Feedback control for some solutions of the sine-Gordon equation

Порубов, А. В.; Fradkov, Alexander L.; Andrievsky, Boris

doi:10.1016/j.amc.2015.07.040

Cited by 16 publications

(7 citation statements)

References 11 publications

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“…To prevent the development of oscillations, one can try to achieve the value of the energy corresponding to the antikink. However, as shown in [14], this does not result in serious deviations in the wave behaviour. One cannot control either the shape or the velocity of the wave.…”

Section: Goal Of Nonlinear Wave Controlmentioning

confidence: 72%

“…For coupled equations, the generation of an optical mode has been suggested as the goal of the control in [13]. Finally, the desired energy of a wave may be the aim of the control [14]. Therefore, the key point in the development of a control algorithm is the formalization of the control goal in the form of a certain goal function.…”

Section: Goal Of Nonlinear Wave Controlmentioning

confidence: 99%

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Control methods for localization of nonlinear waves

Порубов

Andrievsky

2017

Phil. Trans. R. Soc. A.

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A general form of a distributed feedback control algorithm based on the speed-gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions.This article is part of the themed issue 'Horizons of cybernetical physics'.

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Section: Goal Of Nonlinear Wave Controlmentioning

confidence: 72%

Section: Goal Of Nonlinear Wave Controlmentioning

confidence: 99%

Control methods for localization of nonlinear waves

Порубов

Andrievsky

2017

Phil. Trans. R. Soc. A.

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“…A study by Sun [22] proposed an exact solution to Sine-Gordon equations with transcendental characteristics. Here, hypothetical Sine-Gordon equations (u tt = u xx + Sin(u)) [22,23], linear, and quadratic functions are solved simultaneously. The SoTE including Sine-Gordon equations simulates how interferences affect the output wavelength, frequency, and amplitude of the resultant wave function.…”

Section: A Hypothetical Sote Including a Sine-gordon Equationmentioning

confidence: 99%

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

et al. 2021

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A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

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“…Perhaps the first attempt to study energy control approach for distributed systems was made in [15] where a possibility of controlling travelling waves in the sine-Gordon model was demonstrated by simulation. In [16] two energy control algorithms were studied rigorously: one provides spatially distributed control action while another one provides a uniform over the space one.…”

Section: Introductionmentioning

confidence: 99%

Observer-based boundary control of the sine–Gordon model energy

2020

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In this paper the output feedback energy control problem for the sine-Gordon model is studied. An observer for the sine-Gordon equation and a speed-gradient boundary control law for solving this problem are analysed. Explicit inequalities on system's parameters ensuring the exponential decay of the estimation error are obtained. Under an additional assumption the achievement of the control goal is proved. The results of numerical experiments demonstrate that the transient time in energy is close to the transient time in observation error.

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Feedback control for some solutions of the sine-Gordon equation

Cited by 16 publications

References 11 publications

Control methods for localization of nonlinear waves

Control methods for localization of nonlinear waves

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

Observer-based boundary control of the sine–Gordon model energy

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