ANZIAM  J.  46 (2005), 331-340
Stochastic stability of linear systems with semi-Markovian jump parameters

Zhenting Hou
  School of Mathematics
  Central South University
  Changsha
  Hunan 410075
  China
    zthou@csu.edu.cn
Jiaowan Luo
  School of Mathematics and Statistics
  Carleton University
  1125 Colonel By Drive
  Ottawa
  Ontario K1S 5B6
  Canada
    jwluo@math.carleton.ca
and
Peng Shi
  School of Technology
  University of Glamorgan
  Pontypridd
  Wales CF37 1DL
  UK
    pshi@glam.ac.uk


Abstract
Over the past two decades considerable effort has been devoted to problems of stochastic stability, stabilisation, filtering and control for linear and nonlinear systems with Markovian jump parameters, and a number of results have been achieved. However, due to the exponential distribution of the Markovian chain, there are many restrictions on existing results for practical applications. In the present paper, we study systems whose jump parameters are semi-Markovian rather than fully Markovian. We consider only linear systems with semi-Markovian jump parameters, and also study systems with phase-type semi-Markovian jump parameters, because the family of phase-type distributions is dense in the families of all probability distributions on $[0,+\infty)$. Some stochastic stability results are obtained. An example is given to show the potential of the proposed techniques.
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