Comments on “A Controllability Counterexample” and the Continuation Lemma

A Technical Note in this journal vol. 50, no. 6, pp. 840-841, June 2005, by Elliott, gives a bilinear example showing that the Euler discretization of a noncontrollable continuous-time system can be controllable. The example is correct, but there was a flaw in a result of the TN, Lemma 1 (“for discrete-time systems, local controllability implies controllability”) that has independent interest. In this note, the lemma is reformulated as a conjecture for continuous-in-state systems, and it is also proved under additional conditions. For a class of two-dimensional bilinear systems the Euler discretization is shown directly to be small-controllable, a fortiori controllable.