Exponential stability of dynamic equations on time scales
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We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given. 1. Introduction This paper considers the exponential stability of the zero solution of the first-order vector dynamic equation x∆ = f (t,x),
t ≥ 0.
(1.1)
Throughout the paper, we let x(t,t0 ,x0 ) denote a solution of the initial value problem (IVP) (1.1),
x t0 = x0 ,
t0 ≥ 0, x0 ∈ R.
(1.2)
(For the existence, uniqueness, and extendability of solutions of IVPs for (1.1)-(1.2), see [2, Chapter 8].) Also we assume that f : [0, ∞) × Rn → Rn is a continuous function and t is from a so-called “time scale” T (which is a nonempty closed subset of R). Throughout the paper, we assume that 0 ∈ T (for convenience) and that f (t,0) = 0, for all t in the time scale interval [0, ∞) := {t ∈ T : 0 ≤ t < ∞}, and call the zero function the trivial solution of (1.1). If T = R, then x∆ = x and (1.1)-(1.2) becomes the following IVP for ordinary differential equations x = f (t,x),
t ≥ 0,
(1.3)
x t0 = x0 ,
t0 ≥ 0.
(1.4)
Recently, Peterson and Tisdell [7] used Lyapunov-type functions to formulate some sufficient conditions that ensure all solutions to (1.1)-(1.2) are bounded. Earlier, Raffoul [8] used some similar ideas to obtain boundedness of all solutions of (1.3) and (1.4). Here Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 133–144 DOI: 10.1155/ADE.2005.133
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Exponential stability of dynamic equations on time scales
we use Lyapunov-type functions on time scales and then formulate appropriate inequalities on these functions that guarantee that the trivial solution to (1.1) is exponentially or uniformly exponentially stable on [0, ∞). Some of our results are new even for the special cases T = R and T = Z. To understand the notation used above and the idea of time scales, some preliminary definitions are needed. Definition 1.1. A time scale T is a nonempty closed subset of the real numbers R. Since we are interested in the asymptotic behavior of solutions near ∞, we assume that T is unbounded above.
Since a time scale may or may not be connected, the concept of the jump operator is useful. Definition 1.2. Define the forward jump operator σ(t) at t by σ(t) = inf {τ > t : τ ∈ T},
∀t ∈ T,
(1.5)
and define the graininess function µ : T → [0, ∞) as µ(t) = σ(t) − t. Also let xσ (t) = x(σ(t)), that is, xσ is the composite function x ◦ σ. The jump operator σ then allows the classification of points in a time scale in the following way. If σ(t) > t, then we say that the point t is right scattered; while if σ(t) = t then, we say the point t is right dense. Throughout this work, the assumption is made that T has the topology that it inherits from the standard topology on the real numbers R. Definition 1.3. Fix t ∈ T and let x : T → Rn . Define x∆ (t) to be the vector (if it exists) with the property that given > 0, t
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