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The maximum principle and the superdifferential of the value function. (English) Zbl 0684.49009

Probl. Control Inf. Theory 18, No. 3, 151-160, Russian version P1-P10 (1989).
Summary: An important connection between the Pontryagin maximum principle and the Bellman dynamic programming is established. It is proved that adjoint variables appearing in the maximum principle conditions are generalized gradients of the value function, which is the “viscosity” solution of the Hamilton-Jacobi-Bellman equation. This relationship completes the maximum principle conditions to necessary and sufficient optimality conditions.
The paper was initiated by the recent results of W. Clarke and R. Vinter [SIAM J. Control Optimization 25, 1291-1311 (1987; Zbl 0642.49014)] who had obtained a similar connection as a necessary optimality condition.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49L20 Dynamic programming in optimal control and differential games
49J52 Nonsmooth analysis

Citations:

Zbl 0642.49014