Determination of the state transition matrix of exponentially varying systems : algebraic methods†

Abstract

Abstract Necessary and sufficient conditions under which a given square matrix A(t) can be decomposed into the product, exp (A1t)A(0) exp ( — A1l) where Ax is a constant matrix are investigated. The state transition matrix of the state equations [Xdot](t) = A(t)X(t) can be expressed as φ(t.t0) = exp (A1l) exp A2(t —t0) exp (— A1l) with A2δA(0) — A1if and only if A(t) has such a decomposition. In this paper some properties of the system described by [Xdot](t) = A(t)X(t) are examined and necessary and sufficient conditions for A(i) to be decomposable are given; an algorithm for determining A1is also provided