Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. (English) Zbl 0876.49024
Summary: 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 Euler-Lagrange 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.
MSC:
| 49K24 | Optimal control problems with differential inclusions (nec./ suff.) (MSC2000) |
| 34H05 | Control problems involving ordinary differential equations |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
| 49J52 | Nonsmooth analysis |
Keywords:
optimal control; calculus of variations; differential inclusion; Lagrangian; Hamiltonian; maximum principle; nonsmooth analysis; approximate subdifferential; Pontryagin maximum principleReferences:
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