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Rapid stabilization of general linear systems with F-equivalence

Amaury Hayat, Epiphane Loko

Abstract

We study the rapid stabilization of general linear systems, when the differential operator $\mathcal{A}$ has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an $F$-equivalence approach \textcolor{black}{relying on Fredholm transformation} to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew-adjoint systems.

Rapid stabilization of general linear systems with F-equivalence

Abstract

We study the rapid stabilization of general linear systems, when the differential operator has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an -equivalence approach \textcolor{black}{relying on Fredholm transformation} to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew-adjoint systems.
Paper Structure (30 sections, 33 theorems, 240 equations)

This paper contains 30 sections, 33 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Main results
  4. Link with controllability and existing conditions for stabilizability
  5. Application to Schrödinger equation around the ground state
  6. Application to parabolic systems
  7. Heat equation
  8. A nonlinear system: Burgers equation
  9. General diffusion equation
  10. Application to a Gribov operator in the Bargmann space
  11. Strategy and outline
  12. Proof of Theorem \ref{['Theo_main_general']}
  13. Step 1: S is a Fredholm operator
  14. Step 2: $S$ is an isomorphism from $\mathcal{H}^0$ to itself
  15. Step 3: $S$ is an isomorphism from $\mathcal{H}^{r}$ to itself for any $r\in(1/2-\alpha,\alpha-1/2)$
  16. ...and 15 more sections

Key Result

LEMMA 2.1

For any $s \in\mathbb{R}$, and $i \in \{1,...,m\}$ , the function is a norm of $\mathcal{H}^{s}_{i}$ and ($\mathcal{H}^{s}_{i}$,$\|\cdot\|_{\mathcal{H}^{s}_{i}}$) is a Hilbert space, with inner product

Theorems & Definitions (67)

  • LEMMA 2.1
  • LEMMA 2.2
  • LEMMA 2.3
  • DEFINITION 2.4: Exponential stability
  • REMARK 2.5
  • THEOREM 3.1
  • REMARK 3.2: Regularity of the feedback operator
  • THEOREM 3.3
  • REMARK 3.4
  • COROLLARY 3.5: Rapid stabilization
  • ...and 57 more