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Convexity properties of the minimum time function. (English) Zbl 0836.49013
The paper deals with the minimum time optimal control problem governed by the system \begin{aligned} y'(t) & = f \bigl( y(t), u(t) \bigr) \tag{*} \\ y(0) & = x \in \mathbb{R}^n, \quad u : [0, + \infty) \to U, \end{aligned} where $$U \subset \mathbb{R}^m$$ is a compact set. Given a target $$K$$ and denoting by $$\Omega \subset \mathbb{R}^n$$ the set of points which can be driven in a finite time on $$K$$, the minimum time function $$\tau : \Omega \to [0, + \infty)$$ is defined as $\tau (x) : = \inf \biggl\{ T : \bigl( y(t), u(t) \bigr) \text{ solution of } (*),\;y(T) \in K \biggr\}.$ Under a Petrov type controllability assumption, it is proved that the semiconcavity of the distance function from the target (which can be considered as a mild regularity assumption on $$K)$$ implies the semiconcavity of $$\tau$$. For linear control systems a semiconvexity property can also be established.
These conditions can be used to study the structure (Hausdorff dimension estimates, propagation of singularities) of the set of nondifferentiability points of $$\tau$$.
Reviewer: L.Ambrosio (Pavia)

##### MSC:
 49L20 Dynamic programming in optimal control and differential games 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 93C10 Nonlinear systems in control theory
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