About the Differential Inequality for Implicit Control System

Abstract

In this article we study an implicit differential control system described by first order differential equations not solved with respect to the derivative. Existence conditions and solution estimates in the form of theorems on differential inequalities of Chaplygin’s theorem type are obtained. Methods of the theory of multivalued mappings in partially ordered spaces and results on implicit differential inclusions obtained earlier by the author are used. In the first part of the paper we give a statement on the solvability in a partially ordered space of an operator inclusion generated by a multivalued mapping of two arguments, one of which is covering and the other — antitone. The statement has the form of a comparison theorem with the solution of the corresponding operator inequality. In the second part of the paper we consider a boundary value problem for a system of implicit differential inclusions. Solvability conditions (in the class of absolutely continuous functions), estimates of solutions, and conditions for the existence of a solution with the smallest derivative are given. In the third main part, using the results given in the second part, a two-point boundary value problem for an implicit differential control system is investigated. The trajectory is assumed to be absolutely continuous, the control — measurable. Solvability conditions, solution estimates, existence conditions of the solution with the smallest control and with the trajectory having the smallest derivative are obtained.