Long-time behavior and turnpike properties of linear-quadratic graphon mean field control problems

We investigate the asymptotic behavior and turnpike properties of graphon mean field control (GMFC) problems in the linear-quadratic setting.

We consider both a finite-horizon GMFC problem and its associated ergodic counterpart, in which the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous interactions.

The optimal controls and state trajectories for both problems are characterized by systems of Riccati equations together with systems of generalized differential and algebraic equations on suitable Hilbert spaces.

Under a stabilizability condition and appropriate positivity assumptions on the graphon-induced operators, we establish the unique solvability of the ergodic control problem and derive exponential convergence estimates for the finite-horizon system to its stationary limit.

As a consequence, we establish an exponential turnpike property for the optimal pair and prove the convergence of the time-averaged value function for the finite-horizon GMFC problem.