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 \(f(t,x,\dot {x},u)=0\), \(\dot {x}(t)\in B(t) \), \(u(t)\in U(t,x,\dot {x}) \), \(t\in [a,b]\).