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Article Dans Une Revue SIAM Journal on Control and Optimization Année : 2010

Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications

Résumé

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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Dates et versions

hal-00450854 , version 1 (27-01-2010)
hal-00450854 , version 2 (27-08-2010)

Identifiants

  • HAL Id : hal-00450854 , version 1

Citer

Franck Boyer, Florence Hubert, Jérôme Le Rousseau. Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications. SIAM Journal on Control and Optimization, 2010, to appear, pp. ⟨hal-00450854v1⟩

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