Pointwise Stabilization of Discrete-time Stationary Matrix-valued Markovian Chains

We study the pointwise stabilizability of a discrete-time, time-homogeneous, and stationary Markovian jump linear system. By using measure theory, ergodic theory and a splitting theorem of state space we show in a relatively simple way that if the system is essentially product-bounded, then it is pointwise convergent if and only if it is pointwise exponentially convergent.

💡 Research Summary

The paper investigates the pointwise stabilizability of discrete‑time, time‑homogeneous, stationary Markovian jump linear systems. Let S = {S₁,…,S_K} be a finite collection of real d × d matrices and let ξ = (ξₙ)ₙ≥1 be a stationary Markov chain on the finite state space K = {1,…,K} with initial distribution p and transition matrix P. The state evolution is given by
 xₙ = x₀ S_{ξ₁} S_{ξ₂} … S_{ξₙ}, n ≥ 1.
Two notions of convergence are introduced. (i) Pointwise convergence: for each initial vector x ∈ ℝ^{1×d} there exists a set Ωₓ of positive probability such that for every ω ∈ Ωₓ the trajectory x S_{ξ₁}(ω)…S_{ξₙ}(ω) tends to zero as n→∞. (ii) Pointwise exponential convergence: for each x there is a set Ω′ₓ of positive probability on which the Lyapunov exponent
 lim sup_{n→∞} (1/n) log‖x S_{ξ₁}(ω)…S_{ξₙ}(ω)‖
is strictly negative. The paper also defines a stronger “consistent exponential convergence” where a single set of ω’s works for all initial states.

The central hypothesis is essential non‑uniform product boundedness: there exists a measurable function β : Ω→