Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems

Abstract : We consider elliptic operators $A$ on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of $A$ through an observation, with an exponential cost. Following the strategy of G. Lebeau and L. Robbiano (1995), we deduce the construction of a control for the non-selfadjoint parabolic problem $\partial_t u + A u = B g$. In particular, the $L^2$ norm of the control that achieves the extinction of the lower modes of $A$ is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of $A$.
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Journal of Functional Analysis, Elsevier, 2010, 258, pp.2739-2778. 〈10.1016/j.jfa.2009.10.011〉
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Matthieu Léautaud. Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. Journal of Functional Analysis, Elsevier, 2010, 258, pp.2739-2778. 〈10.1016/j.jfa.2009.10.011〉. 〈hal-00421742v2〉

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