Lipschitz stability for determination of states and inverse source problem for the mean field game equations (Q6587552)

This paper deals with mean-field game equations whose principal coefficients depend on time and state variables with a general Hamiltonian and attached non-zero Robin boundary conditions. More precisely, the authors study the inverse source problem for mean-field game equations, i.e. to recover \(\mathcal{L}\), \(H\), \(F\), \(G\) by given data of \(u,v\) in systems of the form\N\begin{align*}\N&-\partial_t u(x,t) + \mathcal{L}u(x,t) + H(x,t,\nabla u,v) = F(x,t) && \in \Omega \times (0,T),\\\N&\partial_tv(x,t) + \mathcal{L}^*v(x,t) - \text{div}(v(x,t) \nabla_p H(x,t,\nabla u,v)) = G(x,t) && \in \Omega \times (0,T),\\\N&u(x,T) = u_T (x,v(x,T)), \quad v(x,0) = v_0(x).\N\end{align*}\NFirst, the authors introduce a Carleman estimate for generalized mean-field game equations. Building on this, they establish as the first main result Lipschitz stability for the inverse source problem of determining \(F\) and \(G\). Notably, this offers an unconditional stability estimate under a non-homogeneous Robin boundary condition, eliminating the need for any boundedness assumptions on \(u\) and \(v\).\N\NAs a second major result, the authors prove a novel global unconditional Lipschitz stability for the inverse source problem. Finally, they explore the application of these findings to the state determination problem for a simplified, yet still nonlinear, mean-field game system.