A parametric convex optimal control problem for a linear system

Abstract

The dependence of the solutions of a terminal optimal control problem on a parameter in the initial state vector is investigated. Attention is devoted mainly to the behaviour of the solution in the neighbourhood of a non-regular point. On the basis of the results, a method is proposed for constructing solutions of the problem for all parameter values. In practical work it is often important to know not only the solution of an optimal control problem for fixed parameter values, but also the dependence of the solution on the parameters, which enables one to estimate how the solution may vary when the parameters fluctuate. In addition, a knowledge of the dependence of the solutions of optimal control problems on the parameters provides the basis for methods of constructing feedback controls [1, 2], as well as stabilization and estimation methods based on the moving horizon strategy [3–5]. Numerical solutions of such problems are generally achieved by continuation of the solution with respect to a parameter [6–9]. The greatest difficulties in applying such methods arise in the case when the “actual” value of the parameter is a non-regular point. Therefore, in most publications devoted to sensitivity analysis and to investigating the parameter-dependence of the solutions, it is assumed that all parameter values are regular, or of degree of non-regularity one. In this paper no such assumptions are made.