Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

Abstract

We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.

Second order mean field games with degenerate diffusion and local coupling

“…where q = r d+η d (14) for any v ∈ L r ((0, T ); W 1,r (T d )) such that T d v(t) dx = 0 a.e. in (0, T ).…”

“…Proof of Proposition 4.2. We follow the argument developed by Graber in [14]. Inequality inf For any (m, w) ∈ K 1 with mH * (·, − w m ) ∈ L 1 , we have, by Lemma 4.3,…”

First Order Mean Field Games with Density Constraints: Pressure Equals Price

“…Note that F * (x, α) ≥ αm − F (x, m) (2.4) and F * (·, α) = 0 for all α ≤ 0. Following the approach of Cardaliaguet-Carlier-Nazaret [12] (see also Cardaliaguet [11], Graber [20] or Cardaliaguet-Graber [13]) it seems that the solution to (1.2) can be obtained as the system of optimality conditions for optimal control problems.…”

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

“…Often this is exploited to obtain results on existence and uniqueness of solutions to mean field games. See, for instance, [15,30,17,18,7]. Here we point out a condition under which the Mean Field Type Control Problem and the Mean Field Game are equivalent for the general linear-quadratic case.…”