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Boundary controllability for a fourth order degenerate parabolic equation with a singular potential

Leandro Galo-Mendoza

TL;DR

The work analyzes null controllability for a one-dimensional, fourth-order degenerate parabolic equation with a singular potential, controlled at the left boundary. It combines a weighted spectral decomposition in terms of Bessel zeros with semigroup theory to prove well-posedness, and uses the Fattorini–Russell moment method to obtain explicit upper and lower bounds on the cost of null controllability. A biorthogonal family is constructed via a Weierstrass product and complex multipliers to enable sharp cost estimates tied to spectral data ($j_{\nu,k}$, $J_{\nu}$). The results cover cases $\alpha+\beta<1$ and $>1$, and a separate treatment for $\alpha+\beta=1$, providing a comprehensive framework for boundary-controllable degenerate fourth-order equations with singular potentials. This advances understanding of control costs in degenerate parabolic settings and highlights the role of spectral geometry in boundary-control problems.

Abstract

In this paper, we prove the null controllability of a one-dimensional fourth-order degenerate parabolic equation with a singular potential. Here, we analyze cases where boundary control conditions are applied at the left endpoint. We utilize a spectral decomposition involving Bessel functions and their zeros in a convenient weighted Sobolev space for a degenerate parabolic operator with specific boundary conditions. We establish the well-posedness of the system using semigroup operator theory. Subsequently, we employ the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. Additionally, we derive a lower estimate of the cost of controllability using a representation theorem for analytic functions of exponential type.

Boundary controllability for a fourth order degenerate parabolic equation with a singular potential

TL;DR

The work analyzes null controllability for a one-dimensional, fourth-order degenerate parabolic equation with a singular potential, controlled at the left boundary. It combines a weighted spectral decomposition in terms of Bessel zeros with semigroup theory to prove well-posedness, and uses the Fattorini–Russell moment method to obtain explicit upper and lower bounds on the cost of null controllability. A biorthogonal family is constructed via a Weierstrass product and complex multipliers to enable sharp cost estimates tied to spectral data (, ). The results cover cases and , and a separate treatment for , providing a comprehensive framework for boundary-controllable degenerate fourth-order equations with singular potentials. This advances understanding of control costs in degenerate parabolic settings and highlights the role of spectral geometry in boundary-control problems.

Abstract

In this paper, we prove the null controllability of a one-dimensional fourth-order degenerate parabolic equation with a singular potential. Here, we analyze cases where boundary control conditions are applied at the left endpoint. We utilize a spectral decomposition involving Bessel functions and their zeros in a convenient weighted Sobolev space for a degenerate parabolic operator with specific boundary conditions. We establish the well-posedness of the system using semigroup operator theory. Subsequently, we employ the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. Additionally, we derive a lower estimate of the cost of controllability using a representation theorem for analytic functions of exponential type.
Paper Structure (9 sections, 14 theorems, 133 equations)

This paper contains 9 sections, 14 theorems, 133 equations.

Table of Contents

  1. Introduction and main results
  2. Functional setting and well-posedness
  3. Existence and uniqueness of solutions to system (\ref{['problem']})
  4. The biorthogonal family and its multiplier
  5. Proof of Theorem \ref{['Teo1']}: Control at the left endpoint
  6. Upper estimate of the cost of the null controllability
  7. Lower estimate of the cost of the null controllability
  8. Comments about case $\alpha+\beta = 1$
  9. Bessel functions

Key Result

Theorem 1

Let $T>0$ and $\alpha,\beta, \mu, \ell, \gamma \in \mathbb{R}$ with $0\leq\alpha<2$, $\mu$, $\ell$ and $\gamma$ satisfying (mucon), (index_l) and (gamma) respectively, and $r\in\{0,1\}$. The next statements hold. where $j_{\nu,2}$ is the second positive zero of the Bessel function $J_\nu$.

Theorems & Definitions (25)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 5
  • proof
  • Definition 6
  • Proposition 7
  • proof
  • Remark 8
  • ...and 15 more