Convexity properties of the minimum time function
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 \(\tau\).
scientific article; zbMATH DE number 778936
scientific article