Control of Cauchy Problem for a Laplacian Operator

Abstract

In this paper we study the control of an ill-posed system relating to the Cauchy problem for an elliptical operator. The control of Cauchy systems for an elliptical operator has already been studied by many authors. But it still seems to be globally an open problem. Of all the studies that have been done on this problem, it is assumed that the set of admissible couple-state must be nonempty to make sense of the problem. This is the case of J. L. Lions in [6] who gave various examples of the admissible set to make a sense of the problem. O. Nakoulima in [9] uses the regularization-penalization method to approach the problem by a sequence of well-posed control problems, he obtains the convergence of the processus in a particular case of the admissible set. G. Mophou and O. Nakoulima in [10] do the same study and obtain the convergence of the processus when the interior of the admissible set is non empty. In this work, we give an approximate solution without an additional condition on the set of admissible couple-state.We propose a method which consists in associating with the singular control problem a "family" of controls of well posed problems. We propose as an alternative the stackelberg control which is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12].