Approximation of Deterministic Mean Field Games with Control-Affine Dynamics

Abstract

We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean field games with control on the acceleration (see Achdou et al. in NoDEA Nonlinear Differ Equ Appl 27(3):33, 2020; Cannarsa and Mendico in Minimax Theory Appl 5(2):221-250, 2020; Cardaliaguet and Mendico in Nonlinear Anal 203: 112185, 2021). We focus our attention on the approximation of such mean field games by analogous problems in discrete time and finite state space which fall in the framework of (Gomes et al. in J Math Pures Appl (9) 93(3):308-328, 2010). For these approximations, we show the existence and, under an additional monotonicity assumption, uniqueness of solutions. In our main result, we establish the convergence of equilibria of the discrete mean field games problems towards equilibria of the continuous one. Finally, we provide some numerical results for two MFG problems. In the first one, the dynamics of a typical player is nonlinear with respect to the state and, in the second one, a typical player controls its acceleration.

Appendices

Appendix I

In this section we prove some technical properties of the semi-discrete value function $v_{k}:\mathbb {R}^d\rightarrow \mathbb {R}$ ($k\in \mathcal {I}$) introduced in Sect. 3.1. In what follows we assume that (H1), (H2) and (H3) are in force.

We start showing the existence of optimal controls for the problem defined by $v_k$ in (3.7).

Lemma 6.2

Given $k \in \mathcal {I}^{*}$ and $x\in \mathbb {R}^d$, there exists ${\bar{\alpha }}\in \mathcal {A}_{k}$ such that $v_k( x)= J_{k, x}({\bar{\alpha }})$. In addition, there exists $\tilde{C}>0$, independent of $\Delta t$, m, k, and x, such that any optimal solution ${{\tilde{\alpha }}}\in \mathcal {A}_k$ satisfies

$$\begin{aligned} \Delta t \sum _{j=k}^{N_t-1}|{{\tilde{\alpha }}}_j|^p\le \tilde{C}. \end{aligned}$$

(6.7)

Proof

Given $k \in \mathcal {I}^{*}$ and $x\in \mathbb {R}^d$, let $(\alpha ^n)_{n\in \mathbb {N}}\subset \mathcal {A}_k$ be a minimizing sequence for $J_{k,x}$. Let $(\gamma ^n)_{n\in \mathbb {N}}\subset \Gamma _{k,x}$ be the sequence of states associated with $(\alpha ^n)_{n\in \mathbb {N}}$ via (3.6) and let ${\bar{\gamma }}^0\in \Gamma _{k,x}$ be the state associated with the null control. By definition of minimizing sequence, (H1)(i), and (H2), for any $\delta >0$, there exists $n_0\in \mathbb {N}$ such that

$$\begin{aligned} \Delta t \underline{\ell }\sum _{j=k}^{N_t-1}\left| \alpha ^n_j\right| ^p-(T-t_k)C_\ell\le & {} J_{k,x}(0)+\delta -g(\gamma ^n_{N_t})\nonumber \\\le & {} (T-t_k)C_\ell +\delta +L_g\left| {\bar{\gamma }}^0_{N_t} - \gamma ^n_{N_t}\right| , \end{aligned}$$

(6.8)

for all $n\ge n_0$. Let us estimate $\left| {\bar{\gamma }}^0_{N_t} - \gamma ^n_{N_t}\right|$. For every $j=k, \dots , N_t-1$ we have

$$\begin{aligned} \begin{array}{lll} \left| {\bar{\gamma }}^0_{j+1} - \gamma ^n_{j+1}\right| &{}\le &{} \left| {\bar{\gamma }}^0_{j} - \gamma ^n_{j}\right| +\Delta t \left| A(t_j,{\bar{\gamma }}^0_{j}) -A(t_j, \gamma ^n_{j})\right| + \Delta t\left| B(t_j, \gamma ^n_{j})\alpha ^n_j\right| \\ \;&{}\le &{}\left( 1+\Delta t L_A\right) \left| {\bar{\gamma }}^0_{j} - \gamma ^n_{j}\right| +\Delta t C_B\left| \alpha ^n_j\right| . \end{array} \end{aligned}$$

