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

Abstract

Abstract It is well known that the state transition matrix ∊ (t,t0)of the state equations X = A(t)X can be expressed as exp (A1t) exp (A2(t − t0)) exp ( − A1 t0rpar; with A2 A,A(0)−A1 if and only if A(t) = exp (A1t)A(0) exp ( − At). Necessary and sufficient conditions assuring the existence of 4, and algebraic methods for its computation have recently been given by the authors. In this paper, under the assumption that eigenvalues of A1, are simple, a different procedure which is simpler as compared to solving an overdetermined system of algebraic equations is given. The procedure is based on a canonical form of the matrix A(t) and also determines whether the state equations can be partitioned into one time-varying and one time-invariant system.