Orbital Decompositions of Control Systems and Multivector Fields

Abstract

We study local decompositions of nonlinear dynamical control systems. Orbital equivalences are used to transform systems into decomposable forms. Similar to the case of a classical decomposition, each orbital decomposition is defined by an invariant distribution. We consider an invariant multivector field composed of fields lying in this distribution. For two orbital decompositions of a system, we define a commutation operation that allows constructing one more system decomposition. The constructions introduced are demonstrated with an example of the Kapitza pendulum system. The analysis of the transformations established for this example leads to a change of variables that transforms the Kapitsa pendulum system into the Hilbert system.