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

Abstract

In this paper, 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, which is necessary for a classical controllability result to work, does not fit such bilinear systems, they can still be controllable. Our approach to prove 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. Examples are given to illustrate the obtained controllability results of this paper.