The space $\mathrm{clcv}(\mathbb R^n)$ with the Hausdorff–Bebutov metric and differential inclusions

The paper is devoted to studying the space of nonempty closed convex (but not necessarily compact) sets in $\mathbb R^n$, a dynamical system of translations, and existence theorems for differential inclusions. This space is made complete by equipping it with the Hausdorff–Bebutov metric. The investigation of these issues is important for certain problems of optimal control of asymptotic characteristics of the controlled system. For example, the problem $\dot x=A(t,u)x$, $(u,x)\in\mathbb R^{m+n}$, $\lambda_n(u(\cdot))\to\min$, where $\lambda_n(u(\cdot))$ – is the maximal Lyapunov exponent of the system $\dot x=A(t,u)x$, leads to a differential inclusion with a noncompact right-hand side.