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Finite-time stabilization via impulse control of degenerate singular parabolic equations

Walid Zouhair, Ghita El Guermai, Ilham Ouelddris

Abstract

This paper examines the impulse controllability of degenerate singular parabolic equations through a modern framework focused on finite-time stabilization. Furthermore, we provide an explicit estimate for the exponential decay of the solution. The proof of our main result combines a logarithmic convexity estimate with specific spectral properties. Finally, we establish the existence and uniqueness of the minimal norm impulse control associated with the system.

Finite-time stabilization via impulse control of degenerate singular parabolic equations

Abstract

This paper examines the impulse controllability of degenerate singular parabolic equations through a modern framework focused on finite-time stabilization. Furthermore, we provide an explicit estimate for the exponential decay of the solution. The proof of our main result combines a logarithmic convexity estimate with specific spectral properties. Finally, we establish the existence and uniqueness of the minimal norm impulse control associated with the system.

Paper Structure

This paper contains 7 sections, 10 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Functional setting
  3. Impulsive approximate controllability
  4. Finite time stabilization
  5. Proof of Theorem \ref{['thm1.4']}
  6. Uniqueness of minimal norm impulse control and its construction
  7. Conclusions and possible extensions

Key Result

Theorem 1.2

Consider $\tau_{k} = \frac{t_{k} + t_{k+1} }{2},$ with $t_{k}=T\left( 1-\dfrac{1}{b^k} \right)$ and $b>1$. Assume that the control subset $\omega$ satisfies cond. Assume that assump1 or assump2 is true. Then, the system 1.1 is finite-time stabilizable. Moreover, there exist positive constants $C,\ma Furthermore, $\lim _{k \rightarrow \infty}\left\|\mathcal{L}_{k}\left(y\left(\tau_{k}\right)\right)

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Corollary 1
  • Lemma 2.5
  • ...and 10 more