Since ${\bar{\gamma }}^0_{k} =\gamma ^n_{k}$, by the discrete Grönwall’s lemma, there exists $C>0$ such that

$$\begin{aligned} \max _{j=k,\dots , N_t}\left| {\bar{\gamma }}^0_{j} - \gamma ^n_{j}\right| \le C \Delta t\sum _{j=k}^{N_t-1}\left| \alpha ^n_j\right| . \end{aligned}$$

(6.9)

Thus, by Young’s inequality, for every $\eta >0$ there exists $C_\eta >0$ such that

$$\begin{aligned} L_gC \Delta t\sum _{j=k}^{N_t-1}\left| \alpha ^n_j\right| \le C_\eta +\eta \Delta t\sum _{j=k}^{N_t-1}\left| \alpha ^n_j\right| ^p. \end{aligned}$$

Taking $\eta <\underline{\ell }$ and combining the above equation with (6.8) and (6.9), we deduce the existence of $\tilde{C}>0$, independent of $\Delta t$, m, k, and x, such that

$$\begin{aligned} \Delta t \sum _{j=k}^{N_t-1}|\alpha _j^n|^p\le \tilde{C}. \end{aligned}$$

Therefore, there exists at least one accumulation point ${\bar{\alpha }}\in \mathcal {A}_k$ of $(\alpha ^n)_{n\in \mathbb {N}}$ and, by the continuity assumptions in (H1)–(H3), we conclude that $v_k(x)= J_{k, x}({\bar{\alpha }})$. Finally, if ${\tilde{\alpha }}$ is any other optimal control for the problem defined by $v_k(x)$, the previous argument shows that (6.7) holds. $\square$

Proof of the Lipschitz property (3.9)

Given $k \in \mathcal {I}^{*}$ and $x\in \mathbb {R}^d$, let ${\bar{\alpha }}\in \mathcal {A}_k$ be an optimal control for $v_k(x)$ and let ${\bar{\gamma }}\in \Gamma _{k,x}$ be the associated state given by (3.6). Let $y\in \mathbb {R}^d$ and let $\zeta \in \Gamma _{k,y}$ be the state associated with ${\bar{\alpha }}$ via (3.6), then (H1)(ii) implies

$$\begin{aligned} \begin{array}{lll} v_k(y)-v_k(x)&{}\le &{}\displaystyle \Delta t\sum _{j=k}^{N_t-1}[\ell (t_j, {\bar{\alpha }}_j, \zeta _j, m(t_j))-\ell (t_j, {\bar{\alpha }}_j, {\bar{\gamma }}_j, m(t_j))]\\ \;&{}\;&{}+g(\zeta _{N_t},m(T))-g({\bar{\gamma }}_{N_t},m(T))\\ \;&{}\le &{}\displaystyle \left( L_\ell \left( T+\Delta t\sum _{j=k}^{N_t}|{\bar{\alpha }}_j|^p\right) +L_g\right) \max \limits _{j=k,\dots , N_t}|\zeta _j-{\bar{\gamma }}_j|. \end{array} \end{aligned}$$

(6.10)

On the other hand, for every $j=k,\dots , N_t-1$,

$$\begin{aligned} |\zeta _{j+1}-{\bar{\gamma }}_{j+1}|\le \left( 1+\Delta t \left[ L_{A} + L_B |\overline{\alpha }_j|\right] \right) |\zeta _{j}-{\bar{\gamma }}_{j}|. \end{aligned}$$

