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Minimax Theory and its Applications 06 (2021), No. 2, 251--280
Copyright Heldermann Verlag 2021

Sharp Estimate of the Cost of Controllability for a Degenerate Parabolic Equation with Interior Degeneracy

Piermarco Cannarsa
Dipartimento di Matematica, Università di Roma "Tor Vergata", 00133 Roma, Italy

Patrick Martinez
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse, France

Judith Vancostenoble
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse, France


This work is motivated by the study of null controllability for the typical degenerate parabolic equation with interior degeneracy and one-sided control: $$ u_t - (\vert x \vert ^\alpha u_x)_x = h(x,t) \chi _{(a,b)} , \quad x\in (-1,1) , $$ with 0 less than $a$ less than $b$ less than $1$. It was proved in \cite{CFM} that this equation is null controllable (in any positive time $T$) if and only if $\alpha$ less than 1, and that the cost of null controllability blows up as $\alpha \to 1^-$. This is related to the following property of the eigenvalues: the gap between an eigenvalue of odd order and the consecutive one goes to $0$ as $\alpha \to 1^-$ (see P.\,Cannarsa, R.\,Ferretti, P.\,Martinez: {\it Null controllability for parabolic operators with interior degeneracy and one-sided control}, SIAM J. Control Optimization 57/2 (2019) 900--924.).\\[1mm] The goal of the present work is to provide optimal upper and lower estimates of the null controllability cost, with respect to the degeneracy parameter (when $\alpha \to 1^-$) and in short time (when $T\to 0^+$). We prove that the null controllability cost behaves as $1/(1-\alpha)$ as $\alpha \to 1^-$ and as $e^{1/T}$ as $T\to 0^+$. Our analysis is based on the construction of a suitable family biorthogonal to the sequence $(e^{\lambda _n t})_n$ in $L^2(0,T)$, under some general gap conditions on the sequence $(\lambda _n)_n$, conditions that are suggested by a motivating example.

Keywords: Controllability, degenerate parabolic equation, biorthogonal family.

MSC: 35K65, 93B05, 93B60, 33C10, 35P10, 30D15.

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