## Abstract

In Mean Field Games of Controls, the dynamics of the single agent is influenced not only by the distribution of the agents, as in the classical theory, but also by the distribution of their optimal strategies. In this paper, we study quasi-stationary Mean Field Games of Controls, in which the strategy-choice mechanism of the agent is different from the classical case: the generic agent cannot predict the evolution of the population, but chooses its strategy only on the basis of the information available at the given instant of time, without anticipating. We prove existence and uniqueness for the solution of the corresponding quasi-stationary Mean Field Games system under different sets of hypotheses and we provide some examples of models which fall within these hypotheses.

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## Acknowledgements

The authors were partially supported by INdAM-GNAMPA Project, codice CUP\(_{E55F22000270001}\). The second author was partially supported also by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”. The authors warmly thank the anonymous referees for useful suggestions and comments to improve the paper.

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## A Appendix: Proofs of Some Results

### A Appendix: Proofs of Some Results

### 1.1 A.1 Proof of Lemma 2.5

### Proof

Set \(b^i(x,a)=b(x,a;\nu _i)\) and \(\ell ^i(x,a)=\ell (x,a;\nu _i)\), \(i=1,2\). We first claim that

where \(C=C_1(1+K+L)\) as in (2.4). Indeed, to prove the claim, it is sufficient to observe that

are a subsolution and a supersolution of the equation satisfied by \(u^\rho _1\). Then estimate (A.1) follows by the comparison principle. The rest of the proof follows the corresponding argument in [15, Thm. 2.2]. \(\square \)

### 1.2 A.2 Proof of Lemma 3.1

### Proof

Given *m* and *p* as in the statement, let \(\Psi :\mathcal{P}(\mathbb {T}^d\times A)\rightarrow \mathcal{P}(\mathbb {T}^d\times A)\) be the map defined by \(\Psi (\mu )=\left( \mathop {Id},\alpha ^*(\cdot ,p(\cdot );\mu )\right) \sharp m\). Given \(\mu _1,\mu _2 \in \mathcal{P}(\mathbb {T}^d\times A)\), by (3.1), we have

where the \(\sup \) here and in the following formulas are taken with respect to 1-Lipschitz functions on \(\mathbb {T}^d\times A\). Hence, by (*H*4), \(\Psi \) is a contraction and therefore there exists a unique fixed point to (3.2).

(*i*). By (3.1), we have

We deduce

and the statement is an easy consequence of the Cauchy-Schwarz inequality applied to (A.2).

(*ii*). Consider \(m_n\) and *m*, \(\mu _n\) and \(\mu \) as in the statement. We have

We denote by \(I_1\) and \(I_2\) the two terms in the last side. Assumption (3.1) ensures

On the other hand, by (3.1), the function \(\alpha ^*(\cdot ,p(\cdot );\mu )\) is Lipschitz continuous with Lipschitz constant \(1+\lambda _1(1+ L_p)\). We deduce

where the last relation is due to the fact that the integrand is a 1-Lipschitz continuous function in *x*. Replacing the last two relations in the previous one we obtain the statement. \(\square \)

### 1.3 A.3 Proof of Lemma 4.2

### Proof

We shall borrow some arguments of [5, Lemma 5.4]; we proceed by contradiction assuming that there exists \(\varepsilon >0\) such that for every \(n\in \mathbb {N}\setminus \{0\}\) there exist \(\mu _n\in C([0,T],\mathcal{P}(\mathbb {T}^d\times A))\), \(\rho _n\in (0,1)\) and \(t_n\in [0,T-h_n]\), with \(h_n\in (0,1/n)\) and

where \(v_n\) and \(w_n\) are the solutions to (4.3) with \(\rho \) replaced by \(\rho _n\) and with \((t,\mu )\) replaced by \((t_n,\mu _n)\) and respectively by \((t_n+h_n,\mu _n)\). Possibly passing to a subsequence, we may assume that the sequence \(\{\rho _n\}\) converges to some value \(\rho \in [0,1]\). We split the rest of the proof according to the fact that \(\rho =0\) or \(\rho \ne 0\).

