Title Existence,uniqueness and continuous dependence of weak solutions of damped sine-Gordon equations (original) (raw)

Optimal parameters for damped sine-Gordon equations with Neumann boundary condition

2011

In this paper we study an identification problem for physical parameters associated with damped sine-Gordon equation with Neumann boundary conditions. The existence, uniqueness, and continuous dependence of weak solution of sine-Gordon equations are established. The method of transposition is used to prove the Gâteaux differentiability of the solution map. The Gâteaux differential of the solution map is characterized. The optimal parameters are established. Computational algorithm and numerical results are presented.

Identification problem for damped sine-Gordon equation with point sources

Journal of Mathematical Analysis and Applications, 2011

We establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gâteaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Fréchet differentiable with respect to the parameters. An expression for the Fréchet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered.

Control for the sine-gordon equation

ESAIM: Control, Optimisation and Calculus of Variations, 2004

In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.

Towards soliton solutions of a perturbed sine-Gordon equation

2005

We give arguments for the existence of {\it exact} travelling-wave (in particular solitonic) solutions of a perturbed sine-Gordon equation on the real line or on the circle, and classify them. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. Such solutions are allowed exactly by the energy balance of these terms, and can

Optimal Parameters for a Damped Sine-Gordon Equation

Journal of the Korean Mathematical Society, 2009

In this paper a parameter identification problem for a damped sine-Gordon equation is studied from the theoretical and numerical perspectives. A spectral method is developed for the solution of the state and the adjoint equations. The Powell's minimization method is used for the numerical parameter identification. The necessary conditions for the optimization problem are shown to yield the bang-bang control law. Numerical results are discussed and the applicability of the necessary conditions is examined.

Perturbation theory for the double sine-Gordon equation

Wave Motion, 2005

This paper presents the perturbation theory for the double-sine-Gordon equation. We received the system of differential equations that shows the soliton parameters modification under perturbation's influence. In particular case λ = 0 the results of the research transform into well-known perturbation theory for the sine-Gordon equation.