Sophie Hall

Sophie Hall, Florian Dörfler, Timm Faulwasser

Generalized Nash equilibria are used in multi-agent control applications to model strategic interactions between agents that are coupled in the cost, dynamics, and constraints, and provide the foundations for game-theoretic MPC (Receding Horizon Games). We study properties of finite-horizon dynamic GNE trajectories from a system-theoretic perspective. We show how strict dissipativity generates the turnpike phenomenon in GNE solutions. Moreover, we establish a converse turnpike result, i.e., the implication from turnpike to strict dissipativity. We derive conditions under which the steady-state GNE is the optimal operating point and, using a game value function, we give a local characterization of the geometry of storage functions. Finally, we design linear terminal penalties that ensure dynamic GNE trajectories applied in open-loop converge to and remain at the steady-state GNE. These connections provide the foundation for future system-theoretic analysis of GNEs similar to those existing in optimal control as well as for recursive feasibility and closed-loop stability results of game-theoretic MPC.

Sophie Hall, Florian Dörfler, Timm Faulwasser

We analyze Receding Horizon Games without any MPC-like terminal ingredients. We show that recursive feasibility can be inferred from the turnpike phenomenon under mild assumptions. Moreover, we prove sufficient conditions for practical asymptotic convergence of the closed-loop trajectories, and we discuss how the gap towards practical asymptotic stability may be closed. We use numerical examples to show that the closed-loop region of attraction around the steady-state GNE shrinks exponentially with the horizon length, a behavior previously known only for model predictive control. Further, we apply a linear end penalty and demonstrate in numerical simulations that it suppresses the leaving arc and ensures asymptotic convergence to the steady-state GNE.