Abstract
In this paper, the problems of the output reachability and a kind of input-output equivalence for a class of nonlinear systems are considered by applying the theory of differential geometry. Because the right hand side of the dynamical equation describing a motion defines a vector field on some manifold, the input-output equivalence can be interpreted in terms of the vector field of the system. The input-output equivalence or the equivalence of the sets of vector fields is defined as follows; if two systems, defined on the same manfold, have the same output equations and the equivalent sets of vector fields, then they produce the same output for the same input. This type of equivalence concept was introduced by Brockett for studying the minimal realization problem of bilinear systems. The systems considered here are defined on a real analytic manifold. The state equations depend on the input variables linearly. Their output equations are analytic with respect to state varables. These systems are the same type studied by Sussman and Jurdgevic, and they are a generalization of bilinear systems. The main results obtained here are the necessary and sufficient conditions for the output reachability and the equivalence of the sets of vector fields. These conditions are algebraic and easily computable. Through these discussions, the concept of the invariant Lie algebra is derined, which corresponds to the invariant subspace of linear systems.
