Shugen Chai

Boundary Feedback Stabilization of Shallow Shells

We consider the stabilization of the shallow shell by boundary feedbacks where the model has a middle surface of any shape. First, we put the shallow shell model in a suitable semigroup scheme. The existence, the uniqueness, and the properties of solutions to the shallow shell are then treated by the semigroup approach and the regularity of elliptic boundary value problems. Finally, we establish the uniform energy decay rate for the shallow shell under some checkable geometric conditions on the middle surface.

Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation

We are concerned with the feedthrough operator for the open-loop system of linear elasticity with Dirichlet boundary control and collocated observation. This system has been known to be regular in the sense of G. Weiss. In this paper, the analytic expression of the corresponding feedthrough operator is presented by means of differential geometry and Fourier transform methods.

Well-Posedness and Exact Controllability of Fourth Order Schrödinger Equation with Boundary Control and Collocated Observation

In this paper, we study the well-posedness and exact controllability of a system described by a fourth order Schrodinger equation on a bounded domain of

Well-posedness and exact controllability of fourth-order Schrödinger equation with hinged boundary control and collocated observation

{\mathbb R}^{n}(n\geqslant 2)

Observability inequalities for thin shells

with boundary control and collocated observation. The Neumann boundary control problem is first discussed. It is shown that the system is well-posed in the sense of D. Salamon. This implies the exponential stability of the closed-loop system under proportional output feedback control. The well-posedness result is then generalized to the Dirichlet boundary control problem. In particular, in order to conclude feedback stabilization from well-posedness, we discuss the exact controllability with the Dirichlet boundary control, which is similar to the Neumann boundary control case. In addition, we show that both systems are regular in the sense of G. Weiss and their feedthrough operators are zero.

Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback

In this paper, we consider the well-posedness and exact controllability of a fourth-order multi-dimensional Schrödinger equation with hinged boundary by either moment or Dirichlet boundary control and collocated observation, respectively. It is shown that in both cases, the systems are well posed in the sense of D. Salamon, which implies that the systems are exactly controllable in some finite time interval if and only if its corresponding closed loop systems under the direct output proportional feedback are exponentially stable. This leads us to discuss further the exact controllability of the systems. In addition, the systems are consequently shown to be regular in the sense of G. Weiss as well, and the feedthrough operators are zero.