Representation Results and Error Estimates for Differential Games with Applications Using Neural Networks

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 network 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 \(O({\Delta t}^{1/2})\) $O({\mathrm{\Delta }t}^{1/2})$ to \(O({\Delta t})\) $O(\mathrm{\Delta }t)$ , 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 the presence of state constraints, using stochastic gradient-type algorithms to deal with the minimax problem.