Pointwise stabilization of discrete-time matrix-valued stationary Markov chains

Let (Ω,F ,P) be a probability space and S = {S1, . . . , SK} a discrete-topological space that consists of K real d-by-d matrices, where K and d both ≥ 2. In this paper, we study the pointwise stabilizability of a discrete-time, time-homogeneous, stationary (p, P)-Markovian jump linear system Ξ = (ξn) n=1 where ξn : Ω → S. Precisely, Ξ is called “pointwise convergent”, if to any initial state x0 ∈ R1×d, there corresponds a measurable set Ωx0 ⊂ Ω with P(Ωx0) > 0 such that x0 ∏n l=1 ξl(ω) → 01×d as n → +∞, ∀ω ∈ Ωx0 ; and Ξ is said to be “pointwise exponentially convergent”, if to any initial state x0 ∈ R1×d, there corresponds a measurable set Ωx0 ⊂ Ω with P(Ω ′ x0) > 0 such that x0 ∏n l=1 ξl(ω) exponentially fast −−−−−−−−−−−−→ 01×d as n → +∞, ∀ω ∈ Ωx0 . Using dichotomy, we show that if Ξ is product bounded, i.e., ∃β > 0 such that ‖ ∏n l=1 ξl(ω)‖2 ≤ β ∀n ≥ 1 and P-a.e. ω ∈ Ω; then Ξ is pointwise convergent if and only if it is pointwise exponentially convergent.