期刊
JOURNAL OF DIFFERENTIAL EQUATIONS
卷 260, 期 4, 页码 3397-3447出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2015.10.039
关键词
Optimal control; Sweeping process; Necessary optimality conditions; Discrete approximations; Variational analysis; Generalized differentiation
类别
资金
- GNAMPA of INdAM
- DFG Research Center MATHEON
- FONDECYT [3140060]
- Basal Projects at CMM, Universidad de Chile
- USA National Science Foundation [DMS-1007132, DMS-1512846]
- Air Force Office of Scientific Research [15RT0462]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1512846] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1007132] Funding Source: National Science Foundation
The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W-1,W-2-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples. (C) 2015 Elsevier Inc. All rights reserved.
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