Global state and feedback equivalence of nonlinear systems (Q1066091)

The problem of finding global state space transformations and global feedback of the form \(u(t)=\alpha (x)+v(t)\) to transform a given nonlinear system \(\dot x=f(x)+g(x)u\) to a controllable linear system on \(R^ n\) or on an open subset of \(R^ n\) is considered here. A complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem are presented. The solution presented here is valid in the \(C^ k\) situation where \(2n+1\leq k\leq \omega\) and agrees with \textit{W. Respondek}'s solution [''Global aspects of linearization, equivalence to polynomial forms and decomposition of nonlinear control systems'', Algebraic and geometric methods in nonlinear control, Proc. CNRS Conf., Paris 1985] in the analytic case which was obtained independently. In a subsequent paper [''Global linearization by feedback and state transformations'', Decision and control, Proc. IEEE Conf. Dec. 1985] the authors extended the results to the multiinput case and to the case of more general feedback (i.e. \(u(t)=\alpha (x)+\beta (x)v(t))\).