Speaker
Description
In this talk, we are interested in the stationary MFG system:
\begin{equation}
\begin{cases}
-\Delta u + H(x,Du) + \lambda u = F[m] & \text{in } \mathbb{T}^d, \
-\Delta m - \mathrm{div}(m H_p(x,Du)) + \lambda m = \lambda m_0 & \text{in } \mathbb{T}^d,
\end{cases}
\end{equation}
where $u$ is the value function and $m$ the distribution of the players. Our goal is to establish the existence of finite element approximations of the solutions to the MFG ssytem and to obtain the associated error estimates.
Our approach consists in reformulating system in the form $F(u,m) = 0$, where $F$ is a nonlinear map defined on a suitably chosen Banach space. In the case where this map is of class $C^1$, we show that the stable solutions, in the sense of Briani-Cardaliaguet, correspond to the regular zeros of $F$, i.e., those for which $dF[u,m]$ is invertible. This property makes it possible to apply the Brezzi-Rappaz-Raviart (BRR) approximation theorem in order to obtain existence and error estimates for the finite element approximations of the stable solutions.
However, the $C^1$ regularity of $F$ requires the Hamiltonian $H$ to be of class $C^2$, which is rather restrictive from the point of view of optimal control. In order to generalize our approach to the more natural class of Hamiltonians with $C^{1,1}$ regularity, we generalize the BRR theorem to the case of maps $F$ that are Lipschitz continuous and satisfy a metric regularity assumption. This last assumption, coming from variational analysis, replaces the invertibility of $dF[\bar{x}]$. This generalization allows us to extend our error estimates for the finite element approximations of the stable solutions to the MFG system to the case of Hamiltonians of class $C^{1,1}$.