A class of singular control problems

Abstract

In this paper we consider the problem of minimizing a class of singular control problems. The main emphasis is given in deriving the equations describing the optimal pairs. The state is governed by a ~th order system of ordinary linear di f ferent ial equations with complex matrixcoefficients in a compact interval, and is subject to a class of approximate boundary conditions. Both the cost functional and state involve very general generalized boundary conditions. The complete proof wi l l appear in [6]. A control problem involving an i n i t i a l condition together with generalized boundary conditions was considered in [3] , for example. The special case of our problem when the cost functional is regular and I= I has been considered in [5]. Since we deal with nonstandard boundary conditions (which become exact boundary conditions in many important cases), a given control generate in f in i te l y many responses, and so the cost functional becomes multi-valued function of control. For this reason we wi l l employ a new method based on the theory of least-squares solutions of a linear relation developed in [8] , [11].