zbMATH — the first resource for mathematics

Lyapunov functionals and asymptotic stability of stochastic delay evolution equations. (English) Zbl 0947.93037
The author studies almost sure decay rates for mild solutions to stochastic evolution equations with delay on a Hilbert space \(H\), by applying the method he recently has developed for strong solutions to a sequence of approximating evolution equations together with a limiting procedure. The investigation yields results of the form \[ \log|\text{mild solution }(t)|/L(t)\leq\text{const}, \] almost surely, \(L(t)\) being an increasing function. Examples are given for \(H= L^2([a, b])\) and \(H= L^2(D)\), \(D\) a bounded domain in a Euclidean space.

93E15 Stochastic stability in control theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
93C25 Control/observation systems in abstract spaces
Full Text: DOI