System-Theoretic Analysis of Dynamic Generalized Nash Equilibria -- Turnpikes and Dissipativity

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.

💡 Research Summary

The paper investigates finite‑horizon dynamic Generalized Nash Equilibrium Problems (GNEPs) that arise in multi‑agent control settings where agents are coupled through costs, dynamics, and constraints. The authors adopt a system‑theoretic viewpoint and establish a rigorous link between strict dissipativity and the turnpike phenomenon for GNE trajectories, mirroring classic results from optimal control theory.

First, the authors formalize the discrete‑time dynamics (x_{k+1}=f(x_k,u_k)) and define each agent’s local cost (\ell_v) together with coupled inequality constraints (g) and individual constraints (h_v). A GNE is a joint decision ((x^\ast,u^\ast)) such that no single agent can lower its accumulated cost by unilateral deviation. A steady‑state GNE ((x_s,u_s)) is both a fixed point of the dynamics and a one‑step GNE of the static game.

The core contribution begins with the definition of a social welfare function (\ell(x_k,u_k)=\sum_v \ell_v(x_k,u_k^v)) and a supply rate (s(x_k,u_k)=\ell(x_k,u_k)-\ell(x_s,u_s)). Strict dissipativity is introduced via a storage function (\Lambda) that satisfies, for all finite horizons and all GNE trajectories, \