On a Control Problem for a System of Implicit Differential Equations

Abstract

We consider the differential inclusion $F(t,x,\dot {x})\ni 0$ with the constraint $\dot {x}(t)\in B(t)$, $t\in [a, b]$, on the derivative of the unknown function, where $F$ and $B$ are set-valued mappings, $F:[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\times \mathbb {R }^m\rightrightarrows \mathbb {R}^k$ is superpositionally measurable, and $B:[a,b]\rightrightarrows \mathbb {R}^n$ is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form $f(t,x,\dot {x},u)=0$, $\dot {x}(t)\in B(t)$, $u(t)\in U(t,x,\dot {x})$, $t\in [a,b]$.