Every control-affine system must be one or the other (not both) of the following:

**Completely integrable:**This means that the Pfaffian form (15.59) can be obtained by differentiating equations of the form with respect to time. This case was interpreted as being trapped on a surface in Section 13.1.3. An example of being trapped on a circle in was given in (13.22).**Nonholonomic:**This means that the system is not completely integrable. In this case, it might even be possible to reach all of , even if the number of action variables is much smaller than , the dimension of .

The notion of integrability used here is quite different from that
required for (14.1). In that case, the state
transition equation needed to be integrable to obtain integral curves
from any initial state. This was required for all systems considered
in this book. By contrast, *complete integrability* implies that
the system can be expressed without even using derivatives. This
means that all integral curves can eventually be characterized by
constraints that do not involve derivatives.

To help understand complete integrability, the notion of an integral
curve will be generalized from one to dimensions. A manifold
is called an *integral manifold* of a set of
Pfaffian constraints if at every , all vectors in the tangent space satisfy the constraints.
For a set of completely integrable Pfaffian constraints, a partition
of into integral manifolds can be obtained by defining maximal
integral manifolds from every . The resulting partition is
called a *foliation*, and the maximal integral manifolds are
called *leaves* [872].

(15.61) |

This is completely integrable because it can be obtained by differentiating the equation of a sphere,

(15.62) |

with respect to time ( is a constant). The particular sphere that is obtained via integration depends on an initial state. The foliation is the collection of all concentric spheres that are centered at the origin. For example, if , then a maximal integral manifold arises for each point of the form . In each case, it is a sphere of radius . The foliation is generated by selecting every .

The task in this section is to determine whether a system is
completely integrable. Imagine someone is playing a game with you.
You are given an control-affine system and asked to determine whether
it is completely integrable. The person playing the game with you can
start with equations of the form
and differentiate them
to obtain Pfaffian constraints. These can
then be converted into the parametric form to obtain the state
transition equation (15.53). It is easy to construct
challenging problems; however, it is very hard to solve them. The
concepts in this section can be used to determine only whether it is
possible to win such a game. The main tool will be the Frobenius
theorem, which concludes whether a system is completely integrable.
Unfortunately, the conclusion is obtained without producing the
integrated constraints
. Therefore, it is important to
keep in mind that ``integrability'' does not mean that *you* can
integrate it to obtain a nice form. This is a challenging problem of
reverse engineering. On the other hand, it is easy to go in the other
direction by differentiating the constraints to make a challenging
game for someone else to play.

Steven M LaValle 2020-08-14