  We study deterministic optimal control problems for differential games with
finite horizon. We propose new approximations of the strategies in feedback
form, and show error estimates and a convergence result of the value in some
weak sense for one of the formulations. This result applies in particular to
neural networks approximations. This work follows some ideas introduced in
Bokanowski, Prost and Warin (PDEA, 2023) for deterministic optimal control
problems, yet with a simplified approach for the error estimates, which allows
to consider a global optimization scheme instead of a time-marching scheme. We
also give a new approximation result between the continuous and the
semi-discrete optimal control value in the game setting, improving the
classical convergence order under some assumptions on the dynamical system.
Numerical examples are performed on elementary academic problems related to
backward reachability, with exact analytic solutions given, as well as a
two-player game in presence of state constraints. We use stochastic gradient
type algorithms in order to deal with the min-max problem.
