The Lyapunov equations and nonuniform exponential stability. (English) Zbl 0893.93031

Buşe, Constantin; Barbu, Dorel

Stud. Cercet. Mat. 49, No. 1-2, 25-31 (1997).

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.

Reviewer: V.Wihstutz (Charlotte)

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)

Keywords:

infinite-dimensional stochastic systems; exponential stability; linear stochastic differential equation; Hilbert space; mild solution; Lyapunov equations