Case \(\rho \ne 0\). Estimates (2.3) and (2.4) and \(\rho >0\) ensure:

Hence, possibly passing to subsequences (that we still denote \(v_n\) and \(w_n\)), we may assume that \(v_n\) and \(w_n\) converge to some function \(\varphi _v\) and respectively \(\varphi _w\) in the topology of \(C^{1}(\mathbb {T}^d)\).

Since \(h_n\rightarrow 0\) as \(n\rightarrow \infty \), possibly passing to a subsequence and without any loss of generality, we assume that both \(\{t_n\}_n\) and \(\{t_n+h_n\}_n\) converge to a common value \(\bar{t}\). By hypothesis (*H*5), there exists \({\bar{H}}(x,p,t)\) such that, as \(n\rightarrow \infty \), \(H(\cdot ,\cdot ;\cdot , \mu _n)\) uniformly converges to \({\bar{H}}\) in \(\mathbb {T}^d\times B(0,\bar{K})\times [0,T]\). In particular, by Ascoli–Arzela theorem, we deduce that \({\bar{H}}\) is continuous on \(\mathbb {T}^d\times B(0,\bar{K})\times [0,T]\) and, exploiting \((H1')\), also that it satisfies

for any \(x_1,x_2\in \mathbb {T}^d\), \(p_1,p_2\in B_{\bar{K}}\) and \(t\in [0,T]\). We denote by \({\bar{u}}\) the unique bounded solution to

By (*H*5) and the continuity of \({\bar{H}}\), for some sequence \(o_n(1)\) with \(\lim _{n}o_n(1)=0\), there holds

for every \((x,p)\in \mathbb {T}^d\times B_{\bar{K}}\). By the Comparison Principle (using the positivity of \(\rho \), the bound (A.4), and the last inequality), we deduce that \(v_n\pm o_n(1)\) are super- and subsolution to (A.5). Letting \(n\rightarrow \infty \), by uniqueness of the solution to (A.5), we obtain \(\varphi _v={\bar{u}}\). Repeating the same arguments for \(w_n\), we obtain \(\varphi _w={\bar{u}}=\varphi _v\). In conclusion, as \(n\rightarrow \infty \), we have

which contradicts our claim (A.3).

Case \(\rho =0\). We introduce the functions \(v^*_n(\cdot ):=v_n(\cdot )-v_n(0)\) and \(w^*_n(\cdot ):=w_n(\cdot )-w_n(0)\). Again by estimates (2.3) and (2.4), we infer

By Ascoli–Arzela theorem, possibly passing to a subsequence, there exist a function \(V\in C^{2,\theta }(\mathbb {T}^d)\) and a constant \(\lambda _v\) such that

By the same arguments as before, relation (A.6) still holds true. Hence, by stability result, we deduce that the function *V* solves

By similar arguments, there exist a function \(W\in C^{2,\theta }(\mathbb {T}^d)\) and a constant \(\lambda _w\) such that

and consequently

By (A.7) and (A.8), the couples \((\lambda _v,V)\) and \((\lambda _w,W)\) are both solutions to the ergodic problem

By the same arguments as those used in the proof of [1, Thm. 4.1], this ergodic problem admits exactly one solution \((\lambda ,u)\in \mathbb {R}\times C(\mathbb {T}^d)\); hence, we have

Finally, as \(n\rightarrow \infty \), we conclude

which contradicts our claim (A.3). \(\square \)

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Camilli, F., Marchi, C. On Quasi-Stationary Mean Field Games of Controls.
*Appl Math Optim* **87**, 47 (2023). https://doi.org/10.1007/s00245-022-09960-2

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DOI: https://doi.org/10.1007/s00245-022-09960-2