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Well-posedness of boundary control systems and application to ISS for coupled heat equations with boundary disturbances and delays

Yassine El Gantouh, Jun Zheng, Guchuan Zhu

Abstract

This paper studies the existence of solutions and, in particular, the well-posedness of a class of boundary control systems. Our main result provides explicit and verifiable conditions on the system data that guarantee continuous dependence of solutions on the initial data and $L^p$-inputs. The proof relies on a new boundedness estimate for the input/output maps of linear time-invariant infinite-dimensional systems with unbounded control and observation operators. The developed technique is applied to derive specific conditions for the exponential input-to-state stability of boundary-coupled heat equations with boundary disturbances and time-delays.

Well-posedness of boundary control systems and application to ISS for coupled heat equations with boundary disturbances and delays

Abstract

This paper studies the existence of solutions and, in particular, the well-posedness of a class of boundary control systems. Our main result provides explicit and verifiable conditions on the system data that guarantee continuous dependence of solutions on the initial data and -inputs. The proof relies on a new boundedness estimate for the input/output maps of linear time-invariant infinite-dimensional systems with unbounded control and observation operators. The developed technique is applied to derive specific conditions for the exponential input-to-state stability of boundary-coupled heat equations with boundary disturbances and time-delays.
Paper Structure (8 sections, 9 theorems, 94 equations, 1 figure)

This paper contains 8 sections, 9 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries on linear infinite-dimensional positive systems
  4. Proof of the main results
  5. Proof of Theorem \ref{['S5.P1']}
  6. Proof of Theorem \ref{['Main1']}
  7. Application: ISS of coupled heat equations with delays and boundary disturbances
  8. Conclusion

Key Result

Theorem 2.1

Let $U$ be a Banach space, $K \in \mathcal{L}(U,V)$, and $p\in [1,\infty)$. Let Assumptions Assp1-Assp2 be satisfied and $q\in [p,\infty]$. Then, the operator ${\mathcal{A}}:{D}({\mathcal{A}})\subset X \to X$ defined by generates a positive $C_0$-semigroup ${\mathcal{T}}:=({\mathcal{T}}(t))_{t\ge 0}$ on $X$. Moreover, for every initial condition $z_0\in X$ and every input function $u\in {L}^q_{lo

Figures (1)

  • Figure 1: Coupled heat equations with boundary disturbances.

Theorems & Definitions (24)

  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1: El3
  • Definition 3.2
  • Definition 3.3
  • Remark 3.1
  • Proposition 3.1: El3
  • ...and 14 more