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Pointwise stabilization of discrete-time matrix-valued stationary Markov chains | | Xiongping Dai
; Yu Huang
; Mingqing Xiao
; | | Date: |
1 Jul 2011 | | Abstract: | Let $(Omega,mathscr{F},mathbb{P})$ be a probability space and
$S={mathrm{S}_1,...,mathrm{S}_K}$ a discrete-topological space that
consists of $K$ real $d$-by-$d$ matrices, where $K$ and $d$ both $ge2$. In
this paper, we study the pointwise stabilizability of a discrete-time,
time-homogeneous, stationary $(p,P)$-Markovian jump linear system
$Xi=(xi_n)_{n=1}^{+infty}$ where $xi_ncolonOmega
ightarrowS$.
Precisely, $Xi$ is called "pointwise convergent", if to any initial state
$x_0inmathbb{R}^{1 imes d}$, there corresponds a measurable set
$Omega_{x_0}subsetOmega$ with $mathbb{P}(Omega_{x_0})>0$ such that ean
x_0{prod}_{ell=1}^nxi_ell(omega) omathbf{0}_{1 imes
d}quad extrm{as}n o+infty,qquadforallomegainOmega_{x_0}; eean and
$Xi$ is said to be "pointwise exponentially convergent", if to any initial
state $x_0inmathbb{R}^{1 imes d}$, there corresponds a measurable set
$Omega_{x_0}^primesubsetOmega$ with $mathbb{P}(Omega_{x_0}^prime)>0$
such that ean
x_0{prod}_{ell=1}^nxi_ell(omega)xrightarrow[]{ extrm{exponentially
fast}}mathbf{0}_{1 imes d}quad
extrm{as}n o+infty,qquadforallomegainOmega_{x_0}^prime. eean Using
dichotomy, we show that if $Xi$ is product bounded, i.e., $existseta>0$
such that ean |{prod}_{ell=1}^nxi_ell(omega)|_2leetaquadforall
nge1 extrm{and}mathbb{P} extrm{-a.e.}omegainOmega; eean then $Xi$ is
pointwise convergent if and only if it is pointwise exponentially convergent. | | Source: | arXiv, 1107.0132 | | Services: | Forum | Review | PDF | Favorites | |
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