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
cannarsa@mat.uniroma2.it
Patrick Martinez
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse, France
patrick.martinez@math.univ-toulouse.fr
Judith Vancostenoble
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse, France
judith.vancostenoble@math.univ-toulouse.fr
[Abstract-pdf]
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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