The ill-posed Cauchy problem by Controllability

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Résumé

In this paper, we are dealing with the ill-posed Cauchy problem for a parabolic operator. To do this, we interpret the problem as an inverse problem, and therefore a controllability problem. This point of view induces a regularization method that makes it possible, on the one hand, to characterize the existence of a regular solution to the problem. On the other hand, this method makes it possible to obtain a singular optimality system for the optimal control, without using any additional assumptions, such as that of non-vacuity of the interior of the sets of admissible controls, an assumption that many analyses have had to use. From this point of view, the regularization method presented here, called controllability method, is original for the analyzed problem.

Mots-clés

Singular distributed system, optimal control, singular optimality system, the ill-posed Cauchy problem, controllability method, inverse problem