Costate estimation for optimal control problems using orthogonal collocation at Gaussian quadrature points

Computing the costate in an optimal control problem is important for verifying the optimality of the solution and performing sensitivity analysis. This dissertation is concerned with the problem of estimating the costate in an optimal control problem using orthogonal collocation at Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) points. First, methods are presented for estimating the costate using orthogonal collocation at the LG or LGR points when the dynamic constraints of the optimal control problem are formulated in integral form. A new continuous-time dual variable called the integral costate is introduced, where the integral costate is the Lagrange multiplier of the integral dynamic constraint. The first-order optimality conditions of the integral form of the optimal control problem are derived in terms of the integral costate. The integral form of the optimal control problem is then discretized using the integral LG and LGR collocation methods and relationship between the discrete form of the integral costate and the costate of the original differential optimal control problem are developed. It is shown that the LGR integration matrix that relates the differential costate to the integral costate is singular while the corresponding LG integration matrix is full rank. The approach developed in this research then provides a way to estimate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. Furthermore, the costate estimates presented in this research result in a set of Karuhn-Kush-Tucker conditions of the nonlinear programming problem which are a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem both in differential and integral forms.;The second part of this research focuses on state inequality path constrained optimal control problems. Problems with active state-inequality path constraints are difficult to solve due to the high-index differential-algebraic equations (DAE) that result from the constraint activity. This DAE index fluctuation in the solution domain results in possible discontinuities in the dual variables which are hard to approximate numerically. Due to these discontinuities, previous costate estimates for direct transcription methods using collocation at LG or LGR points resulted in a transformed adjoint system which was not a discrete approximation to the first-order optimality conditions in the presence of state inequality path constraints. In this research a different set of costate estimates are developed which result in a transformed adjoint system that is a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem. Specifically, a costate estimate using the method of indirect adjoining with continuous multipliers is derived. The equivalence between the first-order optimality conditions of the finite-dimensional nonlinear program and the first-order optimality conditions of the continuous-time optimal control problem ensures convergence of the discrete problem to a local minimum which satisfies the optimality conditions of the original problem. This costate estimate can thus be used to verify the extremality of the approximated solution.