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Journal of Mathematics
Volume 2013, Article ID 895876, 5 pages
Research Article

Optimization of the Forcing Term for the Solution of Two-Point Boundary Value Problems

Dipartimento di Matematica “F. Brioschi”, Modellistica e Calcolo Scientifico (MOX), Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Received 28 August 2012; Accepted 1 January 2013

Academic Editor: Alfredo Peris

Copyright © 2013 Gianni Arioli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a new numerical method for the computation of the forcing term of minimal norm such that a two-point boundary value problem admits a solution. The method relies on the following steps. The forcing term is written as a (truncated) Chebyshev series, whose coefficients are free parameters. A technique derived from automatic differentiation is used to solve the initial value problem, so that the final value is obtained as a series of polynomials whose coefficients depend explicitly on (the coefficients of) the forcing term. Then the minimization problem becomes purely algebraic and can be solved by standard methods of constrained optimization, for example, with Lagrange multipliers. We provide an application of this algorithm to the planar restricted three body problem in order to study the planning of low-thrust transfer orbits.