The turnpike control in stochastic multi-agent dynamics: A discrete-time approach with exponential integrators

In this manuscript, we study the turnpike property in stochastic discrete-time optimal control problems for interacting agents. Extending previous deterministic results, we show that the turnpike property persists in the presence of noise under suitable dissipativity and controllability conditions. To handle the possible stiffness in the system dynamics, we employ for the time discretization integrators of exponential type. Numerical experiments validate our findings, demonstrating the advantages of exponential integrators over standard explicit schemes and confirming the effectiveness of the turnpike control even in the stochastic setting.