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Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. (English) Zbl 1185.35153
Summary: 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 [Commun. Partial Differ. Equations 20, No. 1–2, 335–356 (1995; Zbl 0819.35071)], we deduce the construction of a control for the non-selfadjoint parabolic problem \(\partial _tu+Au=Bg\). 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 \(A\).

35P15 Estimates of eigenvalues in context of PDEs
35K15 Initial value problems for second-order parabolic equations
47F05 General theory of partial differential operators
47A10 Spectrum, resolvent
49J20 Existence theories for optimal control problems involving partial differential equations
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