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The Lyapunov equations and nonuniform exponential stability. (English) Zbl 0893.93031
The authors give necessary and sufficient conditions for exponential stability of the fundamental solution of the linear stochastic differential equation on the Hilbert space $$H$$, $$dy(t)= A(t)y(t)+ B_i(t)y(t)dW^i$$, $$t\geq s$$, $$y(s)\in H_s:= L^2(\Omega, F_s, P;H)$$, with respectively closed and bounded linear operators $$A(t)$$ and $$B_i$$ on $$H$$ such that there is a unique mild solution. Among others, the conditions are stated in terms of solvability of the Lyapunov equations associated with the differential equation.

##### MSC:
 93E15 Stochastic stability in control theory 93C25 Control/observation systems in abstract spaces 93D20 Asymptotic stability in control theory 60H25 Random operators and equations (aspects of stochastic analysis)