Approximate feedback solutions for differential games. Theory and applications

Abstract

Differential games deal with problems involving multiple players, possibly competing, that influence common dynamics via their actions, commonly referred to as strategies. Thus, differential games introduce the notion of strategic decision making and have a wide range of applications. The work presented in this thesis has two aims. First, constructive approximate solutions to differential games are provided. Different areas of application for the theory are then suggested through a series of examples. Notably, multi-agent systems are identified as a possible application domain for differential game theory. Problems involving multi-agent systems may be formulated as nonlinear differential games for which closed-form solutions do not exist in general, and in these cases the constructive approximate solutions may be useful. The thesis is commenced with an introduction to differential games, focusing on feedback Nash equilibrium solutions. Obtaining such solutions involves solving coupled partial differential equations. Since closed-form solutions for these cannot, in general, be found two methods of constructing approximate solutions for a class of nonlinear, nonzero-sum differential games are developed and applied to some illustrative examples, including the multi-agent collision avoidance problem. The results are extended to a class of nonlinear Stackelberg differential games. The problem of monitoring a region using a team of agents is then formulated as a differential game for which ad-hoc solutions, using ideas introduced previously, are found. Finally mean-field games, which consider differential games with infinitely many players, are considered. It is shown that for a class of mean-field games, solutions rely on a set of ordinary differential equations in place of two coupled partial differential equations which normally characterise the problem.