On the Reduced Rank Hypersurface Germ Condition for Near-Controllability of Bilinear Systems

Controllability of nonlinear systems has been extensively studied by using the Lie algebra methods. Although local controllability can be proved, global controllability is in general hard to obtain even for bilinear systems. Recently, a new approach is developed to study controllability of bilinear systems by checking whether the reduced rank points corresponding to the bilinear terms form hyperplanes or hypersurfaces, so that near-controllability, in the global sense, can be algebraically verified once no hypersurface exists. In this paper, we show that there is a flaw in the algorithm for checking the reduced rank hypersurface germ condition (RRHGC) that some special cases are overlooked. We thus propose a new algorithm for testing the RRHGC based on a Darboux-polynomial characterization, which can be used to deal with all cases. An example is provided to demonstrate the proposed new algorithm.