Necessary Optimality Conditions for Nonconvex Differential Inclusions with Endpoint Constraints

Abstract

In this paper, we derive necessary optimality conditions for optimization problems defined by non-convex differential inclusions with endpoint constraints. We do this in terms of parametrizations of the convexified form of the differential inclusion and, under additional assumptions, in terms of the inclusion itself.

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

“…We refer the reader to the survey papers [1][2][3][4][10][11][12][13][14]22,26,30,33,[35][36][37][38]. Now let us explain the principal method that we use to obtain mentioned results.…”

“…The past decade has seen an ever more intensive development of the theory of extremal problems concerned by multivalued mappings with lumped and distributed parameters [4,5,7,9,16,26,28,[33][34][35][36][37][38].…”

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

“…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).…”

“…[3,12,2]), but only progress in nonsmooth analysis provided necessary equipment to tackle the problem and a push for intensive studies (e.g. [5,9,13,14,28,29,30,31,33,39,41,42,49,53] and many more). As a result, a variety of necessary optimality conditions were obtained that either followed the patterns of the classical Euler-Lagrange or Hamiltonian formalisms or were based on the Pontriagin maximum principle for "parametrized" problems of optimal control.…”

“…[53]) where a condition like (1.3) was obtained each time under the additional assumption of the existence of families of some nice (say, smooth) selections of F (t, ·) that typically cannot be verified. A recent example given by Zhu [53] shows that without such assumptions (1.3) may fail to be a necessary condition at all (even for convex valued F ).…”