Stability of solutions of linear systems with an almost-periodic matrix

Exponential dichotomy of linear systems with almost periodic matrices

“…Choosing a subsequence f n k → f 0 , |f 0 | = 1, and passing in (10) with n replaced by n k to the limit as n k → ∞, we obtain (1). This contradicts the hypotheses of the lemma.…”

“…In this article we apply the techniques that are similar to those in [1] to analyzing a more complicated type of behavior of solutions to (1): exponential dichotomy.…”

“…The exponential stability of the solution x = 0 is guaranteed when (3) is nonpositive and not identically zero at every nonzero solution to (1). The justification rests on the properties of some functions on the compact set K = H × S, where H is the hull of A(t), and S is the unit sphere in C N .…”

“…The justification rests on the properties of some functions on the compact set K = H × S, where H is the hull of A(t), and S is the unit sphere in C N . Later the results of [1] were extended to other classes of almost periodic equations: difference, difference-differential, and functional-differential equations [2][3][4][5][6][7][8][9].…”

The direct Lyapunov method for an almost periodic difference system on a compactum

“…In [2,3] it was noted that, with some natural changes in the statement, this result extends to the nonautonomous systemṡ x = f (x, t) provided that the right-hand side and the Lyapunov function ν(x, t) are almost periodic in t (in the case linear systems, the talk is about exponential stability). Later in [4,5], these results were extended to the systems of difference equations…”