% From mormul@duch.mimuw.edu.pl Sat Mar 17 06:10:00 2001
% From: "mormul" <mormul@duch.mimuw.edu.pl>
% Date: Sat, 17 Mar 01 12:05:30 CST
%
% Dear Hector,
%
% may I enclose the Latex file of a proposal for NCN4/Sheffield, submitted on
% March 14 ?
%
% It is longer (2.5 pages), but essentially the same thing sent to you to
% Jerusalem [in Feb., when in Toulouse, I overheard that this year you were
% there] in a big paper letter ('submitted to Nolcos'01').
%
% I added a mention on the origin of the question and extended the references
% part; by Nolcos it was not accepted.
%
% With hearty greetings,
% Piotr
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\documentstyle{article}
\def\n{\noindent}
\title{Minimal nilpotent bases for Goursat distributions in dimensions 
not exceeding 7}
\author{Piotr Mormul\\
\small Institute of Mathematics, Warsaw University, Poland\\
\small e-mail: mormul@mimuw.edu.pl}
\date{}
\begin{document}
\maketitle
\n{\bf Abstract.} Goursat distributions (i.\,e., special rank--2 subbundles 
$D$ in the tangent bundle to an $(r+2)$--dimensional manifold $M$ s.\,t. 
the Lie square $[D,\,D]$ of $D$ is everywhere of rank 3, the Lie square 
of $[D,\,D]$ is everywhere of rank 4, and so on) \,obtained in the 90's 
the well-known presentation in the kinematic model of a car pulling 
$r - 1$ passive trailers, culminating in the work [8] bringing in 
critical angles $a_1 = {\pi\over 2},\:a_2,\:a_3,\dots$ There exist 
also explicit conversions [16], [14] of trailer systems described in 
trigonometric functions into classical (polynomial) local presentation 
of Goursat objects due to Kumpera and Ruiz [9]. 

\n It is known after [8], [5], [11] that germs of Goursat distributions 
on such $M$ can be stratified into $F_{2r-3}$ (Fibonacci number) 
{\it geometric classes} encoded by words of length $r$ over the alphabet 
$\{$G,\,S,\,T$\}$ s.\,t. two first letters are always G and never a T 
goes directly after a G. 

\n We will call {\it tangential\,} classes whose codes possess letters G 
only in the beginning, before the first S (if any) in the code. 

\n In the language of the trailer systems, S corresponds to the right angle 
$(a_1)$ between two neighbouring trailers, while a sequence STT$\dots$T 
inside a code corresponds to the next (i.\,e., closer to the car!) 
neighbouring trailer making angle $a_2$ with the second in the couple 
having the right angle, still next making angle $a_3$ with the first next, 
and so as long as many T's is in the sequence. At the following step this 
angle rule breaks down and now the letters G are written in row until some 
next S (related to a new right angle in the configuration of trailers) 
appears. 
\vskip.1cm
Since the 80's there is interest in control theory [7], [1], [4], [6], 
[10], [12] (and recently also in sub-Riemannian geometry [3], [2], [15]) 
in nilpotent bases for distributions associated to control systems linear 
in controls. In 2000 we proved (answering a 98'\,question of Sussmann) 
that Goursat distributions locally belong to this distinguished class, 
cf. Rem.\,1 in [13]. It appears that a basis naturally prompted by any 
Kumpera-Ruiz coordinates for a Goursat germ is nilpotent. Its real Lie 
algebra we call shortly {\it KR algebra}. 

\n In a nilpotent Lie algebra, the minimal number of Lie multiplications  
yielding always zero is called its {\it order of nilpotency}, see [10], 
p.\,239. \,We gave (workshop Math.\,Control Th.\,and Robotics, Trieste 2000) 
explicit formulas for the order of nilpotency $O_{\rm KR}$ of any KR algebra. 
Recalling, $O_{\rm KR}$ is the last term $d_r$ in the sequence 
$d_1,\,d_2,\dots,\,d_r$ defined only in function of the germ'\,\,geometric 
class encoded as an $r$-letter word $\cal C$: $d_1 = 2$, $d_2 = 3$, 
$d_{j+2} = d_j + d_{j+1}$ when the ($j+2$)-th letter in $\cal C$ is S, 
and otherwise $d_{j+2} = 2d_{j+1} - d_j$. 

\n Comparison with Jean's formulas [8] for the non-holonomy degree 
$d_{\rm H}$ of any germ in the geometric class $\cal C$ yields easily that 
a).\,$d_{\rm H} \le O_{\rm KR}$, and b).\,$d_{\rm H} = O_{\rm KR}$ only 
when $\cal C$ is tangential. 

\n In Trieste a question has been put forward if the KR algebras 
is the last word for the Goursat distributions: whether there exist better 
nilpotent bases with lower nilpotency orders. 

