ON THE STABILITY OF THE BEST REPLY MAP FOR NONCOOPERATIVE DIFFERENTIAL GAMES
Alberto Bressan, Zipeng Wang · Analysis and Applications · 2012
Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls [Formula: see text] is Nash equilibrium solution if [Formula: see text] is the best strategy for Player 1 in reply to [Formula: see text], and [Formula: see text] is the best strategy for Player 2, in reply to [Formula: see text]. The aim of the present note is to investigate the stability of the best reply map: [Formula: see text]. For linear-quadratic games, we derive a condition which yields asymptotic stability, within the class of feedbacks which are affine functions of the state x ∈ ℝn. An example shows that stability is lost, as soon as nonlinear perturbations are considered.