Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions

Abstract

This paper addresses the well-known Goddard problem in the formulation of Seywald and Cliff with the objective to maximize the altitude of a vertically ascending rocket subject to dynamic pressure and thrust constraints. The Goddard problem is used to propose a new method to systematically incorporate the constraints into the system dynamics by means of saturation functions. This procedure results in an unconstrained and penalized optimal control problem which strictly satisfies the constraints. The approach requires no knowledge of the switching structure of the optimal solution and avoids the explicit consideration of singular arcs. A collocation method is used to solve the BVPs derived from the optimality conditions and demonstrates the applicability of the method to constrained optimal control problems.