Stabilizing Model Predictive Control for Generalized Nash Equilibrium Problems using Approximation by α-quasi-GNEPs and Lyapunov End Cost

We study model predictive control (MPC) schemes for non-cooperative dynamic games regarding stabilization. The dynamic games are modelled as generalized Nash equilibrium problems (GNEPs), in which a shared constraint is given as a jointly controlled time-discrete (linear) dynamics. Furthermore, the players’ objectives are interdependent. For the stability analysis, we introduce the class of α-quasi-GNEPs, which approximate the original games. Basing the MPC scheme on these instead, we derive conditions on the players’ objectives which guarantee asymptotic stability in the presence of stabilizing constraints in the form of a Lyapunov terminal cost. Furthermore, we show that a suitable Lyapunov terminal cost can be obtained from a non-game-based MPC scheme. This non-game-based MPC scheme relies on a classical optimal control problem for the aggregated cost. Hence, known results for determining the Lyapunov cost can be applied and carried over to the game-based setting. The theoretical results are complemented by numerical experiments.