\n For germs in tangential classes -- of course not; KR bases are already 
optimal. Up to dimension $r+2 = 5$ all geometric classes are tangential; 
therefore, the question makes sense from dimension 6 onwards. 
\vskip.15cm
In the present contribution we deal with the only non-tangential class in 
dimension 6 and with five such classes existing in dimension 7, and prove 
that 
\vskip.1cm
\n{\bf 1.} In dimension 6, for germs in the class GGSG the nilpotency order 
$O_{\rm KR} = 7$ is minimal possible, notwithstanding the fact that 
$d_{\rm H} = 6$ for these germs. 
\vskip.1cm
\n{\bf 2a.} In dimension 7, for germs in the classes: 

GGSGG \,($d_{\rm H} = 7,\ \,O_{\rm KR} = 9$), 

GGSTG \,($d_{\rm H} = 8,\ \,O_{\rm KR} = 9$), 

GGSGS \,($d_{\rm H} = 11,\ \,O_{\rm KR} = 12$), 

\n their respective KR algebras are of minimal possible nilpotency order.
\vskip.1cm
\n{\bf 2b.} In the same dimension 7, for germs in the classes 

GGGSG \,($d_{\rm H} = 8,\ \,O_{\rm KR} = 10$), 

GGSSG \,($d_{\rm H} = 9,\ \,O_{\rm KR} = 11$), 

\n there is no nilpotent basis of the respective order $d_{\rm H}$.
\vskip.15cm
Therefore, the point 2b is not yet conclusive. Possible bases of 
the respective nilpotency orders $d_{\rm H} + 1$ are not yet excluded. 
\vskip.12cm
\n COROLLARY. In dimension 6 and 7, for all Goursat germs in 
the non-tangential geometric classes, their nilpotent approximations are 
not equivalent to the departure germs. It is so because the nilpotent 
approximation is in itself a Lie algebra always of nilpotency order 
$d_{\rm H}$. Thus those germs are NOT nilpotent in the STRONG sense 
of being equivalent to germ's own nilpotent approximation. 
\vskip.15cm
The weak form of nilpotency -- possession of a nilpotent basis -- appears 
thus much weaker than the strong form of nilpotency of a distribution germ. 
\vskip.3cm
R E F E R E N C E S 
\vskip.15cm
\n[1] A.\,Agrachev, R.\,Gamkrelidze, A.\,Sarychev; Local invariants of 
smooth control systems, {\it Acta Appl. Math. \bf 14} (1989), 191 -- 237. 
\vskip.1cm
\n[2] A.\,Agrachev, J-P.\,\,Gauthier; On subanalyticity of 
Carnot--Caratheodory distances, preprint 25/2000/M, SISSA, Trieste. 
\vskip.1cm
\n[3] A.\,Bella\"iche; in: {\it Sub-Riemannian Geometry}, Birkh\"auser, 
Basel--Berlin 1996, 1 -- 78. 
\vskip.1cm
\n[4] A.\,Bressan; in: {\it Analysis of Controlled Dynamical Systems}, 
Birkh\"auser, Basel--Berlin 1991, 103 -- 117.
\vskip.1cm
\n[5] M.\,Cheaito, P.\,Mormul; {\it ESAIM: Control, Optimisation and 
Calculus of Variations \bf 4} (1999), 137 -- 158. 
\vskip.1cm
\n[6] H.\,Hermes; {\it SIAM Review \bf 33} (1991), 226 -- 238. 
\vskip.1cm
\n[7] H.\,Hermes, A.\,Lundell, D.\,Sullivan; {\it J. Diff. Eqns \bf 55} 
(1984), 385 -- 400. 
\vskip.1cm
\n[8] F.\,Jean; {\it ESAIM: Control, Optimisation and Calculus of 
Variations \bf 1} (1996), 241 -- 266. 
\vskip.1cm
\n[9] A.\,Kumpera, C.\,Ruiz; in: {\it Monge--Amp\`ere Equations and 
Related Topics}, Inst. Alta Math., Rome 1982, 201 -- 248. 
\vskip.1cm
\n[10] G.\,Lafferriere, H.\,Sussmann; in: {\it Nonholonomic Motion Planning}, 
Kluwer, Dordrecht--London 1993, 235 -- 270. 
\vskip.1cm
\n[11] R.\,Montgomery, M.\,Zhitomirskii; Geometric approach to Goursat 
flags, pre-

\n print 1999, available at http://orca.ucsc.edu/$\tilde{\ }$rmont. 
\vskip.1cm
\n[12] R.\,M.\,\,Murray, {\it Math. Control Signals Syst. \bf 7} (1994), 
58 -- 75. 
\vskip.1cm
\n[13] P.\,Mormul; {\it J. Dyn. Control Syst. \bf 6} (2000), 311 -- 330. 
\vskip.1cm
\n[14] W.\,Pasillas-L\'epine, W.\,Respondek; in: {\it Proceedings IEEE}, 
Phoenix (1999), 2914 -- 2919. 
\vskip.1cm
\n[15] Y.\,Sachkov; On nilpotent sub-Riemannian (2,\,3,\,5) problem, 
workshop Math.

\n Control Th.\,and Robotics, Trieste 2000. 
\vskip.1cm
\n[16] O.\,S\"ordalen; in: {\it Proceedings IEEE}, Atlanta (1993), 
382 -- 387. 
\end{document}



















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