Interior feedback stabilization of wave equations with dynamic boundary delay

In this paper we consider an interior stabilization problem for the wave equation with dynamic boundary delay.We prove some stability results under the choice of damping operator. The proof of the main result is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent

Stabilization and Disturbance Rejection for the Beam Equation

Cataloged from PDF version of article.We consider a system described by the Euler–Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate

Stability analysis of laminated beam systems with delay using lyapunov functional

This work is concerned with systems of laminated beams model subject to linear and nonlinear delay feedback. In a dynamic laminated beam, time delay manifests in the form of lags in restoring the desired system stability after perturbations. Four prevalent categories of time delay are considered. For laminated beams with relatively high adhesive stiffness, a constant delay feedback is considered for systems made up of individual beams with same elasticity, and neutral delay otherwise. In systems where delay is significantly due to adhesive softening, distributed delay is considered. Lastly, in structures where the mechanism of dissipating energy is nonlinear, a corresponding nonlinear delay effect is investigated. The mechanism of stabilization mainly relies on the intrinsic structural damping, unlike in previous works where researchers introduced additional dampings such as boundary feedback and material damping. The objective of this work is to establish the asymptotic behavior of a vibrating Timoshenko laminated beam using structural or utmost a single frictional damping in presence of different forms of time delay. The energy method for partial differential equations is the main tool used to establish wellposedness results and asymptotic behavior. The existence and uniqueness of the solution is proved using the linear semi group theory, whereas for energy decay properties, the multiplier technique involving constructing a suitable Lyapunov functional equivalent to the energy is utilized. With appropriate assumptions on the delay weight and wave speeds, it is established that the energy of the system at least decays exponentially due to structural damping. Furthermore, a single additional frictional damping guarantees polynomial decay despite the presence of constant or distributed delay feedback. For nonlinear structural damping, with help of some convexity arguments, general decay result is achieved. In summary, depending on the damping mechanism(s), exponential, polynomial, or general decay results of a laminated beam system subject to different forms of delay feedback are established

On the Stabilization and Stability Robustness Against Small Delays of Some Damped wave equations

Cataloged from PDF version of article.In this note we consider a system which can be modeled by two different one-dimensional damped wave equations in a bounded domain, both parameterized by a nonnegative damping constant. We assume that the system is fixed at one end and is controlled by a boundary controller at the other end. We consider two problems, namely the stabilization and the stability robustness of the closed-loop system against arbitrary small time delays in the feedback loop. We propose a class of dynamic boundary controllers and show that these controllem solve the stabilization problem when the damping cuefMent is nonnegative and stability robustness problem when the damping coefficient is strictly positive

General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback

In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.Comment: 11 page