Generalized solutions of linear control systems

Abstract

In the present paper we consider a linear control system on a finite interval. The main result is Theorem 2, which gives sufficient conditions for the existence of generalized solutions, that steer given object from one point to another. The intensity $I$ (it measures the energy spent to steer the object, and its values are norms of functional belonging to a convenient conjugate space) achieves its smallest value. Since the integrals involved in the solution are extreme points of the unit ball of the corresponding conjugate space, in some cases we can determine these integrals and thus obtain an effective solution of the given system. We illustrate the method with two examples.