Costate Estimation using Multiple-Interval Pseudospectral Methods

A method is presented for costate estimation in nonlinear optimal control problems using multiple-interval collocation at Legendre-Gauss (LG) or Legendre-GaussRadau (LGR) points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuous-time optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuous-time first-order optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuous-time first-order optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multiple-interval collocation approach for estimating costate in continuous-time nonlinear optimal control problems.