The conjugate point condition for smooth control sets

Zeidan, Vera; Zezza, P.

doi:10.1016/0022-247x(88)90085-6

Cited by 42 publications

(45 citation statements)

References 5 publications

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“…On the other hand, when no state constraints are present, the first-order part of this corollary generalizes the results in [MOS98] and [OS95]. When only equality control constraints are present, the statement of Corollary 4.1 has its exact parallel in [ZZ88] for the case where the state is piecewise smooth and the control is piecewise continuous.…”

Section: K (T; δX(t))dµ(t)supporting

confidence: 69%

Optimal Control Problems with Set-Valued Control and State Constraints

Zeidan¹

2003

SIAM J. Optim.

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Abstract. In this paper a general optimal control problem with pure state and mixed controlstate constraints is considered. These constraints are of the form of set-inclusions. Second-order necessary optimality conditions for weak local minimum are derived for this problem in terms of the original data. In particular the nonemptiness of the set of critical directions and the evaluation of its support function are expressed in terms of the given functions and set-valued maps. In order that the Lagrange multiplier corresponding to the mixed control-state inclusion constraint be represented via an integrable function, a strong normality condition involving the notion of the critical tangent cone is introduced.

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Section: K (T; δX(t))dµ(t)supporting

confidence: 69%

Optimal Control Problems with Set-Valued Control and State Constraints

Zeidan¹

2003

SIAM J. Optim.

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“…But, once again, this aspect of the theory varies dramatically between different authors. In [30], for example, it is claimed in an "important remark" that the case when inequality constraints are present is within the scope of the paper, where only the equality constraint case is studied. According to [17], "this is neither clear nor convincing," and the presence of this kind of constraints becomes a crucial feature in the second order conditions obtained.…”

Section: Introductionmentioning

confidence: 99%

“…In [5] and [8] one can find some of the first attempts in this direction but, in [32], it is assured repeatedly that both papers are incorrect (this claim is denied by Bernhard in a reply following [32]). In subsequent papers such as [28], [30], [31] and [33], one encounters different definitions of conjugate points which are not all equivalent even if the problems considered are reduced to the same fixedendpoint problem. In the calculus of variations context, however, all these notions coincide, and in this sense all generalize the classical theory.…”

Section: Introductionmentioning

confidence: 99%

Conjugate journey in optimal control

Rosenblueth¹

2007

imf

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“…One ultimate goal is to obtain results for the optimal control setting by applying the theory for the abstract optimization problem. Another route in accomplishing this goal is to directly study the second variation of the optimal control problem with the given constraints (see e.g., [25], [24]). In the literature, one can find second-order conditions for optimization problems with various types of constraints: equality, inequality, or set valued (e.g., [1], [7], [20]).…”

Section: Introductionmentioning

confidence: 99%

“…In the literature, one can find second-order conditions for optimization problems with various types of constraints: equality, inequality, or set valued (e.g., [1], [7], [20]). On the other hand, there are second-order necessary conditions for optimal control problems with equality or inequality mixed state-control constraints (see [19], [21], [24]), or for the pure control inequality constraints (see [25]). However, none of these results include the pure-state inequality constraint g(t, x(t)) < 0, where /-dependence of g is only upper semicontinuous.…”

Section: Introductionmentioning

confidence: 99%

First- and second-order necessary conditions for control problems with constraints

Zeidan

1994

Trans. Amer. Math. Soc.

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Abstract.Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form G(x, u) e T, where T is a closed convex set of a Banach space with nonempty interior. The inequality constraints g{s, x, u) < 0 depend on a parameter 5 belonging to a compact metric space S. The equality constraints are split into two sets of equations K(x, u) = 0 and H(x, u) = 0 , where the first equation is an abstract control equation, and H is assumed to have a full rank property in u . The objective function is maxteT f(t, x, u) where T is a compact metric space, / is upper semicontinuous in t and Lipschitz in (x, u). The results are in terms of a function a that disappears when the parameter spaces T and S are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.

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The conjugate point condition for smooth control sets

Cited by 42 publications

References 5 publications

Optimal Control Problems with Set-Valued Control and State Constraints

Optimal Control Problems with Set-Valued Control and State Constraints

Conjugate journey in optimal control

First- and second-order necessary conditions for control problems with constraints

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