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

TL;DR

This paper analyzes the properties of dynamic generalized Nash equilibria using system theory, establishing conditions for turnpike behavior, dissipativity, and convergence in multi-agent control scenarios.

Contribution

It introduces a system-theoretic framework for analyzing dynamic GNEs, linking dissipativity with turnpike phenomena and designing penalties for convergence.

Findings

Strict dissipativity induces the turnpike phenomenon in GNE solutions.

A converse result shows turnpike implies strict dissipativity.

Linear terminal penalties can ensure convergence to steady-state GNE.

Abstract

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.