# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021004

## Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

 1 Laboratoire de Mathématiques UMR 6623, Université de Bourgogne Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France 2 University of Sciences and Technology Houari Boumedienne P.O.Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

* Corresponding author: Mokhtari Yacine

Received  July 2020 Revised  October 2020 Published  January 2021

In this paper, we deal with boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domains. By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of the solution are given explicitly.

Citation: Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, doi: 10.3934/eect.2021004
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The curve $(t,\alpha(t))_{t\geq0}$ in red and $(t, \beta(t))_{t\geq0}$ in blue
An example of a boundary curves $(t,\alpha(t)))_{t\geq0}$ and $(t,\beta(t)))_{t\geq0}$ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves
An example of a boundary curves $(t,\alpha(t)))_{t\geq0}$ and $(t,\beta(t)))_{t\geq0}$ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves
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