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The mean field type control theory studies systems of many identical agents interacting via an external media and acting cooperatively in the limiting case when the number of agents tends to infinity. Thus, one can regard a mean filed type control problem as a control problem in the space of probability measures. I would like to discuss the finite-dimensional approximation of the first-order mean field type control problem. In this case the dynamics of each agent is given by the ordinary differential equation. The most natural way here is to consider the finite-particle approximation. However, it does not give an approximation rate. I will talk about the lattice approximation that implies the replacement of the ODE determining the dynamics of each agent by the Markov chain acting on some finite lattice. This approach leads to the mean filed type Markov decision problem. Since the dynamics of distribution of agents in the mean field Markov chain is described by the nonlinear ODE, we obtain the finite-dimensional approximation of the original first-order mean field type control problem. Moreover, the approximation rate can be evaluated using the distance between the ODE providing dynamics in the original mean field type control problem and the Markov chain.

Bio: In 2004, I was graduated from the Ural State University. Three years later, I received my PhD from the Institute of Mathematics and Mechanics (Yekaterinburg, Russia). In 2020, I was awarded by the habitation degree (Dr.Sc) in Higher School of Economics. Nowadays, I am a leading researcher in the Krasovskii Institute of Mathematics and Mechanics. Also I hold a part-time position in the Higher School of Economics (Moscow, Russia). My current research interests include mean field games, mean field type control systems and approximation of control problems.