Convergence of some control problems and Mean Field Games to aggregation and swarming models

Mer, 27/10/2021 - 10:30 / 11:30

104, Viale Romania

Speaker: Martino Bardi , Unipd

Abstract

We consider first  a control problem with cheap controls and large discount factor in the cost functional, modelling a greedy agent who cares only about a very short time-horizon.
We show that in the limit the optimal feedback, suitably rescaled, converges to the gradient of the running cost, so that the limit dynamics follows the steepest descent of the running cost
Next we consider the same asymptotic problem for two classes of Mean Field Games. The first is associated to a stochastic control problem with vanishing noise and control acting on the velocity of the agents, whereas the second is a deterministic MFG with control on the acceleration and a cost related to the Cucker-Smale model of flocking and swarming in animal populations.
For both problems we show that in the limit the distribution of the agents solves a nonlocal nonlinear continuity equation. For the first MFG we also show that the (rescaled) feedback producing the mean field equilibrium converges to the gradient of the running cost, so that the limit dynamics follows the steepest descent of the running cost associated to the limit mass distribution.
In conclusion, our results establish a rigorous connection among Mean-Field Games and various agent-based models such as the Aggregation Equation, some models of crowd dynamics, and the kinetic equations associated to flocking phenomena.
This is joint work with Pierre Cardaliaguet.