By the discrete Grönwall’s lemma, we have

$$\begin{aligned} \max _{j=k,\dots , N_t}|\zeta _j-{\bar{\gamma }}_j|\le \displaystyle e^{\left( L_{A}T+ L_{B}\Delta t \sum \limits _{j=k}^{N_{t}-1}|\bar{\alpha }_j|\right) }|x-y|. \end{aligned}$$

By (6.7), Hölder’s inequality, and (6.10), there exists $L_{v}>0$, independent of x, y, such that $v_k(x)-v_k(y)\le L_v |x-y|$ for all x, $y\in \mathbb {R}^d$, which implies (3.9). $\square$

Lemma 6.3

Given $k \in \mathcal {I}^*$ and $x\in \mathbb {R}^d$, there exists $\alpha _{k,x}\in \mathbb {R}^r$ such that

$$\begin{aligned} v_k(x)\!=\!\Delta t \ell (t_k, \alpha _{k,x}, x,m(t_k))\!+\!v_{k+1}\left( x\!+\!\Delta t[A(t_k, x)\!+\!B(t_k, x)\alpha _{k,x}]\right) \!. \end{aligned}$$

(6.11)

Moreover, there exists $\widehat{C}>0$, independent of $\Delta t$, m, k, and x, such that

$$\begin{aligned} |\alpha _{k,x}|\le \widehat{C}. \end{aligned}$$

(6.12)

Proof

Let $\bar{\alpha }$ be as in Lemma 6.2. Then, by the dynamic programming principle, $\alpha _{k,x}=\bar{\alpha }_k$ satisfies (6.11) and hence

$$\begin{aligned} \begin{array}{l} \Delta t\ell (t_k, \alpha _{k,x}, x, m(t_k))+v_{k+1}\left( x+\Delta t\left[ A(t_k, x)+B(t_k, x)\alpha _{k,x} \right] \right) \\ \quad \le \Delta t\ell (t_k, 0, x, m(t_k))+v_{k+1}\left( x+\Delta tA(t_k, x) \right) . \end{array} \end{aligned}$$

Thus, by (3.9), (H1)(i), and (H3)(ii), we have

$$\begin{aligned} \underline{\ell }|\alpha _{k,x}|^p \le 2C_\ell + L_v C_ B|\alpha _{k,x}| \end{aligned}$$

and the existence of $\widehat{C}>0$ such that (6.12) holds follows from Young’s inequality. $\square$

Proof of the Proposition 3.1

Notice that (H1)(i), with $a=0$, and (H2) imply that

$$\begin{aligned} -C_\ell T+c_{g}\le v^n_{k}(x)\le C_{\ell }T+g({\bar{\gamma }}^{0,n}_{N_t^n},m^n(T))\quad \text {for all }n\in \mathbb {N},\,k\in \mathcal {I}^n,\, x\in \mathbb {R}^d,\nonumber \\ \end{aligned}$$

(6.13)

where ${\bar{\gamma }}^{0,n}$ is defined by (3.6) with $\alpha _j=0$ for all $j=k, \ldots , N_t^n-1$ and $\bar{\gamma }^{0,n}_k=x$. By the definition of ${\bar{\gamma }}^{0,n}$, Remark 2.1(i), and the discrete Grönwall’s lemma, there exists $C>0$, independent of n, k, and x, such that

$$\begin{aligned} |{\bar{\gamma }}^{0,n}_j| \le C(1+ |x|)\quad \text {for all }n\in \mathbb {N},\,k\in \mathcal {I}^n,\,j=k,\ldots ,N^n_t,\;x\in \mathbb {R}^d, \end{aligned}$$

