On the stability of linear impulse stochastic systems. (English. Ukrainian original) Zbl 0941.60073

Swerdan, M. L.; Tsarkov, E. F.

Theory Probab. Math. Stat. 53, 173-181 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 160-167 (1995).

It is investigated the equation \(dx/dt=A(\xi(t))x\) with condition \(x(t_{k}+0)=B(\eta_{k})x(t_{k})\), \(t_{k}=k\Delta,\;k\in N\), where \(\xi(t),\;t\geq 0,\) is a homogeneous Feller Markov process; \(A(y)\) is a continuous matrix-valued function; \(\eta_{k},\;k\in N,\) is a Feller homogeneous Markov chain independent on \(\xi(t)\). Necessary and sufficient conditions for mean-square exponential stability of the trivial solution of the problem are obtained.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34G20	Nonlinear differential equations in abstract spaces
34F05	Ordinary differential equations and systems with randomness

Keywords:

stochastic equation; stability; impulse system