Mean Field Games of Controls with Dirichlet boundary conditions

Abstract

In this paper we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field of control setting, that is in which the dynamics of each agent is affected not only by the average position of the rest of the agents but also by their average optimal choice. This setting allows the modeling of more realistic real-life scenarios in which agents not only will leave the domain at a certain point in time (like during the evacuation of pedestrians or in debt refinancing dynamics) but also act competitively to anticipate the strategies of the other agents. We shall establish the existence of Nash Equilibria for such class of mean-field of controls systems under certain regularity assumptions on the dynamics and the Lagrangian cost. Much of the paper is devoted to establishing several a priori estimates which are needed to circumvent the fact that the mass is not conserved (as we are in a Dirichlet boundary condition setting). In the conclusive sections, we provide examples of systems falling into our framework as well as numerical implementations.

On quasi-stationary Mean Field Games of Controls

“…1) In the first case, we use a continuous dependence estimate for the HJB equation and we assume that the constants that intervene in the regularity of the Hamiltonian with respect to the joint distribution are small. Similar assumptions of smallness also appear in other papers on MFG of Controls (see [2,11]). Note that this smallness assumption does not concern the length of the time interval.…”

“…Since we can bound the latter quantity, using the FP equation, by |t − s| 1 2 , the regularity in time of u follows. For quasi-stationary MFG of Controls, a similar argument fails.…”