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Integral gains for non-autonomous Wazewski systems

Ivan Atamas, Sergey Dashkovskiy, Vitalii Slynko

TL;DR

This work addresses stability of linear non-autonomous Wazewski interconnections on Hilbert spaces by decomposing the system into two subsystems and introducing integral gains to quantify time-varying coupling. A matrix-valued Lyapunov construction is used to derive a small-gain-type condition $r_\sigma(\Pi(t_0))<1$, ensuring stability and, under additional conditions, asymptotic or uniform asymptotic stability. Two illustrative examples demonstrate advantages over classical criteria: removal of a Karafyllis-type condition and handling of unbounded interconnections that challenge prior methods. The results extend to infinite-dimensional settings and lay groundwork for future work on unbounded operators, larger networks, and nonlinear extensions.

Abstract

In this work we consider linear non-autonomous systems of Wazewski type on Hilbert spaces and provide a new approach to study their stability properties by means of a decomposition into subsystems and conditions implied on the interconnection properties. These conditions are of the small-gain type but the appoach is based on a conceptually new notion which we call integral gain. This notion is introduced for the first time in this paper. We compare our approach with known results from the literature and demonstrate advantages of our results.

Integral gains for non-autonomous Wazewski systems

TL;DR

This work addresses stability of linear non-autonomous Wazewski interconnections on Hilbert spaces by decomposing the system into two subsystems and introducing integral gains to quantify time-varying coupling. A matrix-valued Lyapunov construction is used to derive a small-gain-type condition , ensuring stability and, under additional conditions, asymptotic or uniform asymptotic stability. Two illustrative examples demonstrate advantages over classical criteria: removal of a Karafyllis-type condition and handling of unbounded interconnections that challenge prior methods. The results extend to infinite-dimensional settings and lay groundwork for future work on unbounded operators, larger networks, and nonlinear extensions.

Abstract

In this work we consider linear non-autonomous systems of Wazewski type on Hilbert spaces and provide a new approach to study their stability properties by means of a decomposition into subsystems and conditions implied on the interconnection properties. These conditions are of the small-gain type but the appoach is based on a conceptually new notion which we call integral gain. This notion is introduced for the first time in this paper. We compare our approach with known results from the literature and demonstrate advantages of our results.

Paper Structure

This paper contains 6 sections, 2 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Example 1 (from KaT04)
  5. Example 2
  6. Conclusions

Key Result

Theorem 1

Let 1 be a Wazewski system with respect to a selfadjoint solid cone $K$ and written as 2 with $r=2$. Let $\alpha_i, \beta_i, \gamma_i, \delta_i$ be as in 4 and $\phi_i, g_i, q_i, \omega_i, \pi_{ij}$ as in weights_q_i,gains-ii,gains-ij. If the spectral radius of the matrix $\Pi(t_0)\in\mathbb R^{2\ti then solutions to 1 satisfy the estimate where $a_K$ is from nonflat, $q(t):=\min\{q_1(t),q_2(t)\}

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • proof : Sketch
  • Corollary 1
  • proof