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 (u ^* ₁(x), u ^* ₂(x))is Nash equilibrium solution if u ^* ₁ 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 u ^* ₁ . The aim of the present note is to investigate the stability of the best reply map: (u ₁, ₂→R ₁(u ₂)R ₂(u ₁)). 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 ∈ ℝ ⁿ. An example shows that stability is lost, as soon as nonlinear perturbations are considered.