Deterministic mean field games with control on the acceleration

Abstract

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Convergence of some Mean Field Games systems to aggregation and flocking models

“…Existence of a solution to the equilibrium condition (30) for more general MFG systems is obtained in [12], however under a much more restrictive growth condition on F . In addition, [1,12] show that there exists a weak solution to the MFG system of acceleration (24). We postpone the (quite classical) proof of Lemma 2.1 to the next section and proceed with the notion of solution for the kinetic equation (27).…”

“…In [8], the authors study MFG systems of the form $ & %´B t u λ´ν ∆u λ`H px, Du λ , m λ ptqq`λu " 0 in R dˆp 0, T q, B t m λ´ν ∆m λ´d ivpm λ D p Hpx, Du λ , m λ ptqq " 0 in R dˆp 0, T q, u λ pT, xq " u T pxq, m λ p0q " m 0 in R d . (1) Here ν ą 0 is fixed, λ ą 0 is a large parameter which describes the impatience of the players and H " Hpx, p, mq is the Hamiltonian of the problem which includes interaction terms between the players. Under suitable assumptions on the data, [8] states that, as λ tends to 8 and up to subsequences, u λ , Du λ Ñ 0 and m λ converges to a solution of the McKean-Vlasov equation…”

“…For the second result (Theorem 2.2, on the convergence of (4) to (5)), the fact that the coupling function F growths in a quadratic way with respect to the (moment of) the measure prevents us from using fixed point techniques (as in [1,12]) to show the existence of a solution to the MFG system (4) and to obtain estimates on the solution (this would also be the case in the presence of a viscous term). This obliges us to give up the PDE approach of the previous set-up and to use variational techniques, first suggested for MFG problems in [24] and largely developed since then: see, for instance, among many other contributions, [5,14,15,17,26].…”

A Quadratic Mean Field Games Model for the Langevin Equation

“…In the original model introduced in [1,2], an agent can typically act on its velocity (or other first order dynamical quantities) via a control variable. Mean Field Games where agents control the acceleration have been recently proposed in [3][4][5].…”

“…where A t is the the set of the control laws. Formally, the couple (u, m) satisfies the MFG system (see Section 4.1 in [3] for more details)…”