** Geometric control and sub-Riemannian geodesics **

Contents of the course

- The Chow-Rashevsky Theorem
- Sub-Riemannian geodesics
- A closer look at singular curves

References

Here is recommended reading related to the contents of this course.

- Bellaïche: The tangent space in sub-Riemanian geometry
- Montgomery: A tour of subriemannian geometries
- Agrachev-Barilari-Boscain: Introduction to Riemannian and sub-Riemannian geometry, to appear soon
- Jean: Control of nonholonomic systems: from sub-Riemannian geometry to motion planning
- Rifford: Sub-Riemannian Geometry and optimal transport

**1. Control systems **

Example: reversed pendulum on a cart.

In general, a control system is , is the state of the system, is the control, both are functions of time with values in vectorspaces.

Example: given vectorfields , .

** 1.1. Controllability **

Controllability: which states can one reach from a given state ?

Theorem 1 (Chow 1939-Rashevski 1938)Data: vectorfields on a neighborhood. Assume that the Lie algebra generated by these generates the whole tangent space at point . Then every point in a smaller neighborhood of can be reached from .

The conclusion is called *local controllability*. Note that on a connected manifold, local controllability implies global controllability.

** 1.2. Geometric interpretation **

The theorem helps understanding the meaning of Lie brackets: is a direction in which you can move using controls in directions and only. Moving in that direction requires patience: moving aling , , and finally produces a quadrilateral which nearly closes up, but not quite. It is the small defect which is in the direction , so it takes lots of time to go there.

** 1.3. Number of steps **

The assumption in the theorem bears many different names (one in each of the above monographs), let us call it *bracket generating*. Let be a family of vectorfields. Denote by and then, recursively,

Bracket generating means that for some , evaluated at equals the tangent space. The smallest has geometric sgnificance.

** 1.4. Ways of proving the theorem **

In , here is a proof of the theorem. In the obvious commutator of two flows, introduce a small third parameter. Show that one gets a local diffeo.

This works in all dimensions, see Jean’s monograph. It even gives a stronger conclusion known as the ball-box theorem.

**2. Proof of the Chow-Rashevski Theorem **

Nevertheless, I will follow Bellaïche’s proof.

** 2.1. Singular controls **

Introduce the end-point mapping from the space of controls to the space of states. Its differential is given by the variational equation, a linear ODE. We say that a control is *regular* if is a submersion at , *singular* otherwise. In fact, it is merely a property of the trajectory as a curve. Reparametrization (and in particular, time reversal) keeps a trajectory regular or singular.

The *rank* of a singular control is the rank of at .

The Chow-Rashevski Theorem will follow from the following

Proposition 2For a bracket generating family of vectorfields, the end-point map is open.

** 2.2. Density of regularity **

Most controls in are regular. In fact, every ball of contains a regular control. Proof by contradiction. Let maximize in a small ball of , let be its rank. There exists controls such that is an immersion at 0, its image is a submanifold . Then maps into . In particular, the initial vectorfields are tangent to , contradiction.

More generally, the set of regular controls is dense in (concatenate). A theorem of Sontag asserts that is dense in .

** 2.3. The return method **

The idea is that if is regular, concatenated with its time reversal is regular.