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- Title
Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay.
- Authors
Ayadi, Habib; Jlassi, Mariem
- Abstract
In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval |$[0,2\pi ]$| and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of |$L^{2}(0,2\pi)$| -norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.
- Subjects
KORTEWEG-de Vries equation; EQUATIONS of state; BACKSTEPPING control method; EXPONENTIAL stability; NONLINEAR equations
- Publication
IMA Journal of Mathematical Control & Information, 2023, Vol 40, Issue 3, p516
- ISSN
0265-0754
- Publication type
Article
- DOI
10.1093/imamci/dnad016