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 $A$. |