Stability of nonlinear sampled-data control systems

With regard to the asymptotic stability in the large of nonlinear, autonomous sampled-data systems, the following conjecture has been quoted frequently in the literature. Namely, "if the lineafized system is stable for all points of the state space, the original nonlinear system is asymptotically stable in the large" (henceforth abbreviated a.s.i.l.). This paper shows by counter examples that the statement is not true in general. Both total and incremental linearizations are considered and in both cases the conjecture is false. Finally, a sufficient condition for a a.s.i.l, is given.