Mean Field Control with Absorption

In this paper, we study a mean field control problem in which particles are absorbed when they reach the boundary of a smooth domain. The value of the N-particle problem is described by a hierarchy of Hamilton-Jacobi equations which are coupled through their boundary conditions. The value function of the limiting problem, meanwhile, solves a Hamilton-Jacobi equation set on the space of subprobability measures on the smooth domain—that is, the space of nonnegative measures with total mass of at most one. Our main contributions are (i) to establish a comparison principle for this novel infinite-dimensional Hamilton-Jacobi equation and (ii) to prove that the value of the N-particle problem converges in a suitable sense toward the value of the limiting problem as N tends to infinity.

Funding: J. Jackson received financial support from the National Science Foundation (NSF) [Grant DMS-2302703]. P. Cardaliaguet was partially supported by P.S.’s Air Force Office for Scientific Research [Grant FA9550-18-1-0494] and by the Agence Nationale de la Recherche [Project ANR-22-CE40-0010 COSS]. P. E. Souganidis was partially supported by the NSF [Grants DMS-2452972 and DMS-2153822], the Office for Naval Research [Grant N000141712095], and the Air Force Office for Scientific Research [Grant FA9550-18-1-0494].