On linearly perturbed linear systems
The author studies linear systems of differential or difference equations \(x'(t)=A(t)x(t),\) and the linearly perturbed system \(y'(t)=(A(t)+B(t))y(t),\) where the perturbation matrix B(t) is bounded in the operator norm. The main results express the growth constants of the perturbed equation in terms of the growth constants of the unperturbed equation and a bound on the matrix B(t). Assuming that solutions of the unperturbed equation satisfy an estimate of the form \(\| x(t)\| \leq \gamma \| x(s)\| e^{\alpha (t-s)},\) it is shown that solutions of the perturbed equation satisfy the estimate \(\| y(t)\| \leq \gamma e^{(\alpha +\beta \gamma)(t-s)},\) where \(\beta\) is an upper bound for \(\| B\|\) on the given interval, which may be unbounded. The results obtained improve those of Vidyasagar, Boyarsky, and Vannelli in the following aspects: First, it is only assumed that B is bounded, whereas previous work assumed that B(t)\(\to 0\) as \(t\to \infty\). Second a more precise estimate on the rate of growth of solutions is obtained, and finally the methods are elementary. The proof is by an application of Gronwall's inequality, and both the differential and difference equation cases are treated similarly.
scientific article; zbMATH DE number 3957705
scientific article