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

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 Lebeau and Robbiano (1995) [25], we deduce the construction of a control for the non-selfadjoint parabolic problem ∂tu+Au=Bg. In particular, the L2 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.