Control of the Cauchy System for an Elliptic Operator

---

Résumé

In this paper, we are dealing with the ill-posed Cauchy problem for an elliptic operator. A follow-up to a previous paper on the same subject (cf. [4]). Indeed, in [4], we introduced a regularization method, called the controllability method, which allowed us to know how to characterize, on the one hand, the existence of a regular solution to the ill-posed Cauchy problem and, on the other part outside, the optimal solution of the control problem, this without resorting to the Slater type assumption, an assumption to which many analyses had to resort. On occasion, we have dealt with the control problem, with state boundary observation, the problem initially analyzed by J. L. Lions in [6]. The proposed point of view, consisting of the interpretation of the Cauchy system as a system of two inverse problems, then called naturally for conjectures in favor of which the present manuscript wants to constitute an argument. Indeed, we conjectured, in view of the first results obtained, that the proposed method could be improved, from the point of view of the initial interpretation that we had made of the problem. In this sense, we analyze here two other variants (observation of the flow, then distributed observation) of the problem, the results of which confirm the intuition announced in [4]. Those results, it seems to us, are of relevance significant in the analysis of the controllability method introduced in [4].

Mots-clés

Singular distributed system, Optimal control, The ill-posed Cauchy Problem, Controllability, Inverse Problem