Intermediate stability of the solutions to the differential equations system with Markov coefficients (Q2703332)

Let us consider the system of linear differential equations NEWLINE\[NEWLINEdX/dt= A(\zeta(t))X(t), \quad A_{k}=A(\theta_{k}), \quad k=\overline{1,q},NEWLINE\]NEWLINE where \(\zeta(t)\) is a Markov process, defined on some probability space with finite number of states \(\theta_{1},\ldots,\theta_{q}\) and with distribution \(p_{k}(t)= P\{\zeta(t)=\theta_{k}\}\), \(k=\overline{1,q}\) such that NEWLINE\[NEWLINEdp_{k}(t)/dt= \sum_{s=1}^{q}\alpha_{ks}p_{s}(t), \quad \alpha_{ks}\geq 0, \;k\neq s, \;\alpha_{kk}\leq 0, \;\sum_{k=1}^{q}\alpha_{ks}=0,\;k=\overline{1,q}.NEWLINE\]NEWLINE Let the zero solution to the considered system be mean square asymptotic stable. The author obtains sufficient conditions on the matrices \(G_{k}=G(\theta_{k})\), \(k=\overline{1,q}\) for the mean square asymptotic stability of the zero solution to the system \(dX/dt=(A(\zeta(t))+ G(\zeta(t)))X(t)\), \(k=\overline{1,q}\).