(6.14)

which, together with (6.13), implies that $v^n$ is locally bounded, uniformly with respect to n. Therefore, we can define $v^*:[0,T]\times \mathbb {R}^d \rightarrow \mathbb {R}$ and $v_{*}:[0,T]\times \mathbb {R}^d \rightarrow \mathbb {R}$ by

$$\begin{aligned} v^*(t,x)=\limsup _{{\tiny {\begin{array}{c}n\rightarrow \infty \\ t^n_{k(n)}\rightarrow t, \, k(n)\in \mathcal {I}^n\\ x^n\in \mathbb {R}^d\rightarrow x\end{array}}}}v^n_{k(n)}(x^n)\quad \text{ and }\quad v_*(t,x)=\liminf _{{\tiny {\begin{array}{c}n\rightarrow \infty \\ t^n_{k(n)}\rightarrow t, k(n)\in \mathcal {I}^n\\ x^n\in \mathbb {R}^d\rightarrow x\end{array}}}}v^n_{k(n)}(x^n), \end{aligned}$$

for all $(t,x)\in [0,T]\times \mathbb {R}^d$. By [7, Chapter V, Lemma 1.5] we have that $v^*$ and $v_*$ are upper and lower semicontinuous, respectively. We claim that

(a) $v^*(T,x)=v_*(T,x)=g(x,m(T))$ for all $x\in \mathbb {R}^d$.

(b) $v^*$ satisfies, in the viscosity sense (see e.g. [7, Chapter II]),

$$\begin{aligned} -\partial _t v^*(t,x)+ H(t,x,\nabla _xv^*(t,x),m(t))\le 0\quad \text {for all }(t,x)\in (0,T)\times \mathbb {R}^d.\qquad \end{aligned}$$

(6.15)

(c) $v_*$ satisfies, in the viscosity sense,

$$\begin{aligned} -\partial _t v_*(t,x)+ H(t,x,\nabla _{x} v_*(t,x),m(t))\ge 0\quad \text {for all }(t,x)\in (0,T)\times \mathbb {R}^d. \end{aligned}$$

If (a), (b), and (c) hold, then by the comparison principle [27, Theorem 2.1] we obtain that $v^*=v_*=v$ and the result follows from [7, Chapter V, Lemma 1.9]. It remains to prove the claim. The proofs of (b) and (c) being analogous, we only show (a) and (b).

Proof of (a). Let $x\in \mathbb {R}^d$ and $(k(n), x^n)\in \mathcal {I}^n\times \mathbb {R}^d$ be such that $(t^n_{k(n)}, x^n)\rightarrow (T,x)$ as $n\rightarrow \infty$. Denote by ${\widehat{\mathcal {A}}}_{k}^n$ the set defined in (3.5) for $k=0,\ldots ,N^n_t-1$. By Lemma 6.2 and Lemma 6.3, there exists $\alpha ^n\in {\widehat{\mathcal {A}}}_{k(n)}^n$ such that

$$\begin{aligned} v^n_{k(n)}(x^n)=\Delta t_n \sum _{j=k(n)}^{N^n_t-1}\ell (t_j^n, \alpha ^n_j, \gamma ^n_j, m^n(t_j^n))+g\left( \gamma ^n_{N^n_t}, m^n(T)\right) , \end{aligned}$$

where $\gamma ^n$ is the state associated with $\alpha ^n$ via (3.6) and $\gamma ^n_{k(n)}=x^n$. By (H1)(i) we obtain

$$\begin{aligned}{} & {} -C_{\ell } (T-t_{k(n)}^n)+g\left( \gamma ^n_{N^n_t},m^n(T)\right) \le v^n_{k(n)}(x^n)\nonumber \\{} & {} \qquad \le \left( \overline{\ell }\widehat{C}^p+C_\ell \right) (T-t_{k(n)}^n)+g\left( \gamma ^n_{N^n_t},m^n(T)\right) . \end{aligned}$$

(6.16)

By (H3) and Lemma 6.3 we have

$$\begin{aligned} \begin{array}{lll} \left| \gamma ^n_{N^n_t}-x^n \right| &{}=&{}\displaystyle \Delta t_n\sum _{j=k(n)}^{N^n_t-1}\left[ A(t_j^n, \gamma ^n_j)+B(t_j^n, \gamma ^n_j)\alpha ^n_j \right] \\ \;&{}\le &{}(T-t_{k(n)}^n)\left[ C_A \left( 1+ \max _{j}|\gamma ^n_j|\right) +C_B\widehat{C} \right] . \end{array} \end{aligned}$$

