New results in the reduction of linear time-varying dynamical systems

In this paper we consider finite dimensional linear time-varying dynamical systems (LDS) of the form x=A(t)x, x(t0)=x0. Such a system is said to be semiproper (self-commuting) if for all t, ??, A(t)A(??)=A(??)A(t). Using some recent results for obtaining explicit solutions for semiproper systems (see [24], [25], [26], and [27]), we here expand the family of Liapunov reducible systems to include those systems that can be reduced to semiproper ones via what will be called D-similarity transformations. Within this new framework, we define the notion of primary D-similarity transformations, and prove that every LDS that is "well-defined" in a certain sense is reducible by a finite sequence of primary D-similarity transformations. We shall also present an explicit technique for constructing such transformations for LDS with virtually triangular A(t) (i.e. A(t)=LT(t)L-1 for some nonsingular constant matrix L and triangular matrix T(t)). There are difficulties in explicitly obtaining primary D-similarity transformations for the reduction of general LDS. However, in this paper it is shown that, instead of studying such general cases, it suffices to investigate only the reduction of LDS with normal A(t). To achieve these results we treat PA{x}=Ax-x as an operator on a vector space over a differential field, and thereby generalize some familiar results in the theory of matrices over a number field. In particular we introduce the notions of partial spaces, partially linear operators, and LDE operators PA, together with the ideas of D-eigenvalues, D-eigenvectors and D-similarity transformations, all of which are believed to be new.