Saint-Petersburg State University,

28 Universitetsky Ave.,

198504 St.-Petersburg,

RUSSIA

E-mail: skalnitsky@hotmail.com

**Symmetries of Weak-Controllable Systems**

**Abstract:**

In the report we are extending Kobayashi's ideas to proof the completeness
of the nonlinear control system symmetries. The main notion is the property
of weak-controllability. Namely, the set of fields on manifold is weak-controllable
if there is the point from any neighborhood of which we can reach any other
point by trajectories (in spite of controllability, when one can do it
from the point itself). The main result is the fact that if the set of
complete fields is weak-controllable, then all symmetries of it are complete.
This theorem can be regarded as extension of the Palais theorem, stating
the completeness of fields generating by Lie brackets if

Lie algebra is finite-dimensional. Now we can add to this algebra all
symmetries in the case of weak-controllability. In particular, criteria
of controllability obtained in the Nonlinear Control System Theory are
applicable.