Optimal control sensitivity analysis with automatic differentiation
by
Jean-Baptiste Caillau
ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France
Coauthors: Joseph Noailles (ENSEEIHT-IRIT)

The minimum time transfer of a satellite is considered. An optimal control formulation is employed, where the control is the thrust of the engine. Since the initial orbit around the Earth is very eccentric and since we are particularly interested in low thrust transfers, the dynamics is strongly nonlinear. In order to solve the problem numerically, we introduce a parametric technique: the minimum time is then given as the first zero of a so-called non-controllability function \phi. So as to make use of Newton-like algorithms, we need regularity of class C¹ on \phi, defined as the value function of a parameterized family of optimal control problems with fixed final time. To this end, we call upon the sensitivity analysis tools developped by Maurer and Malanowski (1996) in this infinite-dimensional setting. Among the assumptions we need to check, the sufficient second-order optimality condition that generalizes the coercivity condition in finite dimension essentially amounts to finding a bounded solution to the Riccati equation canonically associated with the boundary value problem given by the Pontryagin maximum principle. Because of the complexity of the dynamics (described in terms of orbital coordinates, better suited for numerical purposes) and of the size of the system, the computation of the second member of the Riccati equation is performed using automatic differentiation. The equation is then integrated numerically, allowing us to check the second-order sufficient condition and to give the derivative of our function \phi in closed form. This sensitivity analysis is performed for various thrusts (including very low ones) and the corresponding trajectories, controls and optimal times are given.

Optimal control sensitivity analysis with automatic differentiation

Date received: November 26, 1999