| 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. |
