Trajectories of differential inclusions with state constraints
Solutions of the differential inclusion \[ x'\in F(x),\quad x(0)=x_0,\tag{1} \] are studied that satisfy constraints of the form \[ x(t)\in \Omega \quad \forall \; t\in [0,T].\tag{2} \] Denote by \(\operatorname{co}F\) the convex closure of \(F\) and by \(T_{\Omega }(x)\) the tangent cone to \(\Omega \) at \(x\in \Omega\). Under the conditions that \(F:{\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a Lipschitz continuous compact valued multifunction, \(\Omega \subset {\mathbb R}^n\) is a compact convex set and that the following inward-pointing condition is satisfied \[ \operatorname{co}F(x)\cap \text{int}(T_{\Omega }(x))\neq \emptyset \quad \forall \;x\in \Omega, \] it is proved that, for a given reference solution \(y(.)\) of problem (1) taking values in an \(\varepsilon \)-neighborhood of \(\Omega \), there exists a solution of problem (1)--(2) such that \[ \|x-y\|_{W^{1,1}}\leq C\varepsilon (1+|\ln\varepsilon |). \]
scientific article; zbMATH DE number 5860016
scientific article