Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems

Let S = {S 1, S 2} ⊂ Rd×d have a common, but not necessarily strict, Lyapunov matrix (i.e. there exists a symmetric positive-definite matrix P such that P − S T k PS k ≥ 0 for k = 1, 2). Based on a splitting theorem of the state space Rd (Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several stability criteria for the discrete-time linear switched dynamics xn = S σn · · · S σ1 (x0), x0 ∈ R and n ≥ 1 governed by the switching signal σ : N → {1, 2}. More specifically, let ρ(A) stand for the spectral radius of a matrix A ∈ Rd×d, then the outline of results obtained in this paper are: (1) For the case d = 2, S is absolutely stable (i.e., ‖S σn · · · S σ1‖ → 0 driven by all switching signals σ) if and only if ρ(S 1), ρ(S 2) and ρ(S 1S 2) all are less than 1; (2) For the case d = 3, S is absolutely stable if and only if ρ(A) < 1 ∀A ∈ {S 1, S 2} for l = 1, 2, 3, 4, 5, 6, and 8. This further implies that for any S = {S 1, S 2} ⊂ Rd×d with the generalized spectral radius ρ(S) = 1 where d = 2 or 3, if S has a common, but not strict in general, Lyapunov matrix, then S possesses the spectral finiteness property.