Variational Calculus and Approximate Solution of Optimal Control Problems

Variational calculus is a differential process whereby Taylor series expansions can be developed on a term-by-term basis. Therefore, it can be used to obtain the equations which must be solved for the various-order terms arising from the application of regular perturbation theory to problems involving a small parameter. Variational calculus is summarized and applied to the approximate analytical solution of the optimal control problem. First, the various-order equations are obtained directly for a particular problem. Then, assuming that the zeroth-order solution is almost good enough, the equations for the first-order correction are obtained for the general optimal control problem and applied to the particular problem. The first-order solution is the same as the neighboring extremal for the given value of the parameter.