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 $P(t,x)$ . Then the closed-loop optimal
strategies are given by $u(t)= -R^{-1}B^\ast P(t,x(t)),v(t)= -S^{-1}C^\ast P(t,x(t))$ . 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.