Strong accessibility and integral manifolds of the continuous-time nonlinear control systems

The paper develops the algebraic formalism that is based on the language of ideals and modules associated with the analytic control system given by the ordinary differential equations. Using the Nagano theorem we show how the integral manifolds of Lie algebra of the considered system can be determined by the generators of some ideal of the ring of germs of analytic functions that is invariant with respect to germs of vector fields from the Lie algebra associated with the system. Additionally, basing on the language of ideals and modules the strong accessibility problem is studied. The singular points where the rank of (co)distributions associated to the system is different than at points from their neighbourhoods are considered. Using the germs of one-forms associated with the generators of the ideal one can define the integrable codistribution that in general is not analytic.