Successive Chebyshev pseudospectral convex optimization method for nonlinear optimal control problems

Abstract

This article aims at proposing a successive Chebyshev pseudospectral convex optimization method for solving general nonlinear optimal control problems (OCPs). First, Chebyshev pseudospectral discrete scheme is used to discretize a general nonlinear OCP. At the same time, a convex subproblem is formulated by using the first‐order Taylor expansion to convexify the discretized nonlinear dynamic constraints. Second, a trust‐region penalty term is added to the performance index of the subproblem, and a successive convex optimization algorithm is proposed to solve the subproblem iteratively. Noted that the trust‐region penalty parameters can be adjusted according to the linearization error in iterative process, which improves convergence rate. Third, the Karush–Kuhn–Tucker conditions of the subproblem are derived, and furthermore, a proof is given to show that the algorithm will iteratively converge to the subproblem. Additionally, the global convergence of the algorithm is analyzed and proved, which is based on three key lemmas. Finally, the orbit transfer problem of spacecraft is used to test the performance of the proposed method. The simulation results demonstrate the optimal control is bang‐bang form, which is consistent with the result of theoretical proof. Also, the algorithm is of efficiency, fast convergence rate, and high accuracy. Therefore, the proposed method provides a new approach for solving nonlinear OCPs online and has great potential in engineering practice.

Constrainted optimization problems and optimal taxzations

“…Optimal taxation theory is a major branch in macroeconomic theory and economic theory. Many papers have been published on this topic during last half century (see [7,8,9,11,10,12,13,14,15]). Roughly speaking, for a given society during a given time period, taxation optimization means to design a maximizing utility tax policy subjected to some constraints.…”

“…Meanwhile, the government must consider maximizing the taxpayers' utilities. Precisely speaking, the goals of tax optimizations have three aspects: (1) keep the systems of this government perform well; (2) reduce the size of all taxes of the taxpayer, to minimize possible penalties, and to reduce tax risks; see, e.g., [7,8,9,10]; (3) maximize the utilities of the taxpayers; see, e.g., [11,12,13,14,15].…”

“…Subjected to some economic constraints of the amount of tax revenue decided by the government, the government designs a tax rate function to maximize the social welfare for the whole society, which is the total utilities of the taxpayers. In contrast to [9] with respect to discrete model, in this paper, we study the taxation optimization problem with continuous model. Since the Fan-KKM Theorem has more than one different versions, for easy reference, we briefly review the version of the Fan-KKM Theorem used in this paper.…”

An hp-Legendre Pseudospectral Convex Method for 6-Degree-of-Freedom Powered Landing Problem

“…In various trajectory optimization problems, PCMs are widely employed. Li [8] proposed the successive Chebyshev pseudospectral convex optimization method and successfully addressed spacecraft orbit transfer problems. Yu [9] developed an innovative convex optimization algorithm based on the Chebyshev pseudospectral method, effectively solving reentry trajectory optimization problems.…”