Stabilization for Schrödinger equation with a time delay in the boundary input

Abstract

The stabilization problem of a 1D Schrödinger equation subject to boundary control is concerned in this paper. The control input is suffered from time delay. A "partial state" predictor is designed for the system and undelayed system is deduced. Based on the undelayed system, a feedback control strategy is designed to stabilize the original system. The exact observability of the dual one of the undelayed system is checked. Then it is shown that the system can be stabilized exponentially under the feedback control.

Uniform stabilization of 1-D Schrödinger equation with internal difference-type control

“…Therefore v(x, s, t) satisfies the differential equation and boundary condition in (2). Next we check that w(x, t) also satisfies the equation in (2). Since…”

“…Proof. Assume that (w(x, t), v(x, s, t)) and (w(x, t), z(x, s, t)) are solutions to (2) and 7respectively. According to Theorem 3.2, we have…”

“…Note that in papers mentioned above, the authors mainly applied the collocated feedback to stabilize the systems. Different from the collocated feedback, based on the "partial state" predictor, Cui, Han and Xu [2] designed a dynamical feedback control to stabilize the 1-d Schrödinger equation with boundary differencetype control [19]. Recently, Chen, Xie and Xu [1] applied the approach comes from Feng et al [7] to the multi-dimensional Schrödinger equation with interior pure delay control and obtained the exponential stabilization.…”

“…Design of feedback controller of (2). In this section, we shall design the state feedback control of the system (2). Suppose that (w(x, t), v(x, s, t)) is a solution of (2), we take the control u(x, t) of the form…”

Finite Dimensional Approximation of the Monodromy Operator of a Periodic Delay Differential Equation with Piecewise Constant Orthonormal Functions

“…Linear Periodic Differential Equations (PDDEs) have been of importance for studying problems of vibration, mechanics, astronomy, electric circuits, biology among others in [1] several examples of delay effects on mechanical systems are given, in [2] and [3] effects of the delay in physics and biological processes are considered. Neglecting the fact that interaction between particles does not occur instantaneously sometimes is no longer possible or practical, these finite velocity interactions bring new behaviors that modify significantly the behavior of the system, see for example [4] and [5].…”