Geometry and analysis of control-affine systems: motion planning, heat and Schrödinger evolution

127 Pages·2017·1.99 MB·English

by  Dario Prandi

Geometry and analysis of control-affine systems: motion
planning, heat and Schrödinger evolution
Dario Prandi
To cite this version:
DarioPrandi. Geometryandanalysisofcontrol-affinesystems: motionplanning,heatandSchrödinger
evolution. Optimization and Control [math.OC]. Ecole Polytechnique X, 2013. English. ￿NNT: ￿.
￿pastel-00878567￿
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E´cole doctorale de l’E´cole Polytechnique
Geometry and analysis of control-affine
systems: motion planning, heat and
Schro¨dinger evolution
`
THESE DE DOCTORAT
pr´esent´ee et soutenue publiquement le 23 octobre 2013
pour l’obtention du diplˆome de
´
Docteur de l’Ecole Polytechnique
(sp´ecialit´e math´ematiques appliqu´ees)
par
Dario Prandi
Composition du jury
Directeurs: Ugo Boscain
Fr´ed´eric Jean
Mario Sigalotti
Rapporteurs: Richard Montgomery
Pierre Rouchon
Examinateurs: St´ephane Gaubert
Jean-Paul Laumond
Pierre Pansu
Centre de Math´ematiques Appliqu´ees — UMR 7641

To Meg.
Knowledge and ability were tools,
not things to show off.
– Haruki Murakami, 1Q84

Cette th`ese traite de deux probl`emes qui ont leur origine dans la th´eorie du contrˆole
g´eom´etrique, et qui concernent les syst`emes de contrˆole avec d´erive, c’est-`a-dire de la
forme q˙ = f (q)+ m u f (q). Dans la premi`ere partie de la th`ese, on g´en´eralise le
0 j=1 j j
concept de complexit´e de courbes non-admissibles, d´ej`a bien compris pour les syst`emes
(cid:80)
sous-riemanniens, au cas des syst`emes de contrˆole avec d´erive, et on donne des estima-
tionsasymptotiquesdecesquantit´es. Ensuite, dansladeuxi`emepartie, onconsid`ereune
famille de syst`emes de contrˆole sans d´erive en dimension 2 et on s’int´eresse `a l’operateur
de Laplace-Beltrami associ´e et `a l’´evolution de la chaleur et des particules quantiques
qu’il d´efinit. On ´etudie plus particuli`erement l’effet qu’a l’ensemble ou` les champs de
vecteurs contrˆol´es deviennent colin´eaires sur ces ´evolutions.
This thesis is dedicated to two problems arising from geometric control theory, re-
garding control-affine systems q˙ = f (q)+ m u f (q), where f is called the drift. In
0 j=1 j j 0
the first part we extend the concept of complexity of non-admissible trajectories, well
(cid:80)
understood for sub-Riemannian systems, to this more general case, and find asymptotic
estimates. Then, in the second part of the thesis, we consider a family of 2-dimensional
driftless control systems. For these, we study how the set where the control vector fields
become collinear affects the evolution of the heat and of a quantum particle with respect
to the associated Laplace-Beltrami operator.

Acknowledgements
First and foremost, I would like to thank my advisor, without whom this thesis would
have never saw the light. Ugo, thank you for the trust, for all the opportunities you
gave me, for having taught me what does it mean to do research and how it can be done
literally everywhere. I could not have hoped for a better advisor.
I would like to thank my co-advisors, Fr´ederic Jean and Mario Sigalotti, for their
patience and perseverance. Fred, you taught me a lot all the while showing me that it
is possible to commit to mathematics without letting it devour you. Mario, thank you
for having always been willing to help and to give me advices. Also, your proofreading
taught me the meaning of precision.
IamdeeplyindebitedwithProf. RichardMontgomeryandwithProf. PierreRouchon
for having reviewed this thesis, giving me useful insights. My grateful thanks are also
extended to Prof. Pierre Pansu and to DR St´ephane Gaubert for giving me the honor
of being members of this jury.
A special thanks goes to my family, for still being unconditionally supportive after
8 long years as they were when I was a freshman. I could never have gotten this far
without you.
I am also grateful to Davide Barilari, for being a collegue, a friend, and somewhat of
a mentor. To Mauricio Godoy Molina, for the endless discussions on mathematics and
everything else. To Petri Kokkonen, that always made me feel at home no matter where
or when. And to Marco Penna, for his unconditioned help and friendship.
I want to thank everyone else who helped me taking the best out from these three
years, making of them a special and unforgettable experience. Athena, Cesare, Cosimo,
Daniela, Diana, Fabio, Francesco, Gaspard, Giovanni, Giulio, Mattia, Mauro, Micon,
Miquel, Lucas, Polo, Riccardo(s), Sri, Stefano, Tudor, a big hug to you and to all the
others I could not fit here.
Finally, Margherita, thank you for having always been there. Thank you for having
non stoppingly encouraged me and stood by my side, even when it felt like hell. Thank
you for everything.

Contents
I. Preliminaries 1
1. Introduction 3
1.1. Sub-Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1. Metric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2. Complexity and motion planning . . . . . . . . . . . . . . . . . . . 8
1.1.3. The sub-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4. The Laplace-Beltrami operator in almost-Riemannian geometry . . 12
1.2. Control-affine systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1. H¨older continuity of the value function . . . . . . . . . . . . . . . . 14
1.2.2. Complexity and motion planning . . . . . . . . . . . . . . . . . . . 16
1.3. The Laplace-Beltrami operator on conic and anti-conic surfaces . . . . . . 19
1.3.1. Passage through the singularity . . . . . . . . . . . . . . . . . . . . 21
1.3.2. Stochastic completeness . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4. Perspectives and open problems . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1. Complexity of non-admissible trajectories . . . . . . . . . . . . . . 24
1.4.2. Singular diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Bibliography 27
II. Papers 33
2. H¨older continuity of the value function for control-affine systems 35
3. Complexity of control-affine motion planning 57
4. The heat and Schr¨odinger equations on conic and anticonic surfaces 83
III. Abstracts 111
5. Abstract 113
6. Resum´e 115