Spectral inequalities for the control of linear PDEs

Abstract : Some spectral inequalities were introduced in control theory by David Russell and George Weiss in 1994 [21] as an infinite-dimensional version of the Hautus test for controllability. They are an efficient tool for the control of the linear Schrödinger equation in arbitrary time from a localized source term as proved by Nicolas Burq and Maciej Zworski in 2004 [1] using the unitarity of the Fourier transform in Hilbert spaces. They also allow to analyze which filtering scale is sufficient to discretize this equation in space, as initiated by Sylvain Ervedoza in 2008 [5]. A parallel approach to the control of the linear heat equation in arbitrary time from a localized source term has developed. It starts from another type of spectral inequality introduced by Gilles Lebeau in the late 90s and follows the strategy devised in 1995 by Lebeau and Robbiano [9]. This paper will connect these two spectral approaches, compare the control of the Schrödinger group and the heat semigroup at the level of abstract functional analysis, and illustrate this with examples of PDE problems. It is mainly based on a joint work [4] with Thomas Duyckaerts (Université Paris 13).
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Kaïs Ammari; Gilles Lebeau. PDE's, dispersion, scattering theory and control theory, Jun 2013, Monastir, Tunisia. Société Mathématique de France, Séminaires et Congrès [Seminars and Congresses], 30, pp.81--98, 2017, 〈http://smf4.emath.fr/Publications/SeminairesCongres/2017/30/html/smf_sem-cong_30.php〉
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Luc Miller. Spectral inequalities for the control of linear PDEs. Kaïs Ammari; Gilles Lebeau. PDE's, dispersion, scattering theory and control theory, Jun 2013, Monastir, Tunisia. Société Mathématique de France, Séminaires et Congrès [Seminars and Congresses], 30, pp.81--98, 2017, 〈http://smf4.emath.fr/Publications/SeminairesCongres/2017/30/html/smf_sem-cong_30.php〉. 〈hal-01729608〉

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