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Sharp estimates of the one-dimensional boundary control cost for parabolic systems

Manuel González-Burgos, Lydia Ouaili

Abstract

In this work we present new results on the cost of the boundary controllability of parabolic systems at time $T > 0$. In particular, we will study optimal estimates of the control cost at time $T$ ($T$ small enough) when the eigenvalues of the generator of the $C_0$ semigroup accumulate and do not satisfy a gap condition. The main ingredient we will use is the moment method combined with sharp estimates of the $L^2(0,T; \mathbb C)$-norm of the elements of biorthogonal families to complex exponentials.

Sharp estimates of the one-dimensional boundary control cost for parabolic systems

Abstract

In this work we present new results on the cost of the boundary controllability of parabolic systems at time . In particular, we will study optimal estimates of the control cost at time ( small enough) when the eigenvalues of the generator of the semigroup accumulate and do not satisfy a gap condition. The main ingredient we will use is the moment method combined with sharp estimates of the -norm of the elements of biorthogonal families to complex exponentials.
Paper Structure (8 sections, 16 theorems, 65 equations)

This paper contains 8 sections, 16 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. The one-dimensional heat equation: the moment method
  3. A boundary controllability problem for a parabolic system with complex eigenvalues
  4. The boundary controllability of a phase-field system
  5. The main result: Bounds on biorthogonal families to complex exponentials without gap condition
  6. Control cost for some parabolic systems
  7. The linear phase-field system
  8. A $2 \times 2$ linear coupled parabolic system

Key Result

Theorem 2.1

Let us consider problems f1 and f2. Then, for any $y_0 \in H^{-1} (0, \pi)$ and $\varphi_T \in H_0^{1} (0, \pi)$, one has where $y \in L^2(Q_T) \cap C^0( [0,T] ; H^{-1} (0, \pi) )$ and are, resp., the solutions to f1 and f2 associated to $y_0$ and $v$, and $\varphi_T$, and $\langle \cdot, \cdot \rangle$ stands for the duality product between $H^{-1} (0, \pi)$ and $H_0^1 (0, \pi)$.

Theorems & Definitions (29)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4: Güichal
  • Remark 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Remark 3.1
  • Remark 3.2
  • ...and 19 more