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Sliding-Mode Nash Equilibrium Seeking for a Quadratic Duopoly Game

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstić, Tamer Başar

TL;DR

The paper tackles Nash equilibrium seeking in a quadratic duopoly without relying on a model by introducing a distributed sliding-mode NES that fuses extremum seeking with sinusoidal perturbations and a boundary control law. It treats the payoff map $Q(\Theta)$ as locally quadratic with $Q(\Theta)=y^*+\frac{H}{2}(\Theta-\Theta^*)^2$ and $H<0$, and couples the ES loop to a PDE describing actuation via Kelvin-Voigt damping, using a backstepping transformation to stabilize the closed-loop dynamics. The authors prove that the averaged closed-loop system converges exponentially to a neighborhood of the Nash equilibrium $\Theta^*$, derive residual bounds that shrink with increasing perturbation frequency $\omega$ and perturbation amplitude $a$, and validate the theory through simulations. This approach yields a robust, model-free mechanism for Nash equilibrium seeking in spatially distributed, PDE-governed systems and points to future extensions for more realistic underwater scenarios with external environmental forces.

Abstract

This paper introduces a new method to achieve stable convergence to Nash equilibrium in duopoly noncooperative games. Inspired by the recent fixed-time Nash Equilibrium seeking (NES) as well as prescribed-time extremum seeking (ES) and source seeking schemes, our approach employs a distributed sliding mode control (SMC) scheme, integrating extremum seeking with sinusoidal perturbation signals to estimate the pseudogradients of quadratic payoff functions. Notably, this is the first attempt to address noncooperative games without relying on models, combining classical extremum seeking with relay components instead of proportional control laws. We prove finite-time convergence of the closed-loop average system to Nash equilibrium using stability analysis techniques such as time-scaling, Lyapunov's direct method, and averaging theory for discontinuous systems. Additionally, we quantify the size of residual sets around the Nash equilibrium and validate our theoretical results through simulations.

Sliding-Mode Nash Equilibrium Seeking for a Quadratic Duopoly Game

TL;DR

The paper tackles Nash equilibrium seeking in a quadratic duopoly without relying on a model by introducing a distributed sliding-mode NES that fuses extremum seeking with sinusoidal perturbations and a boundary control law. It treats the payoff map as locally quadratic with and , and couples the ES loop to a PDE describing actuation via Kelvin-Voigt damping, using a backstepping transformation to stabilize the closed-loop dynamics. The authors prove that the averaged closed-loop system converges exponentially to a neighborhood of the Nash equilibrium , derive residual bounds that shrink with increasing perturbation frequency and perturbation amplitude , and validate the theory through simulations. This approach yields a robust, model-free mechanism for Nash equilibrium seeking in spatially distributed, PDE-governed systems and points to future extensions for more realistic underwater scenarios with external environmental forces.

Abstract

This paper introduces a new method to achieve stable convergence to Nash equilibrium in duopoly noncooperative games. Inspired by the recent fixed-time Nash Equilibrium seeking (NES) as well as prescribed-time extremum seeking (ES) and source seeking schemes, our approach employs a distributed sliding mode control (SMC) scheme, integrating extremum seeking with sinusoidal perturbation signals to estimate the pseudogradients of quadratic payoff functions. Notably, this is the first attempt to address noncooperative games without relying on models, combining classical extremum seeking with relay components instead of proportional control laws. We prove finite-time convergence of the closed-loop average system to Nash equilibrium using stability analysis techniques such as time-scaling, Lyapunov's direct method, and averaging theory for discontinuous systems. Additionally, we quantify the size of residual sets around the Nash equilibrium and validate our theoretical results through simulations.
Paper Structure (12 sections, 1 theorem, 37 equations, 5 figures, 1 table)

This paper contains 12 sections, 1 theorem, 37 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Notation and Terminology
  3. Motivating Example for Underwater Search
  4. Problem statement
  5. Basic Gradient-Based Extremum Seeking without PDEs
  6. Scalar Maps with Actuation PDE Dynamics
  7. Trajectory Generation for the Probing Signal
  8. Estimation Errors and Error Dynamics
  9. Boundary Extremum Seeking Control Law
  10. Stability analysis
  11. Simulations
  12. CONCLUSIONS

Key Result

Theorem 1

Consider the control system in Fig. fig:esc, with control law $U(t)$ given in (eq:control_backstepping_3). There exists $\bar{c}^*>0$ such that, $\forall \bar{c} \geq \bar{c}^*$, $\exists \omega^*(\bar{c})>0$ such that, $\forall \omega \geq \omega^*$, and $K>0$ sufficiently large, the closed-loop sy Moreover,

Figures (5)

  • Figure 1: Motivating example - underwater search: $x_e$ represents the relative linear position of the source signal with respect to the sensor.
  • Figure 2: Gradient extremum seeking control loop.
  • Figure 3: The cascade of the PDE dynamics and the ODE integrator.
  • Figure 4: Open-loop eigenvalues for the wave PDE with Kelvin-Voigt damping. The red arrows ($\rightarrow$) indicate the locus as $n$ tends to infinity.
  • Figure 5: Convergence of the state $\alpha(x,t)$ in a three-dimensional space. In blue, we can check $\Theta(t)=\alpha(0,t)$, while in red, we have $\theta(t)=\alpha(D,t)$, both reaching a neighborhood of $\Theta^\ast$.

Theorems & Definitions (1)

  • Theorem 1