Costate Computation by a Chebyshev Pseudospectral Method

Abstract

AMONG the various pseudospectral (PS) methods for optimal control [1], only the Legendre PS method has been mathematically proven to guarantee the feasibility, consistency, and convergence of the approximations [2–5]. As exemplified by its experimental andflight applications in national programs [6–10], it is not surprising that the Legendre PS method has become the method of choice [11–19] in both industry and academia for solving optimal control problems. Efforts to improve the Legendre PS methods by using either other polynomials [20–22] or point distributions [23,24] have not yet resulted in any rigorous framework for convergence of these approximations [24,25]. Compared to Legendre PS methods, Chebyshev PS methods [21,22] for optimal control are somewhat more attractive for a number of reasons. When a function is approximated, it is well known that a Chebyshev expansion is very close to the best polynomial approximation in the infinity norm [26,27]. In addition, Chebyshev polynomials have an attractive computational advantage in terms of the computation of Chebyshev–Gauss–Lobatto (CGL) nodes. Unlike the Legendre–Gauss–Lobatto (LGL) nodes, CGL nodes can be evaluated in closed form [26]. Thus, a Chebyshev PS method offers the possibility of rapid computation because it does not require the use of advanced numerical linear algebra techniques that are necessary for the calculation of LGL nodes [21]. A similar numerical advantage applies to the computation of the derivative via a fast Chebyshev differentiation scheme that is similar to a fastFourier-transform (FFT) computation. In the same spirit, integration is also fast because of the connection between the Clenshaw–Curtis integration and the FFT [27]. Despite these attractive properties, Chebyshev PS methods have not advanced beyond the works of [21,22]. This is, in part, due to the absence of a covector mapping theorem that is crucial for the computation of the costates and other covectors. The computation of costates and other covectors is important in solving practical optimal control problems as it provides ameans for verification and validation of the computed solution [25]. Beyond verification and validation, information about covectors can also be used to facilitate the design of guidance and control algorithms [28]. In this Note, we fill the key gap of costate computation for Chebyshev PSmethods by furthering themethod of Fahroo andRoss [21]. We do this by combining some recent results from Clenshaw– Curtis integration [27], the unification principles proposed by Fahroo and Ross [1,29,30], and the new results of Gong et al. [23].