Costate estimation of state-inequality path constrained optimal control problems using collocation at Legendre-Gauss-Radau points

A method is presented for costate estimation of state-inequality constrained optimal control problems using orthogonal collocation at Legendre-Gauss-Radau points. It is shown that the Lagrange multipliers of the nonlinear programming problem can be accurately mapped to the costates of the continuous-time optimal control problem. The differentiation matrix associated with the costate estimate is singular, whereas the differentiation matrix associated with the state inequality constraint multipliers is invertible. Furthermore, it is shown that the inverse of this differentiation matrix is an integration matrix. Finally, the accuracy of the proposed costate estimate is demonstrated on an optimal control example.