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$k$-Contraction in a Generalized Lurie System

Ron Ofir, Jean-Jacques Slotine, Michael Margaliot

TL;DR

This paper addresses global stability in multi-stable nonlinear systems by introducing $k$-contraction for generalized Lurie systems (GLS). It develops explicit sufficient conditions for $k$-contraction with respect to a state-dependent metric, unifying and extending prior Lurie-system analyses and enabling convergence results for $k=2$ without requiring a unique equilibrium. The main contribution is a tractable, metric-based condition on the GLS Jacobians that guarantees exponential decay of $k$-volumes in the closed loop, with ARI/LMIs recovered in the constant-metric case. The results are demonstrated across tridiagonal, networked, and biochemical circuit models, highlighting practical impact on multi-stable systems, neural networks, and biological control.

Abstract

We derive a sufficient condition for $k$-contraction in a generalized Lurie system~(GLS), that is, the feedback connection of a nonlinear dynamical system and a memoryless nonlinear function. For $k=1$, this reduces to a sufficient condition for standard contraction. For $k=2$, this condition implies that every bounded solution of the GLS converges to an equilibrium, which is not necessarily unique. We demonstrate the theoretical results by analyzing $k$-contraction in a biochemical control circuit with nonlinear dissipation terms.

$k$-Contraction in a Generalized Lurie System

TL;DR

This paper addresses global stability in multi-stable nonlinear systems by introducing -contraction for generalized Lurie systems (GLS). It develops explicit sufficient conditions for -contraction with respect to a state-dependent metric, unifying and extending prior Lurie-system analyses and enabling convergence results for without requiring a unique equilibrium. The main contribution is a tractable, metric-based condition on the GLS Jacobians that guarantees exponential decay of -volumes in the closed loop, with ARI/LMIs recovered in the constant-metric case. The results are demonstrated across tridiagonal, networked, and biochemical circuit models, highlighting practical impact on multi-stable systems, neural networks, and biological control.

Abstract

We derive a sufficient condition for -contraction in a generalized Lurie system~(GLS), that is, the feedback connection of a nonlinear dynamical system and a memoryless nonlinear function. For , this reduces to a sufficient condition for standard contraction. For , this condition implies that every bounded solution of the GLS converges to an equilibrium, which is not necessarily unique. We demonstrate the theoretical results by analyzing -contraction in a biochemical control circuit with nonlinear dissipation terms.
Paper Structure (13 sections, 5 theorems, 56 equations, 3 figures)

This paper contains 13 sections, 5 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compound matrices
  4. $k$-contraction w.r.t. a Riemannian metric
  5. Singular values of the product of two matrices
  6. Generalized Lurie systems
  7. Generalized Lurie system
  8. Main result
  9. Applications
  10. $k$-contraction w.r.t. a state-dependent metric for tridiagonal systems
  11. Networked systems
  12. A biochemical control circuit
  13. Conclusion

Key Result

Lemma 1

Let $\dot \Theta : \Omega \to \mathbb R^{n \times n}$ denote the matrix whose $(i,j)$ entry is $\left(\dot \theta(x)\right)_{ij} := \left(\frac{\partial \theta_{ij}(x)}{\partial x}\right)^\mathsf{T} f(t,x).$ Suppose that for all $t\geq 0,x\in\Omega$. Then for any $t \geq 0,r\in\mathcal{S}^k$, we have

Figures (3)

  • Figure 1: A 3D parallelotope.
  • Figure 2: Block diagram of a generalized Lurie system.
  • Figure 3: Trajectories of \ref{['eq:exa_sm']} from five initial conditions marked by circles. Several equilibrium points are marked by a cross.

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 2 more