Existence and cost of boundary controls for a degenerate/singular parabolic equation

U. Biccari; V. Hernández-Santamaría; J. Vancostenoble

doi:10.3934/mcrf.2021032

Article Contents

2022, Volume 12, Issue 2: 495-530. Doi: 10.3934/mcrf.2021032

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Existence and cost of boundary controls for a degenerate/singular parabolic equation

1.
Chair of Computational Mathematics, Fundación Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, Facultad de Ingeniería, Universidad de Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain
2.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C.P. 04510 CDMX, Mexico
3.
Institut de mathématiques de Toulouse, UMR5219; Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

^* Corresponding author: Umberto Biccari
^* Corresponding author: Umberto Biccari

Received: January 2020

Revised: December 2020

Early access: June 2021

Published: June 2022

This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon). The work of U. B. was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), by the Elkartek grant KK-2020/00091 CONVADP and by the Air Force Office of Scientific Research (AFOSR) under Award NO. FA9550-18-1-0242. The work of V. H.-S. was supported by the programme "Estancias posdoctorales por México" of CONACyT, Mexico

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Abstract

In this paper, we consider the following degenerate/singular parabolic equation

$ \begin{align*} u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u = 0, \qquad x\in (0,1), \ t \in (0,T), \end{align*} $

where $ 0\leq \alpha <1 $ and $ \mu\leq (1-\alpha)^2/4 $ are two real parameters. We prove the boundary null controllability by means of a $ H^1(0,T) $ control acting either at $ x = 1 $ or at the point of degeneracy and singularity $ x = 0 $. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters $ \alpha $ and $ \mu $. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.

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Mathematics Subject Classification: 30E05, 35K65, 35K67, 93B05, 93B60.

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