Applications of Second-Order Necessary and Sufficient Conditions to Optimal Trajectories
Jo, Jang-Won
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https://hdl.handle.net/2142/85124
Description
Title
Applications of Second-Order Necessary and Sufficient Conditions to Optimal Trajectories
Author(s)
Jo, Jang-Won
Issue Date
1997
Doctoral Committee Chair(s)
Prussing, John E.
Department of Study
Aerospace Engineering
Discipline
Aerospace Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Operations Research
Language
eng
Abstract
"A recent advance in sufficient conditions for a weak local minimum in optimal control problems is used to develop a procedure for applying second-order necessary and sufficient conditions for a minimum of a cost functional. For a system with n state variables, an improved Riccati equation solution method is used to transform a test for the unboundedness of a n x n matrix into a test for a scalar being zero. Application to one important second-order necessary and sufficient condition, the Jacobi no-conjugate-point condition, is introduced using the ""Shortest path between two points on a sphere"" problem. Second-order necessary and sufficient conditions are applied to various optimal control problems, including spacecraft trajectory problems with constant thrust acceleration and with time-varying low thrust acceleration for a power-limited rocket engine. A solution that simultaneously maximizes final orbit energy and minimizes propellant consumption is found that satisfies the usual first-order necessary conditions, but is non-optimal. Other example variational problems are investigated: Hamilton's Principle for several dynamic systems, including a circular orbit in an inverse-square gravitational field and projectile motion in a uniform field, as well as a simple example of Zermelo's problem. For those solutions that satisfy first-order necessary conditions but are non-optimal, a Genetic Algorithm is successfully used to find a global near-optimal solution of lower cost."
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