On the analysis of open-loop Nash equilibria admitting a feedback synthesis in nonlinear differential games

Open-loop Nash equilibrium strategies for differential games described by nonlinear, input-affine, systems and cost functionals that are quadratic with respect to the control input are studied. First it is shown that the computation of such strategies hinges upon the solution of a system of nonlinear, time-varying, partial differential equations (PDEs) obtained by building on arguments borrowed from Pontryagin’s Minimum Principle and combined with Dynamic Programming considerations. Then, by relying on a state/costate interpretation of the above characterization, a feedback synthesis of the underlying open-loop strategy is obtained by solving linear first-order PDEs that ensure invariance of certain submanifolds in the state–space of the extended state/costate dynamics. These PDEs are the nonlinear counterpart of the well-known asymmetric Algebraic Riccati Equations arising in the study of linear quadratic Nash games.