Stability criteria for linear Hamiltonian dynamic systems on time scales
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RESEARCH
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Stability criteria for linear Hamiltonian dynamic systems on time scales Xiaofei He1,2, Xianhua Tang1* and Qi-Ming Zhang1 * Correspondence: tangxh@mail. csu.edu.cn 1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410083, Hunan, P.R. China Full list of author information is available at the end of the article
Abstract In this article, we establish some stability criteria for the polar linear Hamiltonian dynamic system on time scales x (t) = α(t)x(σ (t)) + β(t)y(t),
y (t) = −γ (t)x(σ (t)) − α(t)y(t),
t∈Ì
by using Floquet theory and Lyapunov-type inequalities. 2000 Mathematics Subject Classification: 39A10. Keywords: Hamiltonian dynamic system, Lyapunov-type inequality, Floquet theory, stability, time scales
1 Introduction A time scale is an arbitrary nonempty closed subset of the real numbers ℝ. We assume that Ì is a time scale. For t ∈ Ì , the forward jump operator σ : Ì → Ì is defined by σ (t) : inf{s ∈ Ì : s > t} , the backward jump operator ρ : Ì → Ì is defined by ρ(t) : sup{s ∈ Ì : s < t} , and the graininess function μ : Ì → [0, ∞} is defined by μ(t) = s(t) - t. For other related basic concepts of time scales, we refer the reader to the original studies by Hilger [1-3], and for further details, we refer the reader to the books of Bohner and Peterson [4,5] and Kaymakcalan et al. [6]. Definition 1.1. If there exists a positive number ω Î ℝ such that t + nω ∈ Ì for all t ∈ Ì and n Î ℤ, then we call Ì a periodic time scale with period ω. Suppose Ì is a ω-periodic time scale and 0 ∈ Ì . Consider the polar linear Hamiltonian dynamic system on time scale Ì x (t) = α(t)x(σ (t)) + β(t)y(t),
y (t) = −γ (t)x(σ (t)) − α(t)y(t),
t ∈ Ì,
(1:1)
where a(t), b(t) and g(t) are real-valued rd-continuous functions defined on Throughout this article, we always assume that 1 − μ(t)α(t) > 0,
∀t∈Ì
Ì.
(1:2)
and β(t) ≥ 0,
∀ t ∈ Ì.
(1:3)
© 2011 He et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
He et al. Advances in Difference Equations 2011, 2011:63 http://www.advancesindifferenceequations.com/content/2011/1/63
Page 2 of 11
For the second-order linear dynamic equation [p(t)x (t)] + q(t)x(σ (t)) = 0,
t ∈ Ì,
(1:4)
if let y(t) = p(t)xΔ (t), then we can rewrite (1.4) as an equivalent polar linear Hamiltonian dynamic system of type (1.1): x (t) =
1 y(t), p(t)
y (t) = −q(t)x(σ (t)),
t ∈ Ì,
(1:5)
where p(t) and q(t) are real-valued rd-continuous functions defined on 0, and α(t) = 0,
β(t) =
1 , p(t)
Ì
with p(t) >
γ (t) = q(t).
Recently, Agarwal et al. [7], Jiang and Zhou [8], Wong et al. [9] and He et al. [10] established some Lyapunov-type inequalities for dynamic equations on time scales, which generalize the corresponding results on differential and difference equations. Lyapunov-type in
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