Controllability of Continuous-Time Bilinear Systems Without Using the Lie-Algebraic Conditions: A New Perspective

Lie algebra methods are powerful for studying controllability of continuous-time nonlinear systems and most controllability criteria of continuous-time nonlinear systems are Lie-algebraic conditions. However, the Lie-algebraic conditions are in general difficult to verify if the system dimension is high. In this article, we propose a new perspective to study controllability of continuous-time bilinear systems without using the Lie-algebraic conditions. Specifically, we first consider controllability of the bilinear systems in the single-input case under a commutativity condition. We show that, although the Lie algebra rank condition for the homogeneous counterparts, which is necessary for a classical controllability result to work, does not fit such bilinear systems, they can still be controllable. Our approach to proving controllability is using controllability of the discrete-time counterparts of the continuous-time systems, and we derive a necessary and sufficient controllability criterion without the Lie algebra rank condition, which is algebraically verifiable for any finite dimension. More importantly, through this controllability study, we propose a new perspective to deal with the controllability problems of continuous-time bilinear systems by changing the verification of the Lie algebra rank condition to solving two linear algebra problems. That is, we establish a new method beyond Lie algebra methods to controllability of continuous-time bilinear systems. We also generalize the proposed perspective to continuous-time state-affine nonlinear systems to obtain algebraic controllability criteria. Examples are given to illustrate the obtained controllability results of this article.