Control of the Hyperbolic Ill-posed Cauchy Problem by Controllability

---

Résumé

The main purpose of this paper is the control of the hyperbolic ill-posed Cauchy problem. To do this, we adapt to the present case the controllability method previously introduced in the stationary case (Bylli 2023). So 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 analyses have had to use.

Mots-clés

Controllability, Mathematics, Cauchy problem, Well-posed problem, Inverse problem, Regularization (linguistics), Applied mathematics, Optimal control, Cauchy distribution, Control (management), Hyperbolic partial differential equation, Initial value problem, Mathematical optimization, Mathematical analysis, Partial differential equation, Computer science