Floquet theorem for linear implicit nonautonomous difference systems

Abstract

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.

Stability of arbitrarily switched discrete-time linear singular systems of index-1

“…From condition (ii) in Definition 1 we show that ker R , {1,2,.., }. Using similar arguments as in [8][9][10][11] we can prove the following results. Theorem 1 For index-1 SDLS (1), the following assertions hold.…”

The one-step-map for switched singular systems in discrete-time

“…Although the following results are partially obtained by similar arguments as in [2], [3], [14], we will give their proofs here to make our presentation self-contained. Furthermore, these properties also play a crucial rule for the treatment of the inhomogeneous case later.…”

“…Although some authors have already studied discrete-time singular switched (or time-varying) systems (e.g. [15], [2], [16], [3], [26], [29], [31], [28], [6], [7], [27], [1]) it seems that the existence of a one-step-map was not investigated so far and we want to close this gap with this contribution.…”

On stability and Bohl exponent of linear singular systems of difference equations with variable coefficients

“…The following stability notions generalize those for ordinary difference equations. See also [8,10]. Definition 2.7: The zero solution of Equation (1.2) is said to be stable if for any ε > 0 and n 1 ∈ N(n 0 ) there exists a positive constant δ = δ(ε, n 1 ) such that the inequality…”

“…While the theory of DAEs, the continuous-time counterpart of (1.1), has been almost well established, qualitative results in the theory of singular difference equations, particularly those for non-autonomous systems, are very few. Though the first result on LSDEs with variable coefficients was given a long time ago in [3], interesting results on the existence and the stability of solutions have been published only recently, see [7][8][9][10][11][12][13]. Most of the stability and robust stability results for singular difference equations are obtained for autonomous systems, e.g.…”