Permanent Differential Games: Quasi Stationary and Relaxed Steady-State Operations

Abstract

This paper deals with permanent differential games, i.e. differential games the state of which must satisfy a periodicity constraint.

To any given permanent differential game a conventional (non- differential) game of obvious interest can be associated in a straightforward way by simply considering the equilibrium states only of the dynamical system under consideration. In this situation a comparison between the solutions of the former (dynamical) and the latter (static) game is of obvious interest. In particular, the main concern here can be expressed by the following question: Under what conditions (henceforth called dynamic dominance conditions), for a given Pareto-optimal solution of the associated static game, a nonconstant control exists such as to dominate, in the Pareto sense, the constant solution above (which is Pareto-optimal within the class of the constant solutions only)?

The approach taken in the present paper to takle such a kind of question can be considered a standard one in Periodic Optimization Theory [1] and consists in restraining the attention to special classes of control functions (namely quasi-constant and chattering controls) leading to relatively simple and general dynamic dominance conditions, each one of which calls for nothing more than the analysis of a suitable static game.