A novel observer-based approach for the exponential stabilization of the string PDE with cubic nonlinearities

• Published in: Systems & control letters 2022-07, Vol.165, p.105273, Article 105273
• Main Author: Ghaderi, Najmeh
• Format: Article
• Language: English
• Subjects: Boundary control systems; Exponential stability; Nonlinear boundary condition; Observer design; Unstable string PDEs
• ISSN: 0167-6911; 1872-7956
• DOI: 10.1016/j.sysconle.2022.105273

The main objective of this study is to design an observer-based exponential output feedback controller for a one-dimensional vibrating string PDE with in-domain sources, and both anti-damper and anti-spring parameters and also both cubic velocity and nonlinear displacement at the boundary. Due to in-domain sources, the energy injection, cubic displacement and velocity nonlinearity, the unforced system is unstable and may represent a variety of dynamical behaviors. An exponential state feedback controller is first designed and also the well-posedness and the exponential stability of the closed-loop system is investigated. In addition, a novel exponential state observer is constructed, and then an observer-based exponential output feedback controller is proposed. At the end, the well-posedness and the exponential stability of the corresponding closed-loop system are obtained. The validity of the theoretical results is shown by numerical simulations. •Designing an exponential state feedback controller for a vibrating string with indomain sources, negative spring and damper, and two cubic nonlinearities.•Owing to the both negative spring and damper as well as cubic nonlinearities, the unforced plant is unstable and may cause chaotic oscillation and some other dynamical behaviors.•Presenting a new observer to obtain the exponential estimate of the real system containing indomain sources and cubic nonlinearities.•Designing an observer-based output feedback controller with exponential convergence for the underlying string PDE model.•Confirming the efficiency and validity performance of the proposed output feedback control method by simulation results.