(6.17)

On the other hand, arguing as in the proof of (6.14), there exists $C>0$, independent of n, such that

$$\begin{aligned} \max _{j}|\gamma ^n_j|\le C(1+|x^n|), \end{aligned}$$

which, together with (6.17) and the boundeness of $(x^{n})_{n\in \mathbb {N}}$, yields

$$\begin{aligned} \lim _{n\rightarrow \infty }\left| \gamma ^n_{N^n_t}-x^n\right| =0 \end{aligned}$$

and hence

$$\begin{aligned} \lim _{n\rightarrow \infty }g\left( \gamma ^n_{N^n_t}, m^n(T)\right) = g(x,m(T)). \end{aligned}$$

The result follows from the previous equation and (6.16).

Proof of (b). To check that (6.15) holds in the viscosity sense, following the lines of [38, Proposition 4.3], let $\phi \in C^1\left( [0,T]\times \mathbb {R}^d\right)$ and $(t^*, x^*)\in (0,T)\times \mathbb {R}^d$ be such that $v^*-\phi$ has a local maximum in $(t^*, x^*)$. Modifying $\phi$, if necessary, we can assume that $(t^*, x^*)$ is a strict maximum for $v^*-\phi$ on $\overline{\textrm{B}}((t^*, x^*), \delta )$, for some $\delta >0$. Therefore, by [7, Chapter V, Lemma 1.6] there exists a sequence $((k(n), x^n))_{n\in \mathbb {N}} \subset \mathcal {I}^{n,*}\times \mathbb {R}^d$ such that $(t^n_{k(n)}, x^n)\rightarrow (t^*, x^*)$, $v^n_{k(n)}(x^n)\rightarrow v^*(t^*, x^*)$, and

$$\begin{aligned}{} & {} v^{n}_{k}(y)-\phi (t_k^n,y) \le v^{n}_{k(n)}(x^n)-\phi (t_{k(n)}^n,x^n) \nonumber \\{} & {} \qquad \quad \text {for } (k,y)\in \mathcal {I}^n\times \mathbb {R}^d, \; |t_{k}^n-t_{k(n)}^n|\le \delta , \; |y-x^n|\le \delta . \end{aligned}$$

(6.18)

By (6.18), (6.13), (H2), (6.14), and modifying $\phi$ outside $\overline{\textrm{B}}((t^*, x^*), \delta )$, if necessary, we can assume that $\phi \in C^1\left( [0,T]\times \mathbb {R}^d\right)$, $\partial _t \phi$ and $\nabla \phi$ are bounded, and $(k(n), x^n)$ is a global maximum of $\mathcal {I}^n \times \mathbb {R}^d \ni (k,x) \mapsto v^n_{k}(x)-\phi (x)\in \mathbb {R}$. In particular, for all $y\in \mathbb {R}^d$, we have

$$\begin{aligned} v^n_{k(n)+1}\left( y\right) -v^n_{k(n)}\left( x^n\right) \le \phi \left( t^n_{k(n)+1}, y\right) -\phi \left( t^n_{k(n)}, x^n\right) . \end{aligned}$$

