Symmetries of Weak-Controllable Systems
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.