Numerical optimization of solutions to Burgers' problems by means of boundary conditions

A method is proposed for solving an optimal control problem for Burgers' equation. The boundary conditions of the problem are used as a control, and the functional is given by the integral norm of the deviation of the solution to Burgers' equation from a certain experimental function. In the continuous case, the gradient of the functional is obtained in terms of the solution to the conjugate problem. In the discrete case, formulas for the exact gradient of the functional of a discretized problem are used. Various methods of approximation of the direct problem are considered. It is argued which method should be preferred for solving the optimal control problem. It is shown that the choice of an integration scheme can be based solely on the requirement of satisfactory approximation of the original problem. The gradient of the functional of the discretized problem is calculated exactly, and the approximation scheme for the conjugate problem is determined automatically. The results of calculations are presented.