Using (3.8), we obtain

$$\begin{aligned} \begin{array}{lll} 0&{}=&{}\min \limits _{\alpha \in \mathbb {R}^r}\Big \{\Delta t_n \ell \left( t^n_{k(n)}, \alpha , x^n, m^n(t^n_{k(n)})\right) \nonumber \\ {} &{}&{}+v^n_{k(n)+1}\left( x^n+\Delta t_n[A(t^n_{k(n)}, x^n)+B(t^n_{k(n)}, x^n)\alpha ]\right) \Big \} \\ &{}&{} -\, v^n_{k(n)}\left( x^n\right) \\ {} &{}\le &{} \inf \limits _{\alpha \in \mathbb {R}^r}\Big \{\Delta t_n \ell \left( t^n_{k(n)}, \alpha , x^n, m^n(t^n_{k(n)})\right) \nonumber \\ {} &{}&{}+\,\phi \left( t^n_{k(n)+1}, x^n+\Delta t_n[A(t^n_{k(n)}, x^n)+B(t^n_{k(n)}, x^n)\alpha ]\right) \Big \} \\ &{}&{} -\,\phi \left( t^n_{k(n)}, x^n\right) . \end{array} \end{aligned}$$

Since $\nabla \phi$ is bounded, (H1)(i) implies that the last infimum above is reached at some ${\bar{\alpha }}^n\in \mathbb {R}^r$ and there exists $C_{\phi }>0$, independent of n, such that $|{\bar{\alpha }}^n|\le C_\phi$. Therefore, for any $\alpha \in \mathbb {R}^r$ we have

$$\begin{aligned} \begin{array}{lll} 0&{}\le &{} \ell \left( t^n_{k(n)}, {\bar{\alpha }}^n, x^n, m^n(t^n_{k(n)})\right) +\frac{\phi \left( t^n_{k(n)+1}, x^n+\Delta t_n[A(t^n_{k(n)}, x^n)+B(t^n_{k(n)}, x^n){\bar{\alpha }}^n]\right) -\phi \left( t^n_{k(n)}, x^n\right) }{\Delta t_n}\\ \;&{}\le &{} \ell \left( t^n_{k(n)}, \alpha , x^n, m^n(t^n_{k(n)})\right) +\frac{\phi \left( t^n_{k(n)+1}, x^n+\Delta t_n[A(t^n_{k(n)}, x^n)+B(t^n_{k(n)}, x^n)\alpha ]\right) -\phi \left( t^n_{k(n)}, x^n\right) }{\Delta t_n}. \end{array} \end{aligned}$$

Since the sequence $({\bar{\alpha }}^n)_{n\in \mathbb {N}}$ is bounded by $C_\phi$, there exists a subsequence, still denoted by $({\bar{\alpha }}^n)_{n\in \mathbb {N}}$, and $\alpha ^*\in \overline{\textrm{B}}(0, C_\phi )$ such that ${\bar{\alpha }}^n\rightarrow \alpha ^*$ as $n\rightarrow \infty$. Passing to the limit in the previous inequality we obtain

$$\begin{aligned} 0\le & {} \ell \left( t^*, \alpha ^*, x^*, m(t^*)\right) +\partial _t\phi (t^*, x^*)+ \langle \nabla _{x}\phi (t^*, x^*), A(t^*, x^*)+B(t^*, x^*)\alpha ^*\rangle \nonumber \\ \;\le & {} \ell \left( t^*, \alpha , x^*, m(t^*)\right) +\partial _t\phi (t^*, x^*)+\langle \nabla _{x}\phi (t^*, x^*), A(t^*, x^*)+B(t^*, x^*)\alpha \rangle \nonumber \\{} & {} \quad \text {for all } \alpha \in \mathbb {R}^r, \end{aligned}$$

(6.19)

from which we deduce that

$$\begin{aligned} H\left( t^*, x^*, \nabla _{x}\phi (t^*, x^*), m(t^*)\right)= & {} - \ell \left( t^*, \alpha ^*, x^*, m(t^*)\right) -\nabla _{x}\phi (t^*, x^*)\nonumber \\{} & {} \cdot \left[ A(t^*, x^*)+B(t^*, x^*)\alpha ^*\right] . \end{aligned}$$

