Controllability counterexamples of linear systems and bilinear systems arising in Euler discretization

Since controllability of continuous-time nonlinear systems is quite difficult to study, whether it can be explored through their discretized systems becomes an important issue. In this paper, we show that, although controllability of linear time-invariant systems keeps invariant in Euler discretization, controllability of continuous-time nonlinear systems and even linear time-varying systems may be different from controllability of their discretized systems by Euler discretization. In particular, we present arbitrary-finite-dimensional controllability counterexamples through linear time-varying systems and bilinear systems to reveal that controllability of a continuous-time control system may be changed after Euler discretization. Numerical examples are given to demonstrate the presented controllability counterexamples.