Approximation of a Mean Field Game Problem with Caputo Time-Fractional Derivative

A mean field game model in the interpretation of optimal control is investigated theoretically and numerically. This is the problem of minimizing a non-convex and non-coercive objective functional controlled by a coefficient in the Fokker–Planck equation with a time-fractional derivative. The existence of a weak solution to the problem posed is proved under an additional constraint on the smallness of the control function. The differential problem is approximated by the finite difference method, while the state equation is approximated by an efficiently implemented locally one-dimensional scheme. The existence of a solution to the finite-difference optimal control problem without additional constraints on the control function and discretization steps is proved. Stability estimates for the discrete state and adjoint state equations are given. The results of numerical experiments are presented for an applied problem. They demonstrate, among other things, the sensitivity of the solution to the order of the time-fractional derivative.