# zbMATH — the first resource for mathematics

Sufficient stability conditions for stochastic systems with aftereffect. (English. Russian original) Zbl 0824.93069
Differ. Equations 30, No. 4, 509-517 (1994); translation from Differ. Uravn. 30, No. 4, 555-564 (1994).
The stochastic functional differential equation $$dx(t) = [(Vx) (t) + f(t)] dZ(t)$$, $$t \geq 0$$ is under study. Here $$x(t)$$ is an $$n$$- dimensional stochastic process on $$[0,\infty)$$, $$Z$$ is an $$m$$-dimensional semimartingale, $$f(t)$$ is an $$n \times m$$-matrix function on $$[0, \infty)$$, and $$V$$ is a related linear operator. The $$p$$-stability (including asymptotical and exponential) of the trivial solution of the above equation is defined by the behavior of the mean value of $$| x(t)|^ p$$ when $$t$$ tends to infinity. The author finds necessary and sufficient conditions for the $$p$$-stability of the stochastic system. As a special case the stability of a linear stochastic system with distributed delay is studied.

##### MSC:
 93E15 Stochastic stability in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 34K50 Stochastic functional-differential equations