On the Interpretation of Non-anticipative Control Strategies in Differential Games and Applications to Flow Control

Abstract

This paper concerns the notion of feedback strategies in differential games. Classical notions of feedback strategies, based on state feedback control laws for which the corresponding closed loop dynamics uniquely define a state trajectory, are too restrictive for many problems, owing to the absence of minimizing classical feedback strategies or because consideration of classical feedback strategies fails to define, in a useful way, the value of the game. A number of feedback strategy concepts have been proposed to overcome this difficulty. That of Elliot and Kalton, according to which a feedback strategy is a non-anticipative mapping between control functions for the two players, has been widely taken up because it provides a value of the game which connects, via the HJI equation, with other fields of systems science. The non-anticipative mapping approach leaves unanswered, nonetheless, some fundamental questions regarding the representation of optimal feedback strategies. Attempts to solve specific games problems by the method of characteristics, or other techniques, often lead to consideration of optimal feedback strategies in the form of discontinuous state feedback control laws. Such control laws cannot be regarded as classical feedback control strategies because the associated state trajectories are not unique. We provide full details of the proofs of recently announced general theorems that interpret discontinuous state feedback laws as non-anticipative feedback strategies. An application to flow control illustrates the relevance of the theory to process systems engineering.Keywords: Differential Games, Differential Inclusions, Feedback Control.