On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t, x, (x)over dot) (sic) 0 with the constraint (x)over dot (t) is an element of B(t), t is an element of [a, b], on the derivative of the unknown function, where F and B are set-valued mappings, F : [a, b] x R-n x R-n x R-m paired right arrows R-k is superpositionally measurable, and B : [a, b] paired right arrows 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, (x)over dot, u) = 0, (x)over dot (t) is an element of B(t), u(t) is an element of U(t, x, (x)over dot), t is an element of [a, b].