Asymptotic stabilization for a wave equation with periodic disturbance

2019 ◽  
Vol 37 (3) ◽  
pp. 894-917
Author(s):  
Jing Wei ◽  
Hongyinping Feng ◽  
Bao-Zhu Guo
Keyword(s):  
Wave Equation ◽  
Output Feedback Control ◽  
Coupled System ◽  
Original System ◽  
Periodic Signal ◽  
Wave System ◽  
Boundary Stabilization ◽  
Periodic Disturbance ◽  
One Dimensional ◽  
Theoretical Results

Abstract In this paper, we consider boundary stabilization for a one-dimensional wave equation subject to periodic disturbance. By regarding the periodic signal as a boundary output of a free wave equation, we transform the controlled plant into a coupled wave system. We first design a state observer for the coupled system to estimate the disturbance and the system state simultaneously. An output feedback control is then designed to stabilize the original system. As an application, the result is applied to the stabilization of a wave equation with periodic disturbance suffering in output. Finally, some simulations are presented to validate the theoretical results.

Error feedback regulation for 1D anti-stable wave equation

2021 ◽  
pp. 014233122110592
Author(s):  
Zhiyuan Li ◽  
Feng-Fei Jin
Keyword(s):  
Wave Equation ◽  
Tracking Error ◽  
Feedback Regulation ◽  
Original System ◽  
Well Posedness ◽  
Error Feedback ◽  
One Dimensional ◽  
Internal Loop ◽  
Finite Dimensional ◽  
Uniformly Bounded

This paper is concerned with the boundary error feedback regulation for a one-dimensional anti-stable wave equation with distributed disturbance generated by a finite-dimensional exogenous system. Transport equation and regulator equation are introduced first to deal with the anti-damping on boundary and the distributed disturbance of the original system. Then, the tracking error and its derivative are measured to design an observer for both exosystem and auxiliary partial differential equation (PDE) system to recover the state. After proving the well-posedness of the regulator equations, we propose an observer-based controller to regulate the tracking error to zero exponentially and keep the states of all the internal loop uniformly bounded. Finally, some numerical simulations are presented to validate the effectiveness of the proposed controller.

scholarly journals Phonon redshift and Hubble friction in an expanding BEC

2021 ◽  
Vol 10 (3) ◽  
Author(s):  
Stephen Eckel ◽  
Theodore Jacobson
Keyword(s):  
Wave Equation ◽  
Phonon Frequency ◽  
Einstein Condensate ◽  
Perturbation Expansions ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Reduction Methods ◽  
Systematic Perturbation ◽  
Bose Einstein ◽  
Theoretical Results

We revisit the theoretical analysis of an expanding ring-shaped Bose-Einstein condensate. Starting from the action and integrating over dimensions orthogonal to the phonon’s direction of travel, we derive an effective one-dimensional wave equation for azimuthally-travelling phonons. This wave equation shows that expansion redshifts the phonon frequency at a rate determined by the effective azimuthal sound speed, and damps the amplitude of the phonons at a rate given by \dot{\mathcal{V}}/{\mathcal{V}}𝒱̇/𝒱, where \mathcal{V}𝒱 is the volume of the background condensate. This behavior is analogous to the redshifting and ``Hubble friction’’ for quantum fields in the expanding universe and, given the scalings with radius determined by the shape of the ring potential, is consistent with recent experimental and theoretical results. The action-based dimensional reduction methods used here should be applicable in a variety of settings, and are well suited for systematic perturbation expansions.

Microwave heating of materials with low conductivity

1991 ◽  
Vol 433 (1889) ◽  
pp. 479-498 ◽  
Keyword(s):  
Wave Equation ◽  
Microwave Heating ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Coupled System ◽  
Damped Wave Equation ◽  
Governing Equations ◽  
One Dimensional ◽  
Approximate Analytical Solutions ◽  
Very High

The microwave heating of a one-dimensional, semi-infinite material with low conductivity is considered. Starting from Maxwell’s equations, it is shown that this heating is governed by a coupled system consisting of the damped wave equation and a forced heat equation with forcing depending on the amplitude squared of the electric field. For simplicity, the conductivity of the material and the speed of microwave radiation in the material are assumed to have power law dependencies on temperature. Approximate analytical solutions of the governing equations are found as a slowly varying wave. These solutions and the slow equations from which they are derived are found to give criteria for when ‘hotspots' (regions of very high temperature relative to their surroundings) can form. The approximate analytical solutions are compared with numerical solutions of the governing equations.

Nonlinear singular one dimensional thermo-elasticity coupled system and double Laplace decomposition methods

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6269-6280
Author(s):  
Hassan Gadain
Keyword(s):  
Laplace Transform ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Coupled System ◽  
Decomposition Methods ◽  
One Dimensional ◽  
Convergence Proof ◽  
Double Laplace Transform ◽  
Adomian Decomposition

In this work, combined double Laplace transform and Adomian decomposition method is presented to solve nonlinear singular one dimensional thermo-elasticity coupled system. Moreover, the convergence proof of the double Laplace transform decomposition method applied to our problem. By using one example, our proposed method is illustrated and the obtained results are confirmed.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

scholarly journals Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

2020 ◽  
Vol 26 ◽  
pp. 7
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Spectral Properties ◽  
Seismic Waves ◽  
Forced Vibrations ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Spatial Interval ◽  
Rational Multiple ◽  
Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

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