Kun-Yi Yang

We consider a one-dimensional wave equation with local Kelvin-Voigt damping.For the continuous system, Zhang \cite{Zhang2010} has established exponential stabilityunder suitable assumptions on locally degenerate damping coefficients.In this paper, we employ the order-reduction finite difference method,which reformulates the equation as a first-order systemand naturally preserves the damping flux and interface conditions at the discrete level.Under corresponding discrete assumptions,we prove that the semi-discrete scheme generates a family of uniformly exponentially stable $C_0$-semigroups.The proof relies on a discrete frequency-domain multiplier technique.Numerical simulations validate the theoretical analysisand show that the standard finite difference scheme fails to preserve this uniform stability.This work extends the order-reduction methodology to the Kelvin-Voigt damping systemwith interior interface conditions,following previous applications to boundary-damped systems.

In this paper, we consider the stabilization of the coupled heat and wave equations under the static feedback or the dynamic feedback. Moreover, we make the coupled systems discretized by using the finite‐volume approach, and then we consider the stabilized properties of the discrete systems. First, for the coupled system under the static feedback, it is shown that the system is exponentially stable by using the Lyapunov method, and then the corresponding discrete system can be shown to be exponentially stable by constucting the discretized Lyapunov function. Second, for the coupled system under the dynamic feedback, we also show that both of the system and its discrete scheme are exponentially stable. Third, numerical simulations are given to show the effectiveness of the stable controllers.

In this paper, an unstable heat equation where the observation signal suffers from a given time delay has been considered. In order to remove the unstable boundary condition, the original system has been converted into the stabilizing system by backstepping transformation under which the form of the stabilizing controller can be obtained. For the sake of overcoming instability produced by the time delay, firstly, we construct the Luenberger observer and design the predictor. Then, the estimated output feedback controllers have been obtained based on the observer and predictor systems. It is shown that the closed-loop system is stable exponentially. At last, numerical simulations have been given to illustrate the effectiveness of the stabilizing output feedback controllers.

This paper considers a one-dimensional Euler-Bernoulli beam equation where two collocated actuators/sensors are presented at the internal point with pointwise feedback shear force and angle velocity at the arbitrary position ξ in the bounded domain (0,1). The boundary x = 0 is simply supported and at the other boundary x = 1 there is a shear hinge end. Both of the observation signals are subjected to a given time delay τ ( >0). Well-posedness of the open-loop system is shown to illustrate availability of the observer. An observer is then designed to estimate the state at the time interval when the observation is available, while a predictor is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controllers are introduced to guarantee the closed-loop system to be exponentially stable for the smooth initial values when ξ ∈ (0, 1) is a rational number satisfying ξ ≠ 2l∕(2m − 1) for any integers l, m. Simulation results demonstrate that the proposed feedback design effectively stabilizes the performance of the pointwise control system with time delay.

In this paper, we consider stabilization of a 1‐dimensional wave equation with variable coefficient where non‐collocated boundary observation suffers from an arbitrary time delay. Since input and output are non‐collocated with each other, it is more complex to design the observer system. After showing well‐posedness of the open‐loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed‐loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non‐collocated control.

Pointwise stabilisation is proposed in this paper for a string equation where the observation signal is subject to a time delay. Different from the boundary control, the feedback stabiliser is acting at the middle joint of the string. Well-posedness of the open-loop system and solvability of the observer are shown first. An observer system is then designed to estimate the state at the time interval when the observation is available, and a predictor system is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controller is introduced to make the closed-loop system asymptotically stable for the non-smooth initial values and exponentially stable for the smooth initial values, respectively. Simulation results demonstrate that the output feedback based on the observer and predictor effectively stabilises the pointwise control system with time delay.