Automatic solution of optimal-control problems. I. simplest problem in the calculus of variations

摘要

An automatic method for obtaining the numerical solution for the simplest problem in the calculus of variations is described. The nonlinear two-point boundary-value Euler-Lagrange equation is solved using the Newton-Raphson method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the objective function in the calculus-of-variations problem and specify the boundary conditions. None of the derivatives usually associated with the Euler-Lagrange equation and the Newton-Raphson method need be calculated by hand. An example is given with numerical results. The automatic solution of the simplest problem in the calculus of variations in this paper is considered to be the first step in the automatic solution of more general optimal-control problems.