Exact Controllability of Coupled Wave Systems with Transmission Interfaces
Keywords:
Cone, HUM method, Exact controllability, Observability inequality, Transmission problemsAbstract
This work focuses on the analysis of observability and exact controllability for a locally transmitted system, in which an internal control is applied to the second wave problem, which is strongly coupled. First, we establish an observability inequality by employing a result due to A. Haraux [3]. Then, using the Hilbert Uniqueness Method (HUM in short) developed by J. L. Lions [9], we demonstrate that the system is exactly controllable.
[2010] 35Q74, 93D15, 93D15, 74J30
References
[1] Ben Aissa A, Ahmedi W. Stabilization of a locally transmission problems of two strongly-weakly coupled wave systems. Asymptot Anal. 2024;1-31. doi:10.3233/ASY-241939.
[2] Aissa AB. Weak controllability of second order evolution systems and applications. arXiv Prepr. 2013; arXiv:1306.4833.
[3] Haraux A. Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port Math. 1989;46:245-58.
[4] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag; 1983. (Applied Mathematical Sciences; vol. 44).
[5] Wehbe A, Youssef W. Indirect locally internal observability of weakly coupled wave equations. Differ Equ Appl. 2011;3(3):449-62.
[6] Alabau F. Observabilité frontière indirecte de systèmes faiblement couplés. C R Acad Sci Paris Sér I Math. 2001;333(7):645-50.
[7] Alabau-Boussouira F. A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J Control Optim. 2003;42(3):871-906.
[8] Akil M, Hajjej Z. Exponential stability and exact controllability of a system of coupled wave equations by second-order terms (via Laplacian) with only one non-smooth local damping. Math Methods Appl Sci. 2023;1-20.
[9] Lions JL. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués: Perturbations. Paris: Masson; 1988. (Recherches en Mathématiques Appliquées).
[10] Gerbi S, Kassem C, Mortada A, et al. Exact controllability and stabilization of locally coupled wave equations: theoretical results. Z Anal Anwend. 2021;40:67-96.
