Optimal Control

Vinter, R. B.

doi:10.1007/978-0-8176-8086-2

Cited by 256 publications

(388 citation statements)

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“…These relations are referred to as sensitivity relations. For Lipschitz data similar results hold true with superdifferentials replaced by generalized gradients, see [11,24] and also [23] for an extension of the second sensitivity relation to unconstrained differential inclusions. Further results for differential inclusions under general state constraints can be found in [8,9].…”

Section: (T) ∈ F (T X(t))supporting

confidence: 54%

“…In particular, first order necessary conditions for optimality were derived in the form of a maximum principle in [12,19]. We refer to [23] and the bibliography contained therein for further results and comments on maximum principles for nonsmooth control systems and differential inclusions under state constraints. Another related topic under investigation since eighties concerns Hamilton-Jacobi equations under state constraints.…”

Section: Introductionmentioning

confidence: 99%

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On relations of the adjoint state to the value function for optimal control problems with state constraints

Frankowska¹,

Mazzola²

2012

Nonlinear Differ. Equ. Appl.

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Abstract.We consider an optimal control problem under state constraints and show that to every optimal solution corresponds an adjoint state satisfying the first order necessary optimality conditions in the form of a maximum principle and sensitivity relations involving the value function. Such sensitivity relations were recently investigated by P. Bettiol and R.B. Vinter for state constraints with smooth boundary. In the difference with their work, our setting concerns differential inclusions and nonsmooth state constraints. To obtain our result we derive neighboring feasible trajectory estimates using a novel generalization of the so-called inward pointing condition.Mathematics Subject Classification (2010). Primary 49K15, 49Q12.

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Section: (T) ∈ F (T X(t))supporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

On relations of the adjoint state to the value function for optimal control problems with state constraints

Frankowska¹,

Mazzola²

2012

Nonlinear Differ. Equ. Appl.

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“…Since f (x, U ) is convex for all x, from the classical Convergence Theorem (see, e.g., [3, Theorem 1, p. 60]) it follows that z(t) ∈ f (x(t), U ) for a.e. t. It then follows from from Filippov's Selection Theorem (see, e.g., [25,Th. 2.3.13]) that there existsū(·) such that z(t) = f (x(t),ū(t)).…”

Section: Results On the Sweeping Process And Its Regularizationmentioning

confidence: 92%

A Maximum Principle for the Controlled Sweeping Process

Arroud

Colombo

2017

Set-Valued Var. Anal

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Abstract. We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin's Maximum Principle type. The results are also discussed through an example. We combine techniques from [19] and from [6], which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set C(t), but, differently from [6], do not require strict convexity. Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of C(t) is touched. The state space is finite dimensional.

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“…Since N ((w,z); A 0 ) = N (w,z); gph H and N ((w,z); A 1 ) = {0} × N (z; Ω), the equality (18) follows from (20). ✷ Theorem 2 Suppose that Φ has closed range and ker T * ⊂ ker M * .…”

Section: Theorem 1 (See [2 Theorem 42])mentioning

confidence: 99%

Differential Stability of Convex Discrete Optimal Control Problems

Toan

2017

Acta Math Vietnam

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Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen [Appl. Anal. 94, 108-128 (2015)] on differential stability of parametric convex optimization problems under inclusion constraints, we obtain an upper estimate for the subdifferential of the optimal value function of a parametric convex discrete optimal control problem, where the objective function may be nondifferentiable. If the objective function is differentiable, the obtained upper estimate becomes an equality. It is shown that the singular subdifferential of the just mentioned optimal value function always consists of the origin of the dual space.Keywords Parametric convex discrete optimal control problem · Optimal value function · Subdifferentials · Linear operator with closed range · Adjoint operator. Mathematics Subject Classification (2000)

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Optimal Control

Cited by 256 publications

References 0 publications

On relations of the adjoint state to the value function for optimal control problems with state constraints

On relations of the adjoint state to the value function for optimal control problems with state constraints

A Maximum Principle for the Controlled Sweeping Process

Differential Stability of Convex Discrete Optimal Control Problems

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