Second-Order Necessary Conditions and Sufficient Conditions Applied to Continuous-Thrust Trajectories

Abstract

Introduction I N optimal control theory, requiring the first variation of the performance functional to vanish leads to well-known first-order necessary conditions (NC) for an optimal solution.1 These NC allow one to identify candidates for optimality, called stationary or extremal solutions, to distinguish them from solutions that have been proven to be optimal. To determine whether a stationary solution is indeed optimal, one must also test the second-order Jacobi no-conjugate-point NC, which applies if the trajectory is smooth. Also, one can formulate sufficient conditions (SC) that, if satisfied, guarantee that the solution is at least locally optimal. In this Note, a procedure developed by Jo and Prussing2−4 for testing second-order NC and SC is streamlined and applied to an example optimal continuous-thrust trajectory with multiple terminal constraints that yields a different type of result compared to previous examples in Refs. 2, 4, and 5. The procedure is based on earlier work by Wood5,6 that derives new, less restrictive SC for a weak local minimum of the Bolza optimal control problem. However, those SC require that the solution of a matrix Riccati equation be bounded. This is difficult to test numerically because a bounded but rapidly increasing solution can stop the numerical integration and give the false impression that the solution is unbounded. The procedure described and illustrated in this Note replaces the test for an unbounded matrix5,6 by a test for a (scalar) determinant being zero.