Stabilization of One‐Dimensional Schrödinger Equation with Variable Coefficient under Delayed Boundary Output Feedback

Yang, Kun-Yi; Yao, Cui-Zhen

doi:10.1002/asjc.667

Cited by 14 publications

(6 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The system above is considered in the energy state space

scriptH = H_{L}^{1} false(0, 1 false) \times L^{2} false(0, 1 false),

where

H_{L}^{1} false(0, 1 false) = false{f \in H^{1} false(0, 1 false) false| f false(0 false) = 0 false}

and the input (output) space are

U = Y = double-struckC

. The energy of the system is

E (t) = \frac{1}{2} \int_{0}^{1} [w_{x}^{2} (x, t) + a {(x)}^{- 1} w_{t}^{2} (x, t)] d x .

…”

Section: Introductionmentioning

confidence: 72%

See 1 more Smart Citation

Stabilization of a one‐dimensional wave equation with variable coefficient under non‐collocated control and delayed observation

Yang

2017

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The system above is considered in the energy state space

scriptH = H_{L}^{1} false(0, 1 false) \times L^{2} false(0, 1 false),

where

H_{L}^{1} false(0, 1 false) = false{f \in H^{1} false(0, 1 false) false| f false(0 false) = 0 false}

and the input (output) space are

U = Y = double-struckC

. The energy of the system is

E (t) = \frac{1}{2} \int_{0}^{1} [w_{x}^{2} (x, t) + a {(x)}^{- 1} w_{t}^{2} (x, t)] d x .

…”

Section: Introductionmentioning

confidence: 72%

“…This is a generalization of the similar works for the wave equation in Guo et al with constant coefficients. In this sense, it is of great significance for this paper to deal with the systems described by the partial differential equations with variable coefficients …”

Section: Introductionmentioning

confidence: 99%

Stabilization of a one‐dimensional wave equation with variable coefficient under non‐collocated control and delayed observation

Yang

2017

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…Guo et al [14,15] developed an observer-predictor scheme to stabilize the wave equations and Schrödinger equation with time delay in the observation, respectively. Yang and Yao [16] stabilized a Schrödinger equation with variable coefficients and boundary output time delay, by designing observer and predictor for the system. Kafini et al [17] considered a 1D thermoelastic system of Timoshenko type with delay.…”

Section: Introductionmentioning

confidence: 99%

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

Cui

Han

2015

Applicable Analysis

View full text Add to dashboard Cite

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.

show abstract

“…Theory and applications of delay linear and nonlinear Schrödinger equations with the delay term is an operator of lower order with respect to the operator term were widely investigated (see, e.g., [13,17,18,[25][26][27], and the references given therein).…”

Section: Introductionmentioning

confidence: 99%

“…For example, in the article [26], the boundary stabilization of a Schrödinger equation with variable coefficient where the boundary observation suffers from a fixed time delay was studied. This is a generalization of the similar work for the Schrödinger equation in [18] by using the separation principle [17] for constant coefficients.…”

Section: Introductionmentioning

confidence: 99%