A New Approach to Near-Controllability of Bilinear Systems: Addressing Semigroups Through Analyzing the Uncontrollable Regions

Lie algebra methods have been the leading methods in studying the controllability problems of nonlinear systems including bilinear systems over the past decades. However, once semigroups tend to appear, which often occurs on, for instance, bilinear systems with drift, the Lie algebra methods are limited unless assumptions are made to avoid semigroups. In this article, we face the problem of semigroups directly without those assumptions. That is, we propose a new approach to controllability of bilinear systems with drift through analyzing the uncontrollable regions of their corresponding strictly bilinear systems. The key is based on our finding that if the uncontrollable regions breaking the connectedness of the state space are hyperplanes, then they have one-to-one correspondence with the common left real eigenvectors of all coefficient matrices. As a result, we can divide the controllability study into two subtasks: 1) the geometric characterization of the uncontrollable regions for the corresponding strictly bilinear systems; and 2) the recovery of the connectedness by using the drift term for the original systems. If the controllable regions can be connected through the drift term, then the original systems can be proved to be nearly-controllable. Sufficient criteria as well as algorithms for near-controllability of bilinear systems with drift are thus obtained. Examples are given to illustrate the obtained controllability results of this article.