J. Nonl. Evol. Equ. Appl. 2022 (4), pp. 69-88, published on July 31, 2022:

The ill-posed Cauchy problem by Controllability - The parabolic Case

B. A. Guel, O. Nakoulima

Laboratoire MAINEGE, Université Ouaga 3S, Ouagadougou, Burkina Faso

Laboratoire LANIBIO, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso

Received on February 15, 2022, revised version on February 28, 2022
Accepted on February 28, 2022

Communicated by Gaston M. N'Guérékata

Abstract.  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 assumption, such as that of non-vacuity of the interior of the sets of admissible 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.
Keywords: Keywords: Singular Distributed System, Optimal Control, Singular Optimality System, The ill-posed Cauchy Problem, Controllability Method, Inverse Problem.
2010 Mathematics Subject Classification:   35Q93, 35R25, 35R30, 49J20, 93C05, 93C20.

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