On the Equilibria Computation of Set-Valued Lur'e Dynamical Systems

TL;DR

This work develops a resolvent-based framework to compute equilibria of set-valued Lur'ë dynamical systems, enabling zero-finding for inclusions of the form $0\in f(x^*)+B(\mathcal{F}^{-1}+D)^{-1}(Cx^*)$. By exploiting $P$-passivity and maximal monotone operators, the authors derive a practical forward–backward (Tseng-type) algorithm, including a resolvent identity $J_{\gamma\mathcal{B}}(x)=E(\gamma J^E_\mathcal{F}(Ex)+Dx)$ with $E=(\gamma I+D)^{-1}$, and show convergence to equilibria under standard assumptions. They extend the method to a broader class via Algorithm 2 and apply the framework to quasi-variational inequalities and Nash quasi-equilibria, demonstrating equivalences such as $0\in f(x^*)+N_{K(x^*)}(x^*)$ with $K(x)=\Omega- Df(x)$ and providing conditions that relax classical QVI restrictions. A numerical example highlights the method's advantage over explicit schemes in converging to the equilibrium. The results offer a versatile tool for equilibrium computation in control, economics, and game-theoretic contexts, with potential to simplify and accelerate solving QVIs and Nash problems.

Abstract

In this article, we propose an efficient way to compute equilibria of a general class of set-valued Lur'e dynamical systems, which plays an important role in the asymptotical analysis of the systems. Besides the equilibria computation, our study can be also used to solve a class of quasi-variational inequalities. Some examples of finding Nash quasi-equilibria in game theory are given.

Figures (2)

• Figure 1: The explicit scheme does not converge
• Figure 2: The convergence of Algorithm 1

Theorems & Definitions (35)

• Proposition 1
• Proposition 2
• proof
• Remark 1
• Definition 1
• Remark 2
• Remark 3
• Lemma 3
• proof
• Theorem 4