Discrete Carleman Estimates for Elliptic Operators in Arbitrary Dimension and Applications

In arbitrary dimension, we consider the semi-discrete elliptic operator -2 t + AM , where AM is a finite difference approximation of the operator - x((x) x). For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate such as a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano for AM and a null controllability result for the parabolic operator t + AM , for the lower part of the spectrum of AM . With the control function that we construct (whose norm is uniformly bounded) we prove that the L2-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced. Key words. Elliptic operator

• Type: Journal
• Year: 2010
• Where: SIAMCO
• Authors: Franck Boyer, Florence Hubert, Jérôme Le Rousseau
• Keywords: Elliptic Operator, Operator, SIAMCO 2010, Spectral Inequality