Finally, by (6.19), we obtain

$$\begin{aligned} -\partial _t\phi (t^*, x^*)+H\left( t^*, x^*, \nabla _{x}\phi (t^*, x^*), m(t^*)\right) \le 0, \end{aligned}$$

which proves assertion (b). $\square$

Appendix II

Proof of Theorem 2.2

As in Sect. 3.2 we assume that B and A are decomposed as in (3.15) and (3.16), respectively, with $B_1(t,x)\in \mathbb {R}^{r\times r}$ being invertible for all $(t,x)\in [0,T]\times \mathbb {R}^d$. Let $\xi ^{1}$ and $\xi ^{2}$ be two solutions to Problem 2.1 and, for $i=1,2$, define $m^i\in C([0,T];\mathcal {P}_1(\mathbb {R}^d))$ as $m^{i}(t)=e_{t}\sharp \xi ^{i}$ for all $t\in [0,T]$. Given $x\in \mathbb {R}^d$ and $i=1,2$, let us set

$$\begin{aligned} \text {Opt}^{i}(x)=\left\{ \gamma \in W^{1,p}([0,T];\mathbb {R}^d) \, \big | \, \exists \, \alpha \in L^{p}([0,T];\mathbb {R}^r), \; (\gamma ,\alpha ) \; \text {solves}\, ({OC_{x,m^i}}) \right\} . \end{aligned}$$

Observe that, by Remark 2.1(iii), if $\gamma \in \text {Opt}^{i}(x)$, then there exists a unique control $\alpha [\gamma ]\in L^{p}([0,T];\mathbb {R}^r)$, given by (5.24), such that $(\gamma ,\alpha [\gamma ])$ solves $({OC_{x,m^i}})$. Let $\mathbb {R}^d\ni x\mapsto \gamma ^{i,x} \in \Gamma$ be a Borel measurable selection of the set-valued map $\text {Opt}^{i}$. The existence of such a selection follows from exactly the same arguments than those in the proof of [31, Lemma 2.1]. Since $\xi ^i$ solves Problem 2.1, we have $\textrm{supp}(\xi ^i)\subseteq \cup _{x\in \textrm{supp}(m_0)}\text {Opt}^{i}(x)$ and, hence, condition (2.4) implies that $\xi ^i=\gamma ^i\sharp m_0$.

Let us define $J^i:W^{1,p}([0,T];\mathbb {R}^d)\rightarrow \mathbb {R}$ as

$$\begin{aligned} J^i(\gamma )= & {} \int _{0}^{T}\ell (t,\alpha [\gamma ](t), \gamma (t), m^i(t))\textrm{d}t + g(\gamma (T),m^i(T))\\{} & {} \quad \text {for all } \gamma \in W^{1,p}([0,T];\mathbb {R}^d) \end{aligned}$$

and notice that, by (H5), we have

$$\begin{aligned} \int _{\Gamma } \left( J^1(\gamma )-J^2(\gamma ) \right) \textrm{d}(\xi ^1-\xi ^2)(\gamma ) \ge 0. \end{aligned}$$

(7.1)

If $\xi ^1\ne \xi ^2$, then (2.4) implies that

$$\begin{aligned} \int _{\mathbb {R}^d}J^1(\gamma ^{1,x})\textrm{d}m_0(x)<\int _{\mathbb {R}^d}J^1(\gamma ^{2,x})\textrm{d}m_0(x). \end{aligned}$$

Using that $\xi ^i=\gamma ^i\sharp m_0$, the previous condition can be rewritten as

$$\begin{aligned} \int _{\Gamma }J^1(\gamma )\textrm{d}\xi ^1(\gamma )<\int _{\Gamma }J^1(\gamma )\textrm{d}\xi ^2(\gamma ). \end{aligned}$$

(7.2)

Analogously, we obtain

$$\begin{aligned} \int _{\Gamma }J^2(\gamma )\textrm{d}\xi ^2(\gamma )<\int _{\Gamma }J^2(\gamma )\textrm{d}\xi ^1(\gamma ). \end{aligned}$$

(7.3)

Combining (7.2) and (7.3), we obtain a contradiction with (7.1). $\square$