Semiconcavity for optimal control problems with exit time. (English) Zbl 1009.49024

Cannarsa, Piermarco; Pignotti, Cristina; Sinestrari, Carlo

Discrete Contin. Dyn. Syst. 6, No. 4, 975-997 (2000).

Summary: In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of the general form \[ \dot y(t)= f(y(t), u(t)),\quad y(t)\in \mathbb{R}^n,\quad u(t)\in U\subset \mathbb{R}^m. \] However, suitable assumptions are needed relating \(f\) with the running and exit costs.
The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the maximum principle, and to study the singular set of the value function.

MSC:

49L20	Dynamic programming in optimal control and differential games
49K40	Sensitivity, stability, well-posedness

Keywords:

optimal control; dynamic programming; semiconcavity; optimal exit time problem; necessary optimality conditions; maximum principle