Optimal Control

A dynamic system is said to be controlled if it can be acted upon by time-dependent variables, known as commands. Let's illustrate this concept in the case of a spacecraft, described by position and velocity variables (in ${ℝ}^3$ ) h and V, and a mass m > 0, i.e. 7 state variables. The dynamics are, omitting the time argument, $\dot{h}=V$ , $m\dot{V}=F\left(h,\mathrm{\hspace{0.17em}}V\right)+u$ and $\dot{m}=-\mathrm{\hspace{0.17em}}c\left|u\right|$ . Here c is a positive constant and F (h, V) corresponds to the forces of gravity and (where applicable) aerodynamics. The control is the applied force, whose Euclidean norm is denoted by $\left|u\right|$ , subjected to a constraint of the type $\left|u\right|\le U$ . Given a fixed initial point, we seek to reach a target (part of state space) by minimizing a compromise between travel time and energy expended.

For the real-time implementation of a control system, it is necessary to take into account the means of observation and the reconstitution of the state, while considering aspects of signal processing and the choice of control electronics. In contrast, in this article, we consider only the upstream study, in which a deterministic framework is used, and an optimal control is calculated off-line. The shape of the latter can guide the design of the real-time controller.

The presentation will first follow the approach of Lagrange and Pontriaguine, which consists in studying the variations of an optimal trajectory to determine its properties. First- and second-order optimality conditions will be analyzed, in connection with the shooting algorithm, with particular attention...