Properties of a function of a maximum with constraints

Abstract

IN this paper we consider the properties of a function which assumes a maximum value at each point, with a region of control depending on the phase coordinates. The conditions under which the function is continuous and has a derivative in any direction at every point are explained. The form of this derivative is found. The properties of functions which assume a maximum value at every point are widely used in the development of numerical methods in problems of optimal control and in the theory of differential games. For functions of a maximum and a maximum with a constant region of control these properties are considered in [1–3]. In a number of practical cases the region of control depends on the phase coordinates. The question therefore arises of the properties of a function of a maximum with a region of control depending on the phase coordinates.