Near-Controllability of Bilinear Systems Through Near-Transitivity: A New Lie-Algebraic Notion

It is well known that Lie algebra methods are the leading methods for studying controllability of continuous-time nonlinear systems including bilinear systems, where the controllability problems are usually transformed into the transitivity problems of the corresponding Lie algebras. Unfortunately, it is in general a difficult task to check transitivity of Lie algebras, especially in the high-dimensional cases. In this article, we propose a new notion, named near-transitivity. We focus on unconstrained bilinear systems and show that the systems are nearly-controllable if and only if their corresponding Lie algebras are nearly-transitive. That is, even if the Lie algebras are not transitive, they can be nearly-transitive and the systems can still own a very large controllable region nearly covering the whole state space. More importantly, we demonstrate that near-transitivity is easier to check than transitivity. This will be useful in both the theory and applications of controllability of nonlinear systems since verifying near-controllability may suffice for most nonlinear systems. Sufficient algebraic conditions as well as algorithms for checking near-transitivity of unconstrained bilinear systems are presented, which are also generalized to inhomogeneous bilinear systems to derive near-controllability. Furthermore, we apply the presented near-transitivity results to structural bilinear systems to derive necessary and sufficient conditions on structural near-controllability. Examples are given to demonstrate the proposed near-transitivity of this article.