Chapter 4 – Soft-Constrained Feedback Nash Equilibria

Publisher Summary This chapter defines feedback Nash equilibria in indefinite linear quadratic differential games on an infinite time horizon in a deterministic uncertain environment. The relationship between the existence of such equilibria and solutions of sets of algebraic Riccati equations is investigated in the chapter. An assumption is made that the players have a feedback pattern information structure and that they restrict themselves to stationary linear stabilizing feedback strategies. The players cope with the uncertainty by considering an H∞ type performance criterium. For the special one-player case, this results in the study of an indefinite soft-constrained differential game. A converse statement states that if the upper value exists then the involved Riccati equation has a stabilizing solution with the additional property. This theorem is established under a restriction of the set of admissible feedback matrices. This might be viewed as a first step in the direction of obtaining both necessary and sufficient conditions for robust equilibria. It is shown that the soft-constrained equilibria can be interpreted in a stochastic environment as risk-sensitive equilibria.