Numerical Approximation of Exact Controllability for Korteweg-de Vries Equation

A. Salem (Tunisia)

Keywords

Controllability, Kortewegde Vries equation, Numerical approximation, Adjoint system, Hilbert Uniques Method.

Abstract

In this work, the problem of exact controllability of linear Korteweg-de Vries (KdV) equation in bounded domain is considered. Many results of controllability have already been obtained in recent years. In particular, in the case of Dirichlet boundary conditions, L.Rosier [1] have proved the controllability of the linear control system if the length L of the interval is not critical that is if L is not in the set N := 2 π k 2 + kl + l2 3 ; k, l ∈ N∗ ( 1) The first part of this work is a theoretical study of the Korteweg-de Vries equations. In our case, the condition for the control is the difference between the derivate of the solution in the first and last point of space. We prove the ob servability result (as Rosier) which helped us using Hilbert Uniqueness Method (H.U.M) proving the controllability of KdV equation. The second part is a numerical study of the previous result. We apply the H.U.M method used previously theoretically as a calculus algorithm to find a control. We develop the solution of the control problem using the spectral method numerical solution to the control problem. In order to ver ify that spectral method is giving good results, we apply the finite difference method with the control obtained from the spectral method.

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