Syntheses of differential games and pseudo-Riccati equations

Abstract

For differential games of fixed duration of linear dynamical systems with nonquadratic payoff functionals, it is proved that the value and the optimal strategies as saddle point exist whenever the associated pseudo-Riccati equation has a regular solution $P\left(t,x\right)$. Then the closed-loop optimal strategies are given by $u\left(t\right)=-R^{-1}{B}^{\ast }P\left(t,x\left(t\right)\right),v\left(t\right)=-S^{-1}{C}^{\ast }P\left(t,x\left(t\right)\right)$. For differential game problems of Mayer type, the existence of a regular solution to the pseudo-Riccati equation is proved under certain assumptions and a constructive expression of that solution can be found by solving an algebraic equation with time parameter.