One linear set-valued control problem

Abstract

Recently, many authors have considered questions of the existence, uniqueness, and properties of solutions of set-valued differential and integro-differential equations, higher order equations, and have investigated impulse and control systems in the framework of the theory of set-valued equations. Obviously, obtaining all these results would be impossible without the development of the theory of set-valued analysis. In the latter, new definitions of the derivative have appeared for set-valued mappings, which, unlike the previously used Hukuhara derivative, made it possible to differentiate set-valued mappings whose diameter is not only a non decreasing function. As a result, set-valued differential equations were considered whose solutions are set-valued mappings whose diameter is not a monotonic function. This article discusses the new formulation of the optimal control problem (the time-optimality problem) that became possible due to these new derivatives and differential equations, as well as a method for solving this problem.