Conditional Stability and Uniqueness for Determining Coefficient in Some Multidimensional Consolidation Models

Résumé

In this paper, we study an inverse problem of reconstruction spatially varying coefficient in a nonlinear Biot’s consolidation model with the following observation data: both displacement and pressure in a subdomain ω⊂Ω. First the given problem is transformed into an optimization problem by using optimal control framework and we establish the existence of minimizer for the control functional. The solution of the optimization problem is based on a non-linear conjugate gradient method. Moreover, the well-posedness of the adjoint problem and the first order necessary optimality conditions are shown. The convergence proof of the adjoint problem is based on using a general compactness criterion. Based on the necessary optimality condition, we prove the Lipschitz stability and the uniqueness for the inverse problem under some a priori information.

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