Section: Application Domains
Space engineering, satellites, low thrust control
Space engineering is very demanding in terms of safe and highperformance control laws (for instance optimal in terms of fuel consumption, because only a finite amount of fuel is onboard a satellite for all its “life”). It is therefore prone to real industrial collaborations.
We are especially interested in trajectory control of space vehicles using their own propulsion devices, outside the atmosphere. Here we discuss “nonlocal” control problems (in the sense of section 3.1 point 1): orbit transfer rather than station keeping; also we do not discuss attitude control.
In the geocentric case, a space vehicle is subject to
 gravitational forces, from one or more central bodies (the corresponding acceleration is denoted by ${F}_{\mathrm{grav}.}$ below),
 a thrust, the control, produced by a propelling device; it is the $G\phantom{\rule{0.166667em}{0ex}}u$ term below; assume for simplicity that control in all directions is allowed, i.e. $G$ is an invertible matrix
 other “perturbating” forces (the corresponding acceleration is denoted by ${F}_{2}$ below; in simplified models, it is not present). In positionvelocity coordinates, its dynamics can be written as
$\ddot{x}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}{F}_{\mathrm{grav}.}(x,t)\phantom{\rule{0.277778em}{0ex}}+\phantom{\rule{0.277778em}{0ex}}{F}_{2}(x,\dot{x},t)\phantom{\rule{0.277778em}{0ex}}+\phantom{\rule{0.277778em}{0ex}}G(x,\dot{x})\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\parallel u\parallel \le {u}_{\mathrm{max}}.$  (1) 
In the case of a single attracting central body (the earth) and in a geocentric frame, ${F}_{\mathrm{grav}.}$ does not depend on time, or consists of a main term that does not depend on time and smaller terms reflecting the action of the moon or the sun, that depend on time. The second term is often neglected in the design of the control at first sight; it contains terms like atmospheric drag or solar pressure. $G$ could also bear an explicit dependence on time (here we omit the variation of the mass, that decreases proportionally to $\parallel u\parallel $.
Low thrust
Low thrust means that ${u}_{\mathrm{max}}$ is small, or more precisely that the maximum magnitude of $G\phantom{\rule{0.166667em}{0ex}}u$ is small with respect to the one of ${F}_{\mathrm{grav}.}$ (but in general not compared to ${F}_{2}$). Hence the influence of the control is very weak instantaneously, and trajectories can only be significantly modified by accumulating the effect of this low thrust on a long time. Obviously this is possible only because the free system is somehow conservative. This was “abstracted” in section 3.5.
Why low thrust ? The common principle to all propulsion devices is to eject particles, with some relative speed with respect to the vehicle; conservation of momentum then induces, from the point of view of the vehicle alone, an external force, the “thrust” (and a mass decrease). Ejecting the same mass of particles with a higher relative speed results in a proportionally higher thrust; this relative speed (specific impulse, ${I}_{sp}$) is a characteristic of the engine; the higher the ${I}_{sp}$, the smaller the mass of particles needed for the same change in the vehicle momentum. Engines with a higher ${I}_{sp}$ are highly desirable because, for the same maneuvers, they reduce the mass of "fuel" to be taken onboard the satellite, hence leaving more room (mass) for the payload. “Classical” chemical engines use combustion to eject particles, at a somehow limited speed even with very efficient fuel; the more recent electric engines use a magnetic field to accelerate particles and eject them at a considerably higher speed; however electrical power is limited (solar cells), and only a small amount of particles can be accelerated per unit of time, inducing the limitation on thrust magnitude.
Electric engines theoretically allow many more maneuvers with the same amount of particles, with the drawback that the instant force is very small; sophisticated control design is necessary to circumvent this drawback. High thrust engines allow simpler control procedures because they almost allow instant maneuvers (strategies consist in a few burns at precise instants).
Typical problems
Let us mention two.

Orbit transfer or rendezvous. It is the classical problem of bringing a satellite to its operating position from the orbit where it is delivered by the launcher; for instance from a GTO orbit to the geostationary orbit at a prescribed longitude (one says rendezvous when the longitude, or the position on the orbit, is prescribed, and transfer if it is free). In equation (1) for the dynamics, ${F}_{\mathrm{grav}.}$ is the Newtonian gravitation force of the earth (it then does not depend on time); ${F}_{2}$ contains all the terms coming either from the perturbations to the Newtonian potential or from external forces like radiation pressure, and the control is usually allowed in all directions, or with some restrictions to be made precise.

Three body problem. This is about missions in the solar system leaving the region where the attraction of the earth, or another single body, is preponderant. We are then no longer in the situation of a single central body, ${F}_{\mathrm{grav}.}$ contains the attraction of different planets and the sun. In regions where two central bodies have an influence, say the earth and the moon, or the sun and a planet, the term ${F}_{\mathrm{grav}.}$ in (1) is the one of the restricted three body problem and dependence on time reflects the movement of the two “big” attracting bodies.
An issue for future experimental missions in the solar system is interplanetary flight planning with gravitational assistance. Tackling this global problem, that even contains some combinatorial problems (itinerary), goes beyond the methodology developed here, but the above considerations are a brick in this puzzle.
Properties of the control system.
If there are no restrictions on the thrust direction, i.e., in equation (1), if the control $u$ has dimension 3 with an invertible matrix $G$, then the control system is “static feedback linearizable”, and a fortiori flat, see section 3.2. However, implementing the static feedback transformation would consist in using the control to “cancel” the gravitation; this is obviously impossible since the available thrust is very small. As mentioned in section 3.1, point 3, the problem remains fully nonlinear in spite of this “linearizable” structure (However, the linear approximation around any feasible trajectory is controllable (a periodic timevarying linear system); optimal control problems will have no singular or abnormal trajectories.).