On the stability properties of perturbed linear nonstationary systems
The authors discuss the stability properties of perturbed linear nonstationary systems \(\dot x(t)=(A(t)+B(t))x(t)\) with respect to those of the unperturbed linear nonstationary system \(\dot x(t)=A(t)x(t).\) Further, the stability properties of discrete perturbed systems \(x_{k+1}=(A_ k+B_ k)x_ k\) and discrete unperturbed systems \(x_{k+1}=A_ kx_ k\) are investigated. The concepts of stability, asymptotic stability and exponential or geometrical stability are applied. Four theorems are presented and proven for continuous systems. It is found, for example, that the perturbed system is asymptotically stable if the unperturbed system is exponentially stable, A(t) is bounded and B(t)\(\to 0\) as \(t\to \infty\). Six more theorems are found for discrete systems which lead to the following final result: Under some assumptions on \(B_ k\), the perturbed discrete system is geometrically stable iff the corresponding unperturbed system is geometrically stable.
scientific article; zbMATH DE number 3846011
scientific article