New Results in the Reduction of Linear Time-Varying Dynamical Systems

This paper considers finite-dimensional, linear time-varying dynamical systems (LDS) of the form $\dot {\bf x} = {\bf A}(t){\bf x}$, ${\bf x}(t_0 ) = {\bf x}_0 $. Such systems are said to be semiproper (self-commuting) if for all t, $\tau $, ${\bf A}(t){\bf A}(\tau ) = {\bf A}(\tau ){\bf A}(t)$. Using some recent results for obtaining explicit solutions for semiproper systems (see [26]–[29], [34], [35]), the family of Lyapunov reducible systems is expanded to include those systems that can be reduced to semiproper ones via what will be called D-similarity transformations. Within this new framework is defined the notion of primary D-similarity transformations, and every LDS that is “well defined” in a certain sense is proved reducible by a finite sequence of primaryD-similarity transformations. The paper also presents an explicit technique for constructing such transformations for LDS with virtually triangular ${\bf A}(t)$ (i.e., ${\bf A}(t) = {\bf L}{\bf T}(t){\bf L}^{ - 1} $ for some nonsingular constant matrix ${\bf L}$ and triangular matrix ${\bf T}(t)$). There are difficulties in obtaining, explicitly, primary D-similarity transformations for the reduction of general LDS. However, this paper shows that, instead of studying such general cases, it suffices to investigate only the reduction of LDS with normal${\bf A}(t)$. To achieve these results, $\mathcal {P}_{\bf A} \{ {\bf x}\} = {\bf A}{\bf x} - \dot {\bf x}$ is treated as an operator on a vector space over a differential field, and thereby some familiar results in the theory of matrices are generalized over a number field. In particular, the authors introduce the notions of partial spaces, partially linear operators, linear differential equation (LDE) operators $\mathcal {P}_{\bf A} $ and D-similarity transformations, all of which are believed to be new.