ИСТИНА

We consider a problem described by Mean Field Games (MFG) and Optimal Control theory on finite time horizon. This problem consists of a system of PDEs: a Kolmogorov--Fokker--Planck equation, evolving forward in time and a Hamilton--Jacobi--Bellman equation, evolving backwards in time. The difficulties related to the numerical solution come from a turnpike effect. We present a regularization method of this system of PDEs in a form of the extremal problem and introduce its numerical solution at the heart of monotonic schemes. According to special assumptions, PDEs can be reduced to Riccati ODEs. We consider this reduction as a test example for the numerical solution of the extremal problem.