By systematically altering the in-plane flight path and velocity of an aircraft and the rate at which the cable is deployed, the tip of a cable being towed behind an aircraft is shown to rendezvous with a ground-based surface location in minimum time, with minimum control effort required. This is achieved through optimal control, formulated as a nonlinear programming problem using the Legendre-...

Recently, the Legendre pseudospectral (PS) method migrated from theory to flight application onboard the International Space Station for performing a finite-horizon, zeropropellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for infinite-horizon optimal control problems. Motivated by these technicalities, the conc...

A short discussion of optimal control methods is presented including indirect, direct shooting, and direct transcription methods. Next the basics of multiple-interval pseudospectral methods are given independent of the numerical scheme to highlight the fundamentals. The two numerical schemes discussed are the Legendre pseudospectral method with LGL nodes and the Chebyshev pseudospectral method ...

A comparison is made between three pseudospectral methods used to numerically solve optimal control problems. In particular, the accuracy of the state, control, and costate obtained using the Legendre, Radau, and Gauss pseudospectral methods is compared. Three examples with different degrees of complexity are used to identify key differences between the three methods. The results of this study ...

We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with N − 5 unknowns. An efficient and accurate direct solution for algorithms based on the LegendreGalerkin approximations...

A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre–Gauss–Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and ...

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (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 to the costates of the optimal control problem. More precisely...

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional wave equation (FWE) is considered. The fractional derivative is described in the Caputo sense. The method is...