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The cost of boundary controllability for a parabolic equation with inverse square potential. (English) Zbl 1425.93045

Summary: The goal of this paper is to analyze the cost of boundary null controllability for the \(1-D\) linear heat equation with the so-called inverse square potential:
\[u_t - u_{xx} - \frac{\mu}{x^2} u = 0, \qquad x \in (0,1),\ t \in (0,T),\]
where \(\mu\) is a real parameter such that \(\mu \leq 1/4\). Since the works by P. Baras and J. A. Goldstein [North-Holland Math. Stud. 92, 31–35 (1984; Zbl 0566.35035); Trans. Am. Math. Soc. 284, 121–139 (1984; Zbl 0556.35063)], it is known that such problems are well-posed for any \(\mu \leq 1/4\) (the constant appearing in the Hardy inequality) whereas instantaneous blow-up may occur when \(\mu > 1/4\). For any \(\mu \leq 1/4\), it has been proved in [J. Vancostenoble and E. Zuazua, J. Funct. Anal. 254, No. 7, 1864–1902 (2008; Zbl 1145.93009)] (via Carleman estimates) that the equation can be controlled (in any time \(T > 0)\) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at \(x = 1\). The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two parameters \(T > 0\) and \(\mu \in (-\infty, 1/4]\). Our proofs are based on the moment method and very recent results on biorthogonal sequences.

MSC:

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35K05 Heat equation
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References:

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