# Etude asymptotique et transcendance de la fonctionvaleur en contrôle optimal. Catégorie log-exp en géométrie sous-Riemannienne dans le cas Martinet.

Abstract : The main subject of this work is the study and the role of
abnormal trajectories in optimal control theory.

We first recall some fundamental results in optimal control. Then
we investigate the optimality of abnormal trajectories for
single-input affine systems with constraint on the control, first
for the time-optimal problem, and then for any cost, the final
time being fixed or not.
Using such an affine system,
we extend this theory to sub-Riemannian systems of rank 2.
These results show that, under general conditions, an abnormal
trajectory is \it{isolated} among all solutions of the system
having the same limit conditions, and thus is \it{locally
optimal}, until a first \it{conjugate point} which can be
characterized.

Then we investigate the asymptotic behaviour and the regularity
of the value function associated to an analytic affine system
with a quadratic cost. We prove that, if there is no abnormal
minimizer, then the value function is \it{subanalytic and
continuous}. If there exists an abnormal minimizer, the
subanalytic category is not large enough in general, notably in
sub-Riemannian geometry. The existence of an abnormal minimizer
is responsible for \it{non-properness} of the exponential
mapping, which implies a phenomenon of \it{tangency} of the level
sets of the value function with respect to the abnormal
direction. In the single-input affine case, or in the
sub-Riemannian case of rank 2, we describe precisely this
contact, and we get a partition of the sub-Riemannian sphere near
the abnormal into two sectors called \it{$L^\infty$-sector} and
\it{$L^2$-sector}.\\
The question of transcendence is studied in the Martinet
sub-Riemannian case where the distribution is
$\Delta=\rm{Ker }(dz-\f{y^2}{2}dx)$. We prove that for a general
gradated metric of order $0$~:
$g=(1+\alpha y)^2dx^2+(1+\beta x+\gamma y)^2dy^2$,
spheres with small radii \it{are not subanalytic}. In the general
integrable case where $g=a(y)dx^2+c(y)dy^2$, with $a$ and $c$
analytic, Martinet spheres belong to the \it{log-exp category}.
Mots-clés :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00086511
Contributor : Emmanuel Trélat <>
Submitted on : Wednesday, July 19, 2006 - 9:26:20 AM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Monday, April 5, 2010 - 10:08:50 PM

### Identifiers

• HAL Id : tel-00086511, version 1

### Citation

Emmanuel Trélat. Etude asymptotique et transcendance de la fonction
valeur en contrôle optimal. Catégorie log-exp en géométrie sous-Riemannienne dans le cas Martinet.. Mathématiques [math]. Université de Bourgogne, 2000. Français. ⟨tel-00086511⟩

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