Limit Hamilton-Jacobi-Isaacs equations for singularly perturbed zero-sum differential games
In the paper a singularly perturbed (SP) differential game with a fixed end time \(\vartheta\) and a payoff functional of the Bolza type is considered. The SP dynamics consists of slow and fast subsystems. The corresponding Hamilton-Jacobi-Isaacs (HJI) equations for upper and lower value functions of the SP game have terms with multipliers \(1/\varepsilon\) where \(\varepsilon\) is a small parameter. The author applies the method of decomposition to study the convergence in the SP game as \(\varepsilon\to 0\). According to this method, the author deals with an associated fast (AF) game for the fast SP subsystem where slow phase variables are fixed. The AF game is considered on the time interval \([t_0,\vartheta/\varepsilon]\), its payoff is defined as the averaged SP Hamiltonian along the fast trajectory. Under assumptions stated in the paper, letting \(\varepsilon\to 0\), one can obtain the limit Hamiltonian in the subspace of the slow phase variables. A viscosity solution of the corresponding limit HJI equation is an asymptotic for the value function of the SP game. A special case and two examples are considered to illustrate the general results.
scientific article; zbMATH DE number 934215