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Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances

Yongchun Bi, Jun Zheng, Guchuan Zhu

TL;DR

This work addresses input-to-state stabilization for a cascaded system consisting of an $N$-dimensional ODE and a 1-D parabolic PDE under Dirichlet-Robin boundary disturbances and in-domain disturbances. It develops a continuous boundary control via backstepping that decouples the cascade, and proves ISS in the $\sup$-norm using generalized Lyapunov functionals, even in the presence of space-time varying coefficients and boundary disturbances. Well-posedness is established through a lifting approach and analytic semigroup theory, ensuring the target and original systems have unique solutions in continuous function spaces. Numerical simulations validate the theoretical results, showing robust ISS behavior and demonstrating the advantages of avoiding sliding-mode controls in this setting.

Abstract

This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.

Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances

TL;DR

This work addresses input-to-state stabilization for a cascaded system consisting of an -dimensional ODE and a 1-D parabolic PDE under Dirichlet-Robin boundary disturbances and in-domain disturbances. It develops a continuous boundary control via backstepping that decouples the cascade, and proves ISS in the -norm using generalized Lyapunov functionals, even in the presence of space-time varying coefficients and boundary disturbances. Well-posedness is established through a lifting approach and analytic semigroup theory, ensuring the target and original systems have unique solutions in continuous function spaces. Numerical simulations validate the theoretical results, showing robust ISS behavior and demonstrating the advantages of avoiding sliding-mode controls in this setting.

Abstract

This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.
Paper Structure (9 sections, 147 equations, 8 figures)

This paper contains 9 sections, 147 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Controller Design
  4. Well-posedness Analysis
  5. Stability Assessment
  6. Numerical Results
  7. Conclusion
  8. Proof of Proposition kernel exitence (i)
  9. Proof of Proposition equivelent

Figures (8)

  • Figure 1: Signal flow of the ODE-PDE cascade
  • Figure 2: Evolution of $X$, $u$, and $|X(t)|+\sup_{z\in (0,1)}\left|u(z,t)\right|$ for system \ref{['original system']} in open loop with different initial data
  • Figure 3: Evolution of $X$ and $u$ for the closed-loop system \ref{['original system']} with different initial data in the absence of disturbances
  • Figure 4: Evolution of $X$ and $u$ for the closed-loop system \ref{['original system']} in the presence of different internal and in-domain disturbances when $m_0=1$ and $j_1=j_2=0$
  • Figure 5: Evolution of $X$ and $u$ for the closed-loop system \ref{['original system']} in the presence of different internal and in-domain disturbances when $m_0=2$ and $j_1=j_2=0$
  • ...and 3 more figures

Theorems & Definitions (5)

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