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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 860506, 15 pages
http://dx.doi.org/10.1155/2011/860506
Research Article

An LMI Approach to Stability for Linear Time-Varying System with Nonlinear Perturbation on Time Scales

1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
3Center of