Robust stability of linear stochastic systems

Abstract

Consider a linear system whose parameters are disturbed by white noise:

$$ {\Sigma _0}:\,dx\left( t \right) = Ax\left( t \right)dt + {A_1}x\left( t \right)d{w_1}\left( t \right) $$

((26.1))

(Ito stochastic differential equation). Here A,A ₁ ∈ ℝ^n×n and w ₁ is a standard real-valued Wiener process on a probability space (Ω,F,µ) relative to an increasing family (F _t )_t∈ℝ+ of σ-algebras F _t ⊂ F. For every x ⁰ ∈ℝⁿ there exists a unique solution x(t) = x(t,x ⁰) of (26.1) on ℝ₊ = [0,∞) with x(0) = x ⁰.