We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut (xux)x with 0 < x < ` and 2 (0; 2), controlled either by a boundary control acting at x = `, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter . We prove that the control cost blows up with an explicit exponential rate, as eC=((2)2T), when ! 2 and/or T ! 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, G uichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new ne properties of the Bessel functions J of large order (obtained by ordinary dierential equations techniques).

Cannarsa, P., Martinez, P., Vancostenoble, J. (2020). Precise estimates for biorthogonal families under asymptotic gap conditions. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 13(5), 1441-1472 [10.3934/dcdss.2020082].

Precise estimates for biorthogonal families under asymptotic gap conditions

Cannarsa P.
;
2020-01-01

Abstract

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut (xux)x with 0 < x < ` and 2 (0; 2), controlled either by a boundary control acting at x = `, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter . We prove that the control cost blows up with an explicit exponential rate, as eC=((2)2T), when ! 2 and/or T ! 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, G uichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new ne properties of the Bessel functions J of large order (obtained by ordinary dierential equations techniques).
2020
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
Settore MATH-03/A - Analisi matematica
English
Con Impact Factor ISI
Degenerate parabolic equations; null controllability; moment problem; Bessel functions
Cannarsa, P., Martinez, P., Vancostenoble, J. (2020). Precise estimates for biorthogonal families under asymptotic gap conditions. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 13(5), 1441-1472 [10.3934/dcdss.2020082].
Cannarsa, P; Martinez, P; Vancostenoble, J
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/263786
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