Control methods for localization of nonlinear waves (original) (raw)
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Distributed geometric control of wave equation
2008
An approach for the geometric control of a one-dimensional non-autonomous linear wave equation is presented. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then based on geometric control concepts, a distributed control law that enforces stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modeled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation.
Localization of the sine-Gordon equation solutions
Communications in Nonlinear Science and Numerical Simulation, 2016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • The distributed control algorithms allow us to achieve wave localization independent of the shape of the initial condition for the sine-Gordon equation. • The localization of the waves in both directions is achieved by means of a feedforward (nonfeedback) control. • The feedback distributed algorithm provides both localized waves according to analytical solutions and their unidirectional propagation.
ON THE FEEDBACK CONTROL OF THE WAVE EQUATION
Journal of Sound and Vibration, 2000
This paper addresses the problem of design of collocated and non-collocated controllers for a uniform bar without structural damping. The bar whose dynamics are described by the wave equation is required to perform a rest-to-rest maneuver. A time delay controller whose gains are determined using the root-locus technique is used to control the non-collocated system. The e!ect of sensor locations on the stability of the system is investigated when the actuator is located at the one end of the bar. The critical gains which correspond to a pair of poles entering the right-half of the s-plane and the optimal gains corresponding to locating the closed-loop poles at the left extreme of the root-locus for each vibration mode are determined. The gain which minimizes a quadratic cost, in the range of the critical gains, is selected as the optimum gain.
Feedback control for some solutions of the sine-Gordon equation
Applied Mathematics and Computation, 2015
Evolution of an initial localized bell-shaped state for the sine-Gordon equation is considered. It is obtained numerically that variation in the parameters of the localized input gives rise to dierent propagating waves as time goes. The speed gradient feedback control method is employed to achieve unied wave prole weakly dependent on initial conditions. Two speed-gradient like algorithms are developed and compared. It is shown that the algorithm using coecient at the second spatial derivative term in the sine-Gordon equation allows one to generate the same wave with prescribed energy from dierent initial states having dierent energies.
International Journal of Control, 2016
Energy control problems are analyzed for infinite dimensional systems. Benchmark linear wave equation and nonlinear sine-Gordon equation are chosen for exposition. The relatively simple case of distributed yet uniform over the space control is considered. The speed-gradient method (Fradkov,1996) that has already successfully been applied to numerous nonlinear and adaptive control problems is presently developed and justified for the above partial differential equations (PDEs). An infinite dimensional version of the Krasovskii-LaSalle principle is validated for the resulting closed-loop systems. By applying this principle, the closed-loop trajectories are shown to either approach the desired energy level set or converge to a system equilibrium. The numerical study of the underlying closed-loop systems reveals reasonably fast transient processes and the feasibility of a desired energy level if initialized with a lower energy level.
Physical Review E, 2006
We study the evolution of fronts in a nonlinear wave equation with global feedback. This equation generalizes the Klein-Gordon and sine-Gordon equations. Extending previous work, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of oneand two-dimensional fronts, finding a much richer dynamics than for the classical case (with no global feedback), leading in most cases to a localized solution; i.e., the stabilization of one phase inside the other. The nature of the localized solution depends on the strength of the global feedback as well as on other parameters of the model.
Localized solutions of nonlinear network wave equations
Journal of Physics A: Mathematical and Theoretical, 2018
We study localized solutions for the nonlinear graph wave equation on finite arbitrary networks. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the nonlinear graph wave equation to the discrete nonlinear Schrödinger equation and by Fourier analysis. Finally, we examine numerically the condition for localization in the parameter plane, coupling versus amplitude and show that the localization amplitude depends on the maximal normal eigenfrequency.
P S ] 1 6 N ov 2 01 8 Localized solutions of nonlinear network wave equations
2018
We study localized solutions for the nonlinear graph wave equation on finite arbitrary networks. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the nonlinear graph wave equation to the discrete nonlinear Schrödinger equation and by Fourier analysis. Finally, we examine numerically the condition for localization in the parameter plane, coupling versus amplitude and show that the localization amplitude depends on the maximal normal eigenfrequency.
Feedback control of the sine–Gordon antikink
Wave Motion, 2016
A new distributed speed-gradient feedback control algorithm for the sine-Gordon (SG) equation is proposed. It creates the antikink traveling wave mode for a broader class initial conditions compared to the uncontrolled system. In real physical problems it is difficult to provide consistent initial conditions for the secondorder (in time) equations. Therefore for an uncontrolled system even small variations in the initial velocity relative to that of the exact antikink solution of the SG equation give rise to growing oscillations. The control algorithm allows one both to suppress defects and to obtain stable propagation of an antikink in the form of the exact traveling wave solution of the SG equation. In contrast to the existing algorithms the proposed algorithm does not require additional dissipative term for wave generation.
Control of nonlinear distributed parameter systems: Recent results and future research directions
1997
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