Jun-Min Wang

Stabilization and Gevrey Regularity of a Schrödinger Equation in Boundary Feedback With a Heat Equation

We study stability of a Schrödinger equation with a collocated boundary feedback compensator in the form of a heat equation with a collocated input/output pair. Remarkably, exponential stability is achieved for both positive and negative gains, namely, as long as the gain is non-zero. We show that the spectrum of the closed-loop system consists only of two branches along two parabolas which are asymptotically symmetric relative to the line Reλ = -Imλ (the 135° line in the second quadrant). The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class δ >; 2. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.

On the C0-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam

In this paper, we show that a linear unbounded operator associated with an Euler–Bernoulli beam equation under shear boundary feedback generates a C0-semig roup in the underlyingstate Hilbert space. This provides an answer to a longtime unsolved problem due to the lack of dissipativity for the operator. The main steps are a careful estimation of the Green’s function and the verification of the Riesz basis property for the generalized eigenfunctions. As a consequence, we show that this semigroup is differentiable and exponentially stable, which is in sharp contrast to the properties possessed by most feedback controlled beams based on a passive design principle.

Stabilization of an ODE–Schrödinger Cascade

In this paper, we consider a problem of stabilization of an ODE-Schrodinger cascade, where the interconnection between them is bi-directional at a single point. By using the backstepping approach, which uses an invertible Volterra integral transformation together with the boundary feedback to convert the unstable plant into a well-damped target system, the target system in our case is given as a PDE-ODE cascade with exponential stability at the pre-designed decay rate. Instead of one-step backstepping control, which results in difficulty in finding the kernels, we develop a two-step backstepping control design by introducing an intermediate target system and an intermediate control. The exponential stability of the closed-loop system is investigated using the Lyapunov method. A numerical simulation is provided to illustrate the effectiveness of the proposed design.

Dynamic stabilization of an Euler–Bernoulli beam under boundary control and non-collocated observation☆

Abstract We study the dynamic stabilization of an Euler–Bernoulli beam system using boundary force control at the free end and bending strain observation at the clamped end. We construct an infinite-dimensional observer to track the state exponentially. A proportional output feedback control based on the estimated state is designed. The closed-loop system is shown to be non-dissipative but admits a set of generalized eigenfunctions, which forms a Riesz basis for the state space. As consequences, both the spectrum-determined growth condition and exponential stability are concluded.

Exponential Stabilization of Laminated Beams with Structural Damping and Boundary Feedback Controls

We study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two-layered objects. By using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation, we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs. Furthermore, we prove that there is a sequence of generalized eigenfunctions that forms a Riesz basis in the state Hilbert space, and hence the spectrum determined growth condition holds. Furthermore, exponential stability of the closed-loop system can be deduced from the eigenvalue expressions. In particular, the semigroup generated by the system operator is a

Spectral analysis and system of fundamental solutions for Timoshenko beams

C_0

Active disturbance rejection control and sliding mode control of one-dimensional unstable heat equation with boundary uncertainties

-group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis.

Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time

We have found a unified method to analyse Timoshenko beams under various boundary conditions that occurred in practice. Explicit asymptotic expressions for the spectrum are obtained. Our method is very simple but effective because explicit formulas are obtained for the system of fundamental solutions, which are very useful for other purposes such as stability analysis. The eigenfunctions are also shown to form an orthogonal basis.

Riesz Basis Generation of Abstract Second-Order Partial Differential Equation Systems with General Non-Separated Boundary Conditions

In this paper, we are concerned with the boundary stabilization of a one-dimensional unstable heat equation with the external disturbance flowing into the control end. The active disturbance rejection control (ADRC) and the sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an external observer and cancelled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance with the assumption in which the disturbance is supposed to be bounded. The reaching condition, and the existence and uniqueness of the solution for all states in the state space via SMC are established. Simulation examples are presented for both control strategies.

On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type

For a string equation with time delays in the output feedback loop, we study stability and show that the system is a Riesz spectral system and prove that the spectrum-determined growth condition holds for all delays. When the delay is equal to the even multiples of the wave propagation time, we develop the necessary and sufficient conditions for the feedback gain and time delay which guarantee the exponential stability of the closed-loop system. In particular, we show that as the delay of even multiples is increasing to infinity, the stability bound on the feedback gain decays to zero. We also show that whenever the delay is an odd multiple of the wave propagation time, the closed-loop system is unstable. The lack of robustness to a small perturbation in time delay is specifically discussed for the delay equal to two. A numerical simulation for the case of the delay equal to two is presented to illustrate the convergence. Finally, an alternative stability analysis is conducted within the framework of well-posed infinite-dimensional systems.