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Logarithmic Convexity and impulse Approximate Controllability for Degenerate Parabolic Equations with Robin Boundary Conditions

Hind El Baggari, Ilham Ouelddris

TL;DR

This work addresses approximate controllability for a one-dimensional degenerate parabolic equation with Robin boundary conditions at the degeneracy point, subject to an impulsive control in a small region at time \\tau. The authors develop a Carleman commutator-based logarithmic convexity framework to establish a single-time observability inequality for the adjoint system, which they then use to derive null approximate impulse controllability for all final times \\ T, with an explicit cost bound \\mathcal{P}(T,\\epsilon) \\le \\frac{1}{\\epsilon^{\\beta}} e^{\\mathcal{C}(1+1/(T-\\tau))}. The analysis hinges on weighted Sobolev spaces \\ H^1_\\alpha, \\ H^2_\\alpha and a carefully chosen weight \\phi to control boundary terms under Robin conditions. The results provide a constructive, quantitative path to controllability in degenerate parabolic systems with convective boundary effects and impulsive actuation, while highlighting open questions for interior-only control regions.

Abstract

In this work, we investigate the approximate controllability of a class of one-dimensional degenerate parabolic equations with Robin boundary conditions. The degeneracy occurs at one endpoint of the spatial domain, and we apply an impulsive control in a small region at a fixed moment. Our main result establishes an observability inequality for the adjoint system, from which we deduce approximate controllability at final time . The proof relies on a logarithmic convexity argument, developed through a Carleman commutator approach.

Logarithmic Convexity and impulse Approximate Controllability for Degenerate Parabolic Equations with Robin Boundary Conditions

TL;DR

This work addresses approximate controllability for a one-dimensional degenerate parabolic equation with Robin boundary conditions at the degeneracy point, subject to an impulsive control in a small region at time \\tau. The authors develop a Carleman commutator-based logarithmic convexity framework to establish a single-time observability inequality for the adjoint system, which they then use to derive null approximate impulse controllability for all final times \\ T, with an explicit cost bound \\mathcal{P}(T,\\epsilon) \\le \\frac{1}{\\epsilon^{\\beta}} e^{\\mathcal{C}(1+1/(T-\\tau))}. The analysis hinges on weighted Sobolev spaces \\ H^1_\\alpha, \\ H^2_\\alpha and a carefully chosen weight \\phi to control boundary terms under Robin conditions. The results provide a constructive, quantitative path to controllability in degenerate parabolic systems with convective boundary effects and impulsive actuation, while highlighting open questions for interior-only control regions.

Abstract

In this work, we investigate the approximate controllability of a class of one-dimensional degenerate parabolic equations with Robin boundary conditions. The degeneracy occurs at one endpoint of the spatial domain, and we apply an impulsive control in a small region at a fixed moment. Our main result establishes an observability inequality for the adjoint system, from which we deduce approximate controllability at final time . The proof relies on a logarithmic convexity argument, developed through a Carleman commutator approach.
Paper Structure (7 sections, 5 theorems, 115 equations)

This paper contains 7 sections, 5 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Well-Posedness of the system
  3. Logarithmic Convexity via the Carleman commutator approach
  4. The weight function
  5. Proof of the observation inequality
  6. Null approximate impulse controllability
  7. Conclusion

Key Result

Lemma 1.3

Assume that Hypothesis hypo1 holds true. Let us consider $\omega$ a sub-interval of $(0,1)$ that satisfies the geometric condition cond. Then, there exists a positive constant $\mathcal{C}$ and $\rho \in (0,1)$ such that the following observation estimate is satisfied where $u$ is the solution of the following non-impulsive system

Theorems & Definitions (12)

  • Definition 1.1: see Definition 1.2 of qin
  • Definition 1.2
  • Lemma 1.3
  • Proposition 1
  • proof : Proof of Theorem \ref{['thm1']}
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Remark 1
  • ...and 2 more