogo aos teoremas [ [3], Théorème IV.8] e [ [3], Théorème IV.9], respectivamente.

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Section: Propriedades De Multifunçõesunclassified

Sobre Inclusões Dinâmicas em Escalas Temporais

Santos¹

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…Our general reference for this section is [17]; see also [13], Clarke [8,9], Loewen [29] and Vinter [44].…”

Section: Preliminariesmentioning

confidence: 99%

Feedback in state constrained optimal control

2002

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Abstract. An optimal control problem is studied, in which the state is required to remain in a compact set S. A control feedback law is constructed which, for given ε > 0, produces ε-optimal trajectories that satisfy the state constraint universally with respect to all initial conditions in S. The construction relies upon a constraint removal technique which utilizes geometric properties of inner approximations of S and a related trajectory tracking result. The control feedback is shown to possess a robustness property with respect to state measurement error.Mathematics Subject Classification. 49J24, 49J52, 49N55, 90D25.

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“…We shall frequently call upon standard results in nonsmooth calculus (to be found, for example, in [2], [6], [10]). …”

Section: Definitionmentioning

confidence: 99%

“…To justify the last inclusion we applied the "sum rule" [6], taking note of the Lipschitz continuity of L(t, ·, ·) in a neighbourhood of (x(t),ẋ(t)). The differential inclusion for p may therefore be replaced bẏ p(t) ∈ co{η : (η, p(t) + [0,t) γ(s)µ(ds)) ∈ λ∂L(t,x(t),ẋ(t))) + {0} × N R(t) (ẋ(t))}.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…This means, if we can show (2) and we know that F is convex valued, we automatically have that the adjoint arc p also satisfies the conditioṅ p(t) ∈ co{−η : (η,ẋ(t)) ∈ ∂H(t,x(t), p(t))}. (6) This is a strengthened form of the Hamiltonian inclusion involving convexification in only one variable, proved earlier by Loewen and Rockafellar. (See [7] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Necessary conditions for optimal control problems with state constraints

Vinter¹,

Zheng²

1998

Trans. Amer. Math. Soc.

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Abstract. Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild 'one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent 'dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.Proofs make use of recent 'decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

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Optimal Control via Nonsmooth Analysis

Cited by 105 publications

References 0 publications

Sobre Inclusões Dinâmicas em Escalas Temporais

Sobre Inclusões Dinâmicas em Escalas Temporais

Feedback in state constrained optimal control

Necessary conditions for optimal control problems with state constraints

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