Topological equivalence and topological linearization of controlled dynamical systems

The general, differential-equation-independent definition of a continuous-time controlled dynamical system as well as of the state space transformation and static state feedback are introduced. This approach makes it possible to consider transformations that are not smooth and introduce the so-called topological equivalence of controlled dynamical sys­ tems. It is shown that this approach generalizes the usual definitions based on the notion of the smooth ordinary differential equation with the control parameter. Topological equiv­ alence is then used to introduce and investigate the problem of exact topological feedback linearization of a given nonlinear system. Sufficient conditions for the topological linearizability of planar systems are obtained. They particularly show that there do exist smooth systems that are topologically linearizable, but not smoothly linearizable. Finally, we in­ dicate possible application of the topological linearization to the nonsmooth stabilization. Illustrative examples are included.