Computational method for the optimal control problem for the Korteweg-de Vries equation

Pointwise control of the periodic Korteweg-de Vries(K-dV) equation on domain [0, 2π] is considered with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the K-dV equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the K-dV equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parametrization by using the Runge-Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.