Deterministic mean field games with control on the acceleration

Abstract

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton–Jacobi equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x,v)\in {\mathbb {R}}^N\times {\mathbb {R}}^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

Appendix

Let us now consider the second order MFG system: for a positive number $\sigma$,

$$\begin{aligned} \left\{ \begin{array}{lll} (i)\ -\partial _t u-\sigma \Delta _{x,v} u-v\cdot D_xu+\frac{1}{2} \vert D_vu\vert ^2-\frac{1}{2} \vert v \vert ^2-l(x,v) =F[m](x, v),\ &{} \text {in }{\mathbb {R}}^{2N}\times (0,T),\\ (ii)\ \partial _t m-\sigma \Delta _{x,v} m-{{\,\mathrm{div}\,}}_v (m D_v u)-v\cdot D_xm=0, &{} \text {in }{\mathbb {R}}^{2N}\times (0,T),\\ (iii)\ m(x,v, 0)=m_0(x,v),\quad u(x,v,T)=G[m(T)](x, v),&{} \text {on }{\mathbb {R}}^{2N}. \end{array}\right. \nonumber \\ \end{aligned}$$

(6.1)

We aim at proving the existence and uniqueness of a classical solution to system (6.1). We shall see that these results are byproducts of the estimates that we have already used above in the vanishing viscosity limit. More precisely, the properties obtained in Sect. 4 will play a crucial role in what follows.

Theorem 6.1

Under our standing assumptions, there exists a classical solution to problem (6.1). Moreover, if the coupling costs F and G satisfy (2.3), the solution is unique.

Proof

Our arguments are reminiscent of those used in the proof of [12, Theorem 3.1]. We introduce

$$\begin{aligned} {\mathcal {C}}:=\left\{ m\in C^0([0,T]; {\mathcal {P}}_1({\mathbb {R}}^{2N})):\quad m(0)=m_0\right\} \end{aligned}$$

which is a non-empty closed and convex subset of $C^0([0,T]; {\mathcal {P}}_1({\mathbb {R}}^{2N}))$. We define a map ${\mathcal {T}}$ as follows: for any $m\in {\mathcal {C}}$, let u be the unique solution to (6.1)-(i) and $u(x,v,T)=G[m(T)](x, v)$ found in Lemma 4.1; we set ${\mathcal {T}}(m)=\mu$ where $\mu$ is the unique solution to (6.1)-(ii) and $m(x,v, 0)=m_0(x,v)$ found in Lemma 4.2. Lemma 4.3 ensures that ${\mathcal {T}}$ maps ${\mathcal {C}}$ into itself.

By the same arguments as in the proof of Theorem 2.1, the map ${\mathcal {T}}$ is continuous with respect to the norm of $C([0,T]; {\mathcal {P}}_1({\mathbb {R}}^{2N}))$ and it is compact. Hence, Schauder fixed point theorem ensures the existence of a fixed point m for ${\mathcal {T}}$. Let u denote the corresponding solution to (6.1)-(i) and -(iii). By Lemma 4.1 and Lemma 4.2 again, u and m are regular. In conclusion, (u, m) is the desired solution to (6.1).

Let us now prove the uniqueness part of the statement. Let $(u_1,m_1)$ and $(u_2,m_2)$ be two solutions; set ${\overline{u}}=u_1-u_2$. Our aim is to follow the arguments in the proof of Proposition 2.1. To this end, it is enough to prove that ${\overline{u}}$ is an admissible test-function for $m_1$ and $m_2$. Indeed, for any $R>1$, let $\phi _R$ be a cut-off function in ${\mathbb {R}}^{2N}$ defined by $\phi _R(x,v):=\phi _1(x/R,v/R)$ where $\phi _1$ is a $C^2$ function such that $\phi _1=1$ in $B_1$, $\phi _1=0$ outside $B_{2}$. Clearly,

$$\begin{aligned}&D_{x,v}\phi _R=0 \quad \text {outside }\overline{B_{2R}{\setminus } B_R},\quad \Vert D_{x,v}\phi _R\Vert _\infty \le C/R\nonumber \\&\quad \text {and }\Vert \Delta _{x,v}\phi _R\Vert _\infty \le C/R^2. \end{aligned}$$

(6.2)

Using $\phi _R {\overline{u}}$ as test-function in (6.1)-(ii) with $(u,m)=(u_i,m_i)$ for $i=1$ or $i=2$, we get

$$\begin{aligned} \nonumber 0= & {} \iint _{{\mathbb {R}}^{2N}\times [0,T]} m\left[ -\phi _R \partial _t {\overline{u}}-\sigma \Delta _{x,v} (\phi _R {\overline{u}})+ D_v u\cdot D_v(\phi _R {\overline{u}})+v\cdot D_x(\phi _R {\overline{u}})\right] \, dxdvdt\\\nonumber&+\int _{{\mathbb {R}}^{2N}}m(x,v,T)\phi _R \left( G[m_1(T)](x,v)-G[m_2(T)](x,v)\right) dxdv-\int _{{\mathbb {R}}^{2N}}m_0\phi _R {\overline{u}}dxdv\\ \nonumber= & {} \iint _{{\mathbb {R}}^{2N}\times [0,T]} m\phi _R\left( 2 v\cdot D_x {\overline{u}}+\frac{|D_v u_2|^2-|D_v u_1|^2}{2} +F[m_1] -F[m_2]+D_vu\cdot D_v{\overline{u}} \right) \, dxdvdt\\\nonumber&+\int _{{\mathbb {R}}^{2N}}m(x,v,T)\phi _R \left( G[m_1(T)](x,v)-G[m_2(T)](x,v)\right) dxdv-\int _{{\mathbb {R}}^{2N}}m_0\phi _R {\overline{u}}dxdv\\\nonumber&+ \iint _{{\mathbb {R}}^{2N}\times [0,T]} m\left( -\sigma {\overline{u}} \Delta _{x,v} \phi _R -2\sigma D_{x,v}\phi _R\cdot D_{x,v}{\overline{u}}\right) \, dxdvdt\\&+\iint _{{\mathbb {R}}^{2N}\times [0,T]} m\left( {\overline{u}} D_v u\cdot D_v\phi _R +{\overline{u}} v\cdot D_x\phi _R\right) \, dxdvdt \end{aligned}$$

(6.3)

where the second equality is due to equation (6.1)-(i). Since $m>0$ and $m\in L^\infty ((0,T);{\mathcal {P}}_2({\mathbb {R}}^{2N}))$ (see Lemma 4.2 and Lemma 4.3), by the estimates on $u_i$ in Lemma 4.1, the dominated convergence theorem ensures that as $R\rightarrow \infty$, the first two lines in right hand side of (6.3) converge to

$$\begin{aligned}&\iint _{{\mathbb {R}}^{2N}\times [0,T]} m\\ {}&\quad \times \left[ 2 v\cdot D_x {\overline{u}}-\frac{|D_v u_1|^2}{2} +\frac{|D_v u_2|^2}{2} +F[m_1] -F{\mathbb {R}}+D_vu\cdot D_v{\overline{u}}\right] \, dxdvdt\\&+\int _{{\mathbb {R}}^{2N}}m(x,v,T) \left( G[m_1(T)](x,v)-G[m_2(T)](x,v)\right) dxdv-\int _{{\mathbb {R}}^{2N}}m_0 {\overline{u}}dxdv; \end{aligned}$$

hence, it remains to prove that the last two lines in the right hand side of (6.3) converge to 0. Indeed, again by Lemmas 4.1, 4.2 and 4.3, and by our estimates (6.2), the dominated convergence theorem yields

$$\begin{aligned} \iint _{{\mathbb {R}}^{2N}\times [0,T]} m\left( -\sigma {\overline{u}} \Delta _{x,v} \phi _R -2\sigma D_{x,v}\phi _R\cdot D_{x,v}{\overline{u}}\right) \, dxdvdt\rightarrow 0. \end{aligned}$$

Let us now address to the last integral in the right hand side of (6.3): the properties in (6.2) entail

$$\begin{aligned} \left| m\left( {\overline{u}} D_v u\cdot D_v\phi _R +{\overline{u}} v\cdot D_x\phi _R\right) \right| \le C m(1+|v|^2) \chi _R \end{aligned}$$

where $\chi _R$ is the characteristic function of $B_{2R}{\setminus } B_R$. Moreover, since $m\in L^\infty ((0,T);{\mathcal {P}}_2({\mathbb {R}}^{2N}))$, the right hand side in the last inequality belongs to $L^1$ independently of R. Therefore, again by the dominated convergence theorem, we get that as $R\rightarrow \infty$ the last integral in (6.3) converges to 0. $\square$