A numerical approach to an optimal boundary control of the viscous Burgers' equation

The dynamics of the forced Burgers' equation subject to Dirichlet boundary conditions using boundary control is analyzed with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient method is suggested to solve the optimal boundary control of the Burgers' equation. The solution method involves the transformation of the original problem into one with homogeneous boundary conditions. This modifies the problem from one in which there are boundary controls to one in which there are distributed controls. The Modal space technique is applied on the distributed controls of the forced Burgers' equation to generate a low-dimensional dynamical systems. The time-variant controls are approximated by a finite term of the Fourier series whose coefficients and frequencies giving optimal solutions are to be determined, thereby converting the optimal control problem into mathematical programming problem. The approximate solution space based on the control parameterization is obtained by using the Runge-Kutta method. Numerical simulations for the boundary controls are presented for various target functions to assess the efficiency of the proposed method.