Regularity of solution maps of differential inclusions under state constraints
The paper concerns the Lipschitzian properties of the set-valued map \({\mathcal S}^K_{[t_0,T]}\) which assigns to every point \(\xi_0\in K\) the set of solutions \(x:[t_0,T]\to\mathbb R^n\) to the system \[ x'(t)\in F(t,x(t)), \quad x(t)\in K,\;x(t_0)=\xi_0. \] Here, the \(t\)-measurable, \(x\)-lipschitzean set-valued map \(F:[t_0,T]\times\mathbb R^n\to\mathbb R^n\) has (possibly unbounded) closed convex images and satisfies an inward pointing condition on the sufficiently smooth boundary of the closed set \(K\subseteq\mathbb R^n\). The paper extends and simplifies the approach developed in \textit{P. Bettiol, P. Cardaliaguet,} and \textit{M. Quincampoix} [Int. J. Game Theory 34, No.4, 495--527 (2006; Zbl 1103.49019)]. The result is applied to investigate Lipschitzian properties of the value function for the constrained Bolza problem of optimal control theory.
scientific article; zbMATH DE number 5146248