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Asymptotic stability for functional differential equations. (English) Zbl 0805.34068

The authors consider the system of functional differential equations with finite delay (1) \(x'(t) = f(t,x_ t)\), where \(f : [0, \infty) \times C_ H \to \mathbb{R}^ m\) is continuous and takes bounded sets into bounded sets and \(f(t,0) = 0\). Here \((C, \| \cdot \|)\) is the Banach space of continuous functions \(\varphi : [ - h,0] \to \mathbb{R}^ m\) with the supremum norm, \(h\) is a non-negative constant, \(C_ H\) is the open \(H\)- ball in \(C\), and \(x_ t(s) = x(t+s)\) for \(-h \leq s \leq 0\). Criteria for various types of stability (asymptotic stability, partial stability, uniform asymptotic stability) are obtained via Lyapunov’s direct method. The following corollary is representative of their results: “Suppose there is a Lyapunov functional \(V\), wedges \(W_ i\), positive constants \(K\) and \(J\), a sequence \(\{t_ n\} \uparrow \infty\) with \(t_ n - t_{n-1} \leq K\) such that (i) \(V(t_ n, \varphi) \leq W_ 2 (\| \varphi \|)\), (ii) \(V_{(1)}' (t,x_ t) \leq -W_ 3 (| x(t) |)\) if \(t_ n - h \leq t \leq t_ n\), (iii) \(| f(t, \varphi) | \leq J(t + 1) \ln (t + 2)\) for \(t \geq 0\) and \(\| \varphi \|<H\). Then \(x=0\) is asymptotically stable.” An example is given that illustrates the main results. The authors note that the conclusion that the zero solution of the equation considered is asymptotically stable can not be obtained by the known stability criteria.
Reviewer: V.Petrov (Plovdiv)

MSC:

34K20 Stability theory of functional-differential equations
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