Active Optimal Control of the KdV Equation Using the Variational Iteration Method

The optimal pointwise control of the KdV equation is investigated with an objective of minimizing a given performance measure. The performance measure is specified as a quadratic functional of the final state and velocity functions along with the energy due to open- and closed-loop controls. The minimization of the performance measure over the controls is subjected to the KdV equation with periodic boundary conditions and appropriate initial condition. In contrast to standard optimal control or variational methods, a direct control parameterization is used in this study which presents a distinct approach toward the solution of optimal control problems. The method is based on finite terms of Fourier series approximation of each time control variable with unknown Fourier coefficients and frequencies. He’s variational iteration method for the nonlinear partial differential equations is applied to the problem and thus converting the optimal control of lumped parameter systems into a mathematical programming. A numerical simulation is provided to exemplify the proposed method. A modal for planar, unidirectional waves propagating in shallow water was originally introduced by Korteweg and de Vries in 1895 � 1� . The modal is expressed by a third-order nonlinear partial differential equation called KdV equation. The KdV equation has been at the center of naval science studies and other physical phenomena such as weakly nonlinear long waves for the last 150 years. Therefore, solving and controlling the behavior of the KdV equation have great implications. Review of new techniques such as variational approaches, parameter-expanding methods, and parameterized perturbation method for nonlinear problems is presented by He in � 2� , and a detailed study of He’s approaches is given in � 3� . In the literature, there are a considerable number of numerical and theoretical aspects of the KdV equation. A survey of results for the KdV equation is given in � 4� . Existence and uniqueness of the

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