Direct trajectory optimization and costate estimation via an orthogonal collocation method

A direct trajectory optimization and costate estimation by means of an orthogonal collocation method is discussed. The Gauss pseudospectoral method has been described for solving optimal control problems numerically. The continuous time optimal problem in a direct method is transcribed to a nonlinear programming problem (NLP), which can be solved numerically by well-developed algorithms that attempt to satisfy a set of conditions associated with the NLP. The KKT conditions from the NLP obtained through the Gauss pseudospectral discretization are identical to the variational conditions of the continuous time optimal control problem discretized through the Gauss pseudospectral method. The KKT mutipliers of the NLP can be used to obtain an accurate estimate of the costate at both the Legendre-Gauss points and the boundary points. The results thereby viability of the Gauss pseudospectral method as a means of obtaining accurate solutions to continuous-time optimal control problems.