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Controllability for a 2x2 nonlinear degenerate parabolic system via one boundary control force

Margarita Arias, Abdelkarim Hajjaj, Amine Sbai

Abstract

In this paper we study the local boundary controllability for a non linear system of two degenerate parabolic equations with a control acting on only one equation. We analyze boundary null controllability properties for the linear system via the moment method by Fattorini and Russell, together with some results on biorthogonal families. Moreover, we provide an estimate on the null-control cost. This estimate let us prove a local exact boundary controllability result to zero of the nonlinear system following the iterative method from Lebeau and Robbiano as in \cite{Burgos_2020, Liu_2012}.

Controllability for a 2x2 nonlinear degenerate parabolic system via one boundary control force

Abstract

In this paper we study the local boundary controllability for a non linear system of two degenerate parabolic equations with a control acting on only one equation. We analyze boundary null controllability properties for the linear system via the moment method by Fattorini and Russell, together with some results on biorthogonal families. Moreover, we provide an estimate on the null-control cost. This estimate let us prove a local exact boundary controllability result to zero of the nonlinear system following the iterative method from Lebeau and Robbiano as in \cite{Burgos_2020, Liu_2012}.

Paper Structure

This paper contains 14 sections, 13 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Functional framework
  4. Well-posedness
  5. Spectral properties of the operators
  6. Spectral properties of scalar degenerate operators
  7. Spectral properties of vectorial degenerate operators
  8. Boundary controllability of the linear system \ref{['bound-sys']}
  9. Boundary controllability of the non linear system
  10. Null controllability of the non-homogeneous system \ref{['bound-nonhomog-sys']}
  11. Proof of Theorem \ref{['thm-local-exact-control']}
  12. Appendix
  13. Proof of Proposition \ref{['well.posed.transp']}
  14. Spectral properties

Key Result

Theorem 1.1

Let $\alpha \in [0,2)$ and denote $\mu_1$ and $\mu_2$ the eigenvalues of $A$. Then system bound-sys is null controllable at any time $T >0$ if and only if rank holds and also Moreover, there exist two positive constants $C_0$ and $M$ independent of $T$, such that for any $T>0$ there is a bounded linear operator $\mathcal{C}_{T}^{(0)}: H_{\alpha}^{-1}(0,1)^2 \rightarrow L^{2}(0, T)$ satisfying an

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.2
  • ...and 11 more