On the Equivalence of Controlled Systems with a Continuous Symmetry Group

The problem of global transformation of nonintegrable control dynamical systems to a standard one is considered with the help of Lie Group theory methods. It is assumed that a Lie group (Lie algebra) is known only for the standard dynamical systems. The condition of global equivalence of nonlinear dynamical systems is expressed in terms of solvable Lie algebras. The class of dynamical systems equivalent to the standard one is determined by the use of the Cartan criterion for the solvability of a Lie algebra. Group theory methods can be used to simplify the analysis of controlled dynamic systems if the symmetry group is known from a priori considerations. Otherwise, it is necessary to find preliminary infinitesimal operators of the Liegroup, i.e., a Lie algebra. In fact, this is not simpler than to integrate immediately the initial equations. In the present paper the problem is formulated in a different manner so as to avoid this difficulty. Let us define a class of standard controlled systems whose symmetry group (or more exactly, symmetry Lie algebra) is known. The symmetry Lie algebra is obtained for the controlled systems which are equivalent to the standard ones After that the class of dynamic systems, invariant with respect to the known symmetry group, is determined.