The ill-posed Cauchy problem by Controllability the elliptic case

Abstract

In this paper, we are dealing with the ill-posed Cauchy problem for an elliptic 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 resorting to any additional assumption, such as that of non-vacuity of the interior of the sets of admissibles controls, an assumption that many analysis have had to use. From this point of view, the regularization method presented here, called controllability method, is original for the analyzed problem.