System-Theoretic Analysis of Dynamic Generalized Nash Equilibria -- Turnpikes and Dissipativity
For researchers in multi-agent control and game-theoretic MPC, this work lays foundational system-theoretic connections for GNE, analogous to those in optimal control, enabling future stability and feasibility results.
This paper provides a system-theoretic analysis of dynamic generalized Nash equilibria (GNE), establishing that strict dissipativity implies the turnpike phenomenon and proving a converse result. It also designs linear terminal penalties ensuring open-loop GNE trajectories converge to the steady-state GNE.
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.