Procedure for applying second-order conditions in optimal control problems

A recent advance in sufficient conditions for a weak local minimum in the Bolza optimal control problem is used to develop a practical procedure for applying second-order necessary conditions and sufficient conditions. For a system with n state variables, a transition matrix method is used to transform a test for the unboundedness of an n x n matrix solution of a Riccati equation into a test for a scalar being zero. This allows routine testing of second-order conditions, including the Jacobi no-conjugate-point necessary condition. Four example problems are? analyzed: a simple minimum-time problem, the shortest path between two points on a sphere, a multiobjective spacecraft trajectory optimization, and an application of Hamilton's Principle to a circular orbit in an inverse-square gravitational field. In those examples for which second-order conditions are violated and an analytical solution does not exist, a genetic algorithm is used to determine a near-optimal solution.