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Local integral input-to-state stability for non-autonomous infinite-dimensional systems

Yongchun Bi, Panyu Deng, Jun Zheng, Guchuan Zhu

TL;DR

The paper tackles LiISS analysis for non-autonomous infinite-dimensional systems with time-varying coefficients and superlinear terms. It develops generalized comparison principles for nonlinear ODEs and uses them to formulate a LiISS Lyapunov framework in Banach spaces, with Hilbert-space results for LiISS-LFs via a time-varying operator $P(t)$ yielding a quadratic Lyapunov functional $V(t,x)=\langle P(t)x,x\rangle$. The main contributions are the LiISS Lyapunov theorem in Banach spaces, sufficient conditions for LiISS-LFs in Hilbert spaces, and two concrete examples (an ODE and a PDE) that illustrate the method and the use of interpolation inequalities to handle superlinear growth. Through these results, the authors provide a tractable approach to LiISS analysis that avoids the sometimes intractable requirement of proving $0$-UGASs, with numerical demonstrations supporting the theory. The work advances stability analysis for non-autonomous, infinite-dimensional systems and suggests future work on boundary-control cases and broader applications.

Abstract

In this paper, we prove comparison principles for nonlinear differential equations with time-varying coefficients and develop Lyapunov analytical tools for the integral input-to-state stability (iISS) analysis of nonlinear non-autonomous infinite-dimensional systems, which involve nonlinearities satisfying a superlinear growth, {bringing} difficulties to the iISS {analysis.} Specifically, our approach starts by establishing several forms of comparison principles for a wide range of ordinary differential equations having time-varying coefficients and superlinear terms, paving the way to conduct iISS assessment for general nonlinear non-autonomous infinite-dimensional systems within the Lyapunov stability framework. Then, by using the comparison principles, we prove a local {iISS} {(LiISS)} Lyapunov theorem for the nonlinear non-autonomous infinite-dimensional systems in the framework of Banach spaces. {Furthermore,} we provide sufficient conditions of the existence of a local iISS Lyapunonv functional (LiISS-LF) and construct LiISS-LFs for the systems in the framework of Hilbert spaces. Finally, we preset two examples to illustrate the proposed {Lyapunov} method for the LiISS analysis: one is to show how to obtain the LiISS of a nonlinear finite-dimensional system with time-varying coefficients and superlinear terms under linear state feedback control law while another one is to show how to employ the interpolation inequalities to handle superliner terms and establish the LiISS-LF for a class of multi-dimensional parabolic equations with space-time-varying coefficients. To demonstrate the validity of the results, numerical experiments are also conducted to verify the LiISS of these two classes of systems.

Local integral input-to-state stability for non-autonomous infinite-dimensional systems

TL;DR

The paper tackles LiISS analysis for non-autonomous infinite-dimensional systems with time-varying coefficients and superlinear terms. It develops generalized comparison principles for nonlinear ODEs and uses them to formulate a LiISS Lyapunov framework in Banach spaces, with Hilbert-space results for LiISS-LFs via a time-varying operator yielding a quadratic Lyapunov functional . The main contributions are the LiISS Lyapunov theorem in Banach spaces, sufficient conditions for LiISS-LFs in Hilbert spaces, and two concrete examples (an ODE and a PDE) that illustrate the method and the use of interpolation inequalities to handle superlinear growth. Through these results, the authors provide a tractable approach to LiISS analysis that avoids the sometimes intractable requirement of proving -UGASs, with numerical demonstrations supporting the theory. The work advances stability analysis for non-autonomous, infinite-dimensional systems and suggests future work on boundary-control cases and broader applications.

Abstract

In this paper, we prove comparison principles for nonlinear differential equations with time-varying coefficients and develop Lyapunov analytical tools for the integral input-to-state stability (iISS) analysis of nonlinear non-autonomous infinite-dimensional systems, which involve nonlinearities satisfying a superlinear growth, {bringing} difficulties to the iISS {analysis.} Specifically, our approach starts by establishing several forms of comparison principles for a wide range of ordinary differential equations having time-varying coefficients and superlinear terms, paving the way to conduct iISS assessment for general nonlinear non-autonomous infinite-dimensional systems within the Lyapunov stability framework. Then, by using the comparison principles, we prove a local {iISS} {(LiISS)} Lyapunov theorem for the nonlinear non-autonomous infinite-dimensional systems in the framework of Banach spaces. {Furthermore,} we provide sufficient conditions of the existence of a local iISS Lyapunonv functional (LiISS-LF) and construct LiISS-LFs for the systems in the framework of Hilbert spaces. Finally, we preset two examples to illustrate the proposed {Lyapunov} method for the LiISS analysis: one is to show how to obtain the LiISS of a nonlinear finite-dimensional system with time-varying coefficients and superlinear terms under linear state feedback control law while another one is to show how to employ the interpolation inequalities to handle superliner terms and establish the LiISS-LF for a class of multi-dimensional parabolic equations with space-time-varying coefficients. To demonstrate the validity of the results, numerical experiments are also conducted to verify the LiISS of these two classes of systems.
Paper Structure (10 sections, 10 theorems, 82 equations, 11 figures)

This paper contains 10 sections, 10 theorems, 82 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Comparison principles for nonlinear ODEs with time-varying coefficients
  3. LiISS of non-autonomous infinite-dimensional systems
  4. LiISS of non-autonomous infinite-dimensional systems
  5. The LiISS Lyapunov theorem in the framework of Banach spaces
  6. Constructing LiISS-LFs in the framework of Hilbert spaces
  7. Illustrative examples
  8. Local stabilization of a nonlinear ODE system under linear state feedback control
  9. LiISS of time-varying parabolic PDEs with superlinear terms
  10. Conclusion

Key Result

Lemma 1

Assume that $\alpha,g \in C(\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})$ with $\lim\limits_{s\rightarrow +\infty}\int_{0}^sg(\tau) {\rm{d}}\tau=+\infty$. For $T\in \mathbb{R}_{>0}$ and $y_0\in \mathbb{R}_{\geq 0}$, if $y$ is nonnegative and absolutely continuous on $[0,T]$ and satisfies the following then there is a function $\beta \in\mathcal{K}\mathcal{L}$ such that Particularly, when $\alpha(y)

Figures (11)

  • Figure 1: Evolution of $x$ and ${|x|}$ for system \ref{['nonlinear ODE']} in open loop with different small initial data.
  • Figure 2: Evolution of $x$ for system \ref{['nonlinear ODE']} under the control law \ref{['control law']} with different small initial data.
  • Figure 3: Evolution of $x$ for system \ref{['nonlinear ODE-closed loop']} with different disturbances when ${(x_1(0),x_2(0))^{\top}=(0.1,0.25)^{\top}}$.
  • Figure 4: Evolution of $x$ for system \ref{['nonlinear ODE-closed loop']} with different disturbances when ${(x_1(0),x_2(0))^{\top}=(0.2,0.5)^{\top}}$.
  • Figure 5: Evolution of ${|x|}$ for system \ref{['nonlinear ODE-closed loop']} with different disturbances and relatively small initial data: solid curve for ${(x_1(0),x_2(0))^{\top}=(0.1,0.25)^{\top}}$ and dashed curve for ${(x_1(0),x_2(0))^{\top}=(0.2,0.5)^{\top}}$.
  • ...and 6 more figures

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • ...and 4 more