On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t,x,ẋ)∋ 0 with the constraint ẋ(t)∈ B(t) , t∈ [a, b] , on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×ℝ^n×ℝ^n×ℝ ^m⇉ℝ^k is superpositionally measurable, and B:[a,b]⇉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,ẋ,u)=0 , ẋ(t)∈ B(t) , u(t)∈ U(t,x,ẋ) , t∈ [a,b] .