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