An hp-adaptive pseudospectral method is presented for numerically solving optimal control problems. The method presented in this paper iteratively determines the number of segments, the width of each segment, and the polynomial degree required in each segment in order to obtain a solution to a userspecified accuracy. Starting with a global pseudospectral approximation for the state, on each iteration the method determines locations for the segment breaks and the polynomial degree in each segment for use on the next iteration. The number of segments and the degree of the polynomial on each segment continue to be updated until a user-specified tolerance is met. The terminology 'hp' is used because the segment widths (denoted h) and the polynomial degree (denoted p) in each segment are determined simultaneously. It is found that the method developed in this paper leads to higher accuracy solutions with less computational effort and memory than is required in a global pseudospectral method. Consequently, the method makes it possible to solve complex optimal control problems using pseudospectral methods in cases where a global pseudospectral method would be computationally intractable. Finally, the utility of the method is demonstrated on a variety of problems of varying complexity. Determining locations of new segments or increase in number of collocation pointsLet be a user-defined tolerance and assume that the maximum entry of Equation (27) is greater than . In this case, the segment either needs to be divided into more segments or the degree of

A method is presented for direct trajectory optimization and costate estimation using global collocation at Legendre-Gauss-Radau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsingular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finite-horizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally, * M.S. Student, Dept. it is shown that a previously developed Radau collocation method, which is restricted to infinite-horizon problems, is subsumed by the method presented in this paper. The results of this paper show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions to general finite-horizon optimal control problems.

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An algorithm is described to solve multiple-phase optimal control problems using a recently developed numerical method called the Gauss pseudospectral method. The algorithm is well suited for use in modern vectorized programming languages such as FORTRAN 95 and MATLAB. The algorithm discretizes the cost functional and the differential-algebraic equations in each phase of the optimal control problem. The phases are then connected using linkage conditions on the state and time. A large-scale nonlinear programming problem (NLP) arises from the discretization and the significant features of the NLP are described in detail. A particular reusable MATLAB implementation of the algorithm, called GPOPS, is applied to three classical optimal control problems to demonstrate its utility. The algorithm described in this article will provide researchers and engineers a useful software tool and a reference when it is desired to implement the Gauss pseudospectral method in other programming languages.

The problem of small spacecraft minimum-fuel heat-rate-constrained aeroassisted orbital transfer between two low Earth orbits with inclination change is considered. Assuming impulsive thrust, the trajectory design is described in detail and the aeroassisted orbital transfer is posed as a nonlinear optimal control problem. The optimal control problem is solved using an hp-adaptive pseudospectral method, and the key features of the optimal trajectories are identified. It was found that the minimum impulse solutions are obtained when the vehicle enters the atmosphere exactly twice. Furthermore, even for highly heat-rate-constrained cases, the final mass fraction of the vehicle was fairly large. Finally, the structural loads on the vehicle were quite reasonable, even in the cases where the heating rate was unconstrained.