Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

Abstract

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.

Optimal control of differential inclusions on manifolds

“…Inequality (27) then implies that p is everywhere nonzero. Finally, p satisfies boundary conditions (12) and maximum principle (13). It follows once more that x * is not controllable relative to C and this completes the proof.…”

“…Taking the limit in (30), and (31) we see that p also satisfies the boundary conditions (12) and maximum principle (13). Finally, we must have p(t) ≥ 1 + k 2 F −1/2 for some t ∈ [0, T ] and it follows from (27) that p is everywhere nonzero, hence x * is not controllable relative to C.…”

“…Different variants of the following theorem appeared in mid-1990s (see [13,14,28,20]). The work of Ioffe and Rockafellar in [14] in fact provides a more general necessary optimality condition than that given in next Theorem 3.1.…”

“…This early work assumed that F was convex-valued. Subsequent developments of optimality conditions for this dynamic optimization problem are related to work by Clarke, Ioffe, Mordukhovich, Rockafellar, Vinter [13,14,22,23,28,30] and others and are largely concerned with a relaxation of the assumption of convexity of values of F and a replacement of the original Hamiltonian inclusion by more refined ones (for example, by a smaller "partially convexified" variant).…”

Nonsmooth Problems of Calculus of VariationsviaCodifferentiation

“…These problems were first studied in the works of Rockafellar [21][22][23]. After these works many different approaches were suggested to studying nonsmooth problems of the calculus of variations (cf., for details, [5][6][7]14,15,[17][18][19]25,26]); however, all existing approaches have some disadvantages, that make their practical applications quite difficult.…”