From Optimal Control to Non-cooperative Differential Games: a Homotopy Approach

Abstract: We propose a new approach to the study of Nash equilibrium solutions to non-cooperative differential games. The original problem is embedded in a one-parameter family of differential games, where the parameter θ ∈ [0, 1] accounts for the strength of the second player. When θ = 0, the second player adopts a myopic strategy and the game reduces to an optimal control problem for the first player. As θ becomes strictly positive, Nash equilibrium solutions can be obtained by studying a bifurcation problem for the corresponding system of Hamilton-Jacobi equations.