Determination of the Transitivity of Bilinear Systems

Abstract

We consider a bilinear control system on $\mathbb{R}_0^n = \mathbb{R}^n - \{ 0\} $\[\frac{{dx}}{{dt}}(A_0 + \sum\limits_{i = 1}^r {u_i (t)A_i } )x,\] where $x \in \mathbb{R}^n ,A_0$, $A_1 , \cdots ,A_r $ are $n \times n$ real matrices, $\mathfrak{g}$ is the Lie algebra they generate, and $u_1 (t), \cdots ,u_r (t)$ are real valued control functions. Although there exists a standard rank condition in terms of the Lie algebra g which Sussman and Jurdjevic have shown to be sufficient to guarantee accessibility, it is primarily of theoretical interest, being essentially impossible to apply to given data. In this paper the authors investigate the possibility of developing algorithms involving only rational computations on the matrices, which would determine whether the rank condition is everywhere satisfied, i.e. whether the bilinear system has the accessibility or, in some instances, controllability property. This is equivalent to determining whether or not the matrix Lie group G generated by $\exp (tA_i )$, $i = 0,1, \cdots ,r$, is or is not transitive on $\mathbb{R}_0^n $. The possible transitive Lie groups have been classified by one of the authors. Using this classification, it is shown that there do exist various sequences of rational operations on the matrices $A_0 ,A_1 , \cdots ,A_r $ which enable one in a finite number of steps to decide whether or not the system has the accessibility-transitivity property.