A Measurable Space as Uncertainty Set

Abstract

The purpose of this chapter is to extend the possibilities of the Maximum Principle approach for the class of Min-Max Control Problems dealing with the construction of the optimal control strategies for uncertain systems given by a system of ordinary differential equations with unknown parameters from a given compact measurable set. The problem considered belongs to the class of optimization problems of the Min-Max type. Below, a version of the Robust Maximum Principle applied to the Min-Max Mayer problem with a terminal set is presented. A fixed horizon is considered. The main contribution of this material is related to the statement of the robust (Min-Max) version of the Maximum Principle formulated for compact measurable sets of unknown parameters involved in a model description. It is shown that the robust optimal control, minimizing the worst parametric value of the terminal functional, maximizes the Lebesgue–Stieltjes integral of the standard Hamiltonian function (calculated under a fixed parameter value) taken over the given uncertainty parametric set. In some sense, this chapter generalizes the results given in the previous chapters in such a way that the case of a finite uncertainty set is a partial case of a compact uncertainty set, supplied with a given atomic measure.