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System-Theoretic Analysis of Dynamic Generalized Nash Equilibrium Problems -- Turnpikes and Dissipativity

Sophie Hall, Florian Dörfler, Timm Faulwasser

TL;DR

This work analyzes open-loop trajectories in finite-horizon dynamic generalized Nash equilibrium problems (GNEPs) from a system-theoretic lens, introducing dissipativity-based turnpike analysis to GNEPs. It proves that strict dissipativity with respect to the steady-state GNE $(x_s,u_s)$ induces a measure turnpike and, under a local minimality assumption, establishes a converse result. The authors define a game value function $V_N(x)$ whose gradient ties to the initial dual data and show how the storage function gradient relates to the steady-state multipliers, linking GNEP optimality to classical optimal-control concepts. They propose per-agent linear end penalties, including a learning scheme to adapt these penalties online, to suppress leaving arcs and ensure convergence to $(x_s,u_s)$ in open-loop trajectories. Numerical simulations on coupled GNEP examples validate turnpike behavior and demonstrate the effectiveness of linear penalties and penalty learning in enforcing steady-state operation.

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. We study the properties of open-loop 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 GNE open-loop trajectories 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.

System-Theoretic Analysis of Dynamic Generalized Nash Equilibrium Problems -- Turnpikes and Dissipativity

TL;DR

This work analyzes open-loop trajectories in finite-horizon dynamic generalized Nash equilibrium problems (GNEPs) from a system-theoretic lens, introducing dissipativity-based turnpike analysis to GNEPs. It proves that strict dissipativity with respect to the steady-state GNE induces a measure turnpike and, under a local minimality assumption, establishes a converse result. The authors define a game value function whose gradient ties to the initial dual data and show how the storage function gradient relates to the steady-state multipliers, linking GNEP optimality to classical optimal-control concepts. They propose per-agent linear end penalties, including a learning scheme to adapt these penalties online, to suppress leaving arcs and ensure convergence to in open-loop trajectories. Numerical simulations on coupled GNEP examples validate turnpike behavior and demonstrate the effectiveness of linear penalties and penalty learning in enforcing steady-state operation.

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. We study the properties of open-loop 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 GNE open-loop trajectories 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.
Paper Structure (13 sections, 1 theorem, 64 equations, 6 figures, 1 algorithm)

This paper contains 13 sections, 1 theorem, 64 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setting: Finite-Horizon Generalized Nash Equilibrium Problem
  3. Dissipativity and Turnpike Relations for GNEPs
  4. Dissipativity of GNEPs
  5. A converse turnpike result for GNEPs
  6. Optimal operation at steady-state GNE
  7. Optimality Interpretation of GNEs
  8. The game value function of GNEPs
  9. Sensitivity analysis of the storage function
  10. Turnpike leaving arc and linear end penalties
  11. Learning the linear end penalty
  12. Simulation Study
  13. Conclusion

Key Result

Proposition 1

Suppose the set $\mathcal{S}_{\infty}^{GNE}(\mathbf{x}) \not= \emptyset$, then if GNEP eq:MPCPerAgent is strictly dissipative with respect to $(x_s,u_s)$ following Definition dfn:StrictDiss, it is optimally operated at the steady-state GNE $(x_s,u_s)$ and suboptimally operated off steady state.

Figures (6)

  • Figure 1: Schematic of a state trajectory exhibiting the turnpike property for different horizon lengths.
  • Figure 2: Overview of implications between strict dissipativity, turnpike and optimal operation at the steady-state GNE.
  • Figure 3: Finite-horizon and steady-state GNEP with GNE-KKT system
  • Figure 4: Open-loop GNE trajectories of \ref{['eq:GNEP_example']} without terminal penalty.
  • Figure 5: Open-loop GNE trajectories of \ref{['eq:GNEP_example']} with a $\lambda_s\, x_N$ penalty.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 1: Generalized Nash equilibrium
  • Definition 2: Steady-state GNE
  • Remark 1: Existence & computation
  • Definition 3: Strict dissipativity of GNEPs
  • Remark 2: Why restrict dissipativity to $\mathcal{S}^{\text{\tiny GNE}}_N?$
  • Definition 4: Measure turnpike in GNEPs
  • Remark 3: Turnpike with strict $x$ dissipativity
  • Definition 5: $\varepsilon$-GNE chen2021distributed
  • Definition 6: Optimal operation of RHG
  • Proposition 1
  • ...and 1 more