On the Stability of the Best Reply Map for Noncooperative Differential Games

Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls (u1(x), u ∗ 2(x)) is Nash equilibrium solution if u1 is the best strategy for Player 1 in reply to u ∗ 2, and u ∗ 2 is the best strategy for Player 2, in reply to u1. Aim of the present note is to investigate the stability of the best reply map: (u1, u2) 7→ (R1(u2), R2(u1)). 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 ∈ IR. An example shows that stability is lost, as soon as nonlinear perturbations are considered.