Almost Sure Stability of Linear Stochastic Systems via Lyapunov Exponents Method

Abstract

In this paper we study the almost sure (sample path) stability of linear stochastic systems governed by the models: {x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> )x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ϵ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> {x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t+1</sub> = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> (ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> )x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ϵ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> (1) where {ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> : t ʧ#; 0} is a time homogeneous ergodic Markov process taking values in a measurable state space ε. Of particular interest is the case when ϵ is a discrete set with finite cardinality. In this situation, we discuss the problem of almost sure stability in the context of the computation of the Lyapunov spectrum for linear stochastic systems of the form (1). The paper contains a brief survey of techniques for almost sure stability of the model (1), a description of the socalled Lyapunov exponent (or Lyapunov spectrum) approach to stochastic stability, some results for the computation of the Lyapunov exponents for two-dimensional systems, and several examples to illustrate the application of the theory and computations presented.