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On Robust Fixed-Time Stabilization of the Cauchy Problem in Hilbert Spaces

Moussa Labbadi, Christophe Roman, Yacine Chitour

Abstract

This paper presents finite-time and fixed-time stabilization results for inhomogeneous abstract evolution problems, extending existing theories. We prove well-posedness for strong and weak solutions, and estimate upper bounds for settling times for both homogeneous and inhomogeneous systems. We generalize finite-dimensional results to infinite-dimensional systems and demonstrate partial state stabilization with actuation on a subset of the domain. The interest of these results are illustrated through an application of a heat equation with memory term.

On Robust Fixed-Time Stabilization of the Cauchy Problem in Hilbert Spaces

Abstract

This paper presents finite-time and fixed-time stabilization results for inhomogeneous abstract evolution problems, extending existing theories. We prove well-posedness for strong and weak solutions, and estimate upper bounds for settling times for both homogeneous and inhomogeneous systems. We generalize finite-dimensional results to infinite-dimensional systems and demonstrate partial state stabilization with actuation on a subset of the domain. The interest of these results are illustrated through an application of a heat equation with memory term.
Paper Structure (13 sections, 7 theorems, 85 equations, 11 figures, 1 table)

This paper contains 13 sections, 7 theorems, 85 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Preliminaries
  4. Main results
  5. Full state fixed time, or solution with compact support with uniform upper bound.
  6. Partial state fixed time, or partial compact support with uniform upper bound
  7. Proof of the mains results
  8. Application heat with memory term
  9. Problem statement and result
  10. Numerical simulation
  11. Comparison of fixed-time and finite-time stabilization for the objective function
  12. Conclusions and future work
  13. Boundedness Is Insufficient for the Existence of Strong Solutions

Key Result

Theorem 1

Let $A$ be a maximal monotone relation on a Hilbert space $\mathtt{H}$ and $f \in W^{1,1}(0,T;\mathtt{H})$. Then for all $X_0 \in \mathcal{D}(A)$, there exists a unique $X(t): \mathbb{R}^+ \to \mathtt{H}$ such that $X(0)=X_0$, $\forall t\geq 0$, $X(t)\in \mathcal{D}(A)$, for a.e. $t\in[0,T)$, $\frac

Figures (11)

  • Figure 1: Distributed state $v(t,\cdot)$ for values $1$
  • Figure 2: Distributed control $U(t,\cdot)$ for values $1$
  • Figure 3: Objective $\log(\left\Vert X \right\Vert(t))$ for values $1$
  • Figure 4: Distributed state $v(t,\cdot)$for values $2$
  • Figure 5: Distributed control $U(t,\cdot)$ for values $2$
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Theorem 21 and subsequent remark in brezis1971monotonicity
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2: Theorem 1.3 barbu1993analysis
  • Theorem 3: Theorem 1.5 in barbu1993analysis
  • Definition 4
  • ...and 6 more