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Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications

Abstract : In arbitrary dimension, we consider the semi-discrete elliptic operator $- \d_t^2 + \Am$, where $\Am$ is a finite difference approximation of the operator $-\nabla_x (\Gamma(x) \nabla_x)$. 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 $A^m$ and a null controllability result for the parabolic operator $\partial_t + A^m$, for the lower part of the spectrum of $A^m$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-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.
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Franck Boyer, Florence Hubert, Jérôme Le Rousseau. Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (8), pp. 5357-5397. ⟨10.1137/100784278⟩. ⟨hal-00450854v2⟩

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