Notes of Ludovic Rifford lecture nr 1

Geometric control and sub-Riemannian geodesics

Contents of the course

  1. The Chow-Rashevsky Theorem
  2. Sub-Riemannian geodesics
  3. 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 {\dot{x}=f(x,u)}, {x} is the state of the system, {u} is the control, both are functions of time with values in vectorspaces.

Example: given vectorfields {X_1,\ldots,X_m}, {f(x,u)=\sum u_i X_i(x)}.

1.1. Controllability

Controllability: which states can one reach from a given state {x_1} ?

Theorem 1 (Chow 1939-Rashevski 1938) Data: vectorfields {X_1,\ldots,X_m} on a neighborhood. Assume that the Lie algebra generated by these generates the whole tangent space at point {x_1}. Then every point in a smaller neighborhood of {x_1} can be reached from {x_1}.

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: {[X,Y]} is a direction in which you can move using controls in directions {X} and {Y} only. Moving in that direction requires patience: moving aling {X}, {Y}, {-X} and finally {-Y} produces a quadrilateral which nearly closes up, but not quite. It is the small defect which is in the direction {[X,Y]}, 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 {\mathcal{F}=\{X_1,\ldots,X_m\}} be a family of vectorfields. Denote by {Lie^{1}(\mathcal{F})=span(\mathcal{F})} and then, recursively,

\displaystyle  \begin{array}{rcl}  Lie^{k+1}=Lie^{k}(\mathcal{F})\cup\{[X,Y]\,;\,Y\in Lie(\mathcal{F}),\,Y\in Lie^{k}(\mathcal{F})\}. \end{array}

Bracket generating means that for some {r\in{\mathbb N}}, {Lie^{k}(\mathcal{F})} evaluated at {x_1} equals the tangent space. The smallest {r} has geometric sgnificance.

1.4. Ways of proving the theorem

In {{\mathbb R}^3}, 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 {E} 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 {u} is regular if {E} is a submersion at {u}, 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 {u} is the rank of {E} at {u}.

The Chow-Rashevski Theorem will follow from the following

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

2.2. Density of regularity

Most controls in {L^2} are regular. In fact, every ball of {L^2} contains a regular control. Proof by contradiction. Let {u} maximize {rank(DE)} in a small ball of {L^2}, let {d<n} be its rank. There exists controls {v_i} such that {\lambda\mapsto E(u+\sum\lambda_{i=1}^d v_i)} is an immersion at 0, its image is a submanifold {N}. Then {E} maps into {N}. In particular, the initial vectorfields {X_i} are tangent to {N}, contradiction.

More generally, the set of regular controls is dense in {L^2} (concatenate). A theorem of Sontag asserts that is dense in {C^\infty}.

2.3. The return method

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

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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