## Notes of Ludovic Rifford’s lecture nr 4

Open problems

1. The Sard conjecture
2. Regularity of geodesics
3. Small balls

1. The Sard conjecture

1.1. Statement

Theorem 1 (Morse 1939 for ${p=1}$, Sard 1942) If ${f:{\mathbb R}^d\rightarrow{\mathbb R}^p}$ is of class ${C^k}$,

$\displaystyle \begin{array}{rcl} k\geq\max\{1,d-p+1\}\quad \Rightarrow\quad \mathcal{L}^p(\textrm{critical values})=0, \end{array}$

and this is sharp (Whitney).

Does this theorem generalize to the endpoint map of a smooth control system ?

Conjecture. The set of all positions at time ${t}$ of singular paths starting at ${x}$ has measure zero.

Remark. There are examples of smooth (even polynomial) functions on ${L^2}$ which do not satisfy Sard’s theorem. The only infinitesimal version of Sard’s theorem is Smale’s for Fredholm maps.

Conjecture is open for Carot groups (which may be harder).

1.2. Positive cases

Fat distributions have no singular curves but constants.

For rank two distributions in dimension 3, singular curves are contained in the Martinet surface which is known to be countably 2-rectifiable. Conjecturally, the singular values of the endpoint map have Hausdorff dimension ${\leq 1}$. Generically, the horizontal curves on the Martinet surface form a foliation whose singularity are either saddles or foci. At foci, the length of leaves is infinite, so one can ignore them.

1.3. The minimizing Sard conjecture

Let ${S}$ denote the set of points joined to ${x}$ by a minimizing geodesic which is singular. Let ${S_s\subset S}$ denote the set of points joined to ${x}$ by a minimizing geodesic which is singular and not the projection of a normal extremal.

The following partial result turns out to be rather easy.

Proposition 2 (Rifford-Trélat, Agrachev) ${S}$ has empty interior.

Lemma 3 Assume that there is a function ${\phi:M\rightarrow{\mathbb R}}$ such that

1. ${\phi}$ is differentiable at ${y}$,
2. ${\phi(y)=d(x,y)^2}$ and ${d(x,y)^2>\phi(z)}$ for all neigboring ${z\not=y}$.

Then there is a unique minimizing geodesic between ${x}$ and ${y}$, which is the projection of a normal extremal ${\psi}$ such that ${\psi(1)=(y,D_y\psi}$.

\proof

Let ${v}$ be the control of some minimizing geodesic. For ${u\in L^2}$ close to ${v}$,

$\displaystyle \begin{array}{rcl} \|u\|_{L^2}^2 =C(u)\geq e(x,E^x(u)), \end{array}$

with equality at ${u=v}$. By assumption, ${e(x,E^x(u))\geq\phi(E^x(u))}$, with equality at ${u=v}$. Therefore ${v}$ minimizes ${C(u)-\phi(E^x(u))}$ in a neighborhood of ${v}$, and it is locally unique. So there is ${p\in T^*_y M}$ such that ${p\cdot D_vE^x=D_vC}$, ${v}$ is normal, q.e.d.

\proof

of Proposition. Any continuous function has a smooth (even constant) support function at a dense set of points, q.e.d.

Question. Can one improve this to full measure ?

2. Regularity of minimizers

Projections of normal extremals are smooth.

Question. Are abnormal minimizing geodesics of class ${C^1}$ ?

2.1. Partial results

Theorem 4 (Monti-Leonardi) Consider an equiregular (${Lie^k}$ all have constant dimension) distribution. Assume that ${[Lie^k,Lie^\ell]\subset Lie^{k+\ell+1}}$. Then curves with a corner cannot be minimizing.

Theorem 5 (Süssmann) If data are real analytic, singular controls are real analytic on an open dense subset of their interval f definition.

This comes from sub-analytic geometry.

3. Small balls

Question. Are small spheres homeomorphic to spheres ?

It is true in Carnot groups.

Yuri Baryshnikov claims that the answer is yes in the contact case, but the proof does not seem to be correct.

In the absence of abnormal geodesics, then almost every sphere at ${x}$ is a Lipschitz submanifold.