Representations of solutions, translation formulae and asymptotic behavior in discrete linear systems and periodic continuous linear systems

Abstract

We give a method for studying of asymptotic behavior of solutions to a periodic continuous linear system. It is based on a representation of solutions given in the paper, which is a reformation of the variation of constants formula into the sum of a $\tau$-periodic function and an exponential-like function. By using such representations, the set of initial values is completely classified according to the asymptotic behavior of the solutions to the continuous system. In particular, the set of initial values of bounded solutions is precisely determined. To give the representation for the continuous system, we will establish translation formulae by comparing two representations of solutions to a discrete linear system. These two representations are deeply related to the binomial coefficients, the Bernoulli numbers and the Stirling numbers.