A perturbation treatment for optimal slightly nonlinear systems with linear control and quadratic criteria

Abstract

Recently, it has been discovered that symbolic algebraic manipulations can be performed on the computer with input/output data in symbolic form. Accordingly, and for slightly nonlinear two-point boundary-value problems, it is feasible to obtain approximate analytical solutions in the form of power series in a small parameter. In these solutions, the boundary values are presented in a literal form. In this paper, the Lie canonical transformation is applied to derive approximate optimal solutions for slightly nonlinear systems with quadratic criteria. The transformation generator is determined by simple partial differential equations of the first order. To determine the arbitrary constants of the transformation in terms of the two-point boundary values, inversion of a vectorial power series in a small parameter is required, and a recursive algorithm for this inversion is given. To express the final solution in terms of these boundary values, a substitution of the inverted vectorial power series into another vectorial power series is also necessary, and a recursive algorithm for this substitution is presented.