Semi-groupes intégraux de SL(2,$${\mathbb{R}})$$. Application à la théorie du contrôle. (Integral semigroups on SL(2,$${\mathbb{R}})$$. Application to control theory).(French)Zbl 0602.93005

The subject of this work is the controllability problem for a family F of right invariant vector fields on the standard Lie group SL(2,$${\mathbb{R}})$$. In particular, it is proven that F is controllable if and only if one of the following two cases occurs:
1) the cones generated by F and -F coincide and $$Lie(F)=sl(2,{\mathbb{R}});$$
2) the cones generated by F and -F differ but the cone generated by F contains a compact element.
The main result is applied to the controllability problem on $$GL^+(2,{\mathbb{R}})$$, on $$S^ 1$$ and on $${\mathbb{R}}^ 2\setminus \{0\}$$.
Reviewer: A.Bacciotti

MSC:

 93B05 Controllability 93B03 Attainable sets, reachability 93C10 Nonlinear systems in control theory 22E99 Lie groups