Controllability, extremality, and abnormality in nonsmooth optimal control

Warga, J.

doi:10.1007/bf00934445

Cited by 64 publications

(40 citation statements)

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“…if a certain linearized inclusion is λ-locally controlable around the null solution for every λ ∈ ∂h(z(T )), where ∂h(.) denotes Clarke's generalized Jacobian of the locally Lipschitz function h. The key tools in the proof of our result is a continuous version of Filippov's theorem for mild solutions of semilinear differential inclusions obtained in [2] and a certain generalization of the classical open mapping principle in [14].…”

Section: Introductionmentioning

confidence: 99%

On controllability for nonconvex semilinear differential inclusions

Cernea¹

2013

J. Nonlinear Sci. Appl.

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Section: Introductionmentioning

confidence: 99%

On controllability for nonconvex semilinear differential inclusions

Cernea¹

2013

J. Nonlinear Sci. Appl.

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“…Applying this result to the collection of all selections of F (convex valued and satisfying the Lipschitz condition), Kaśkosz and Lojasiewicz obtained in [28] another necessary condition for inclusion constrained problems, and Zhu [53] extended their result to nonconvex inclusions (also bounded and Lipschitz) by elaborating on a controllability theorem of Warga [52]. An obvious drawback of these conditions is the absence of any analytic mechanism for obtaining selections (even in the case of a convex valued inclusion).…”

Section: Introductionmentioning

confidence: 98%

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

Ioffe

1997

Trans. Amer. Math. Soc.

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Abstract. We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of EulerLagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

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“…We are concerned with necessary conditions for a solution to P. If F(t, x) admits a parametrization, i.e. there exist a set U and a function f(t, x, u) such that F(t, x)=[ f(t, x, u) : u # U] and if certain regularity conditions are satisfied then the maximum principle and its various generalizations (see [26,30]) concerning the control system defined by ( f, U ) provide necessary conditions for a solution to P. However, for a nonconvex-valued multifunction F it is very difficult to determine whether such a parametrization exists. On the other hand, a variant of Lojasiewicz'es parametrization theorem [17] shows that under fairly general conditions such a parametrization exists when F is convex-valued (Section 4).…”

Section: Introductionmentioning

confidence: 99%

“…We then follow the road mapped by Warga [26,30]. Denote by S(U ) the (compact convex) set of relaxed controls corresponding to U.…”

Section: Introductionmentioning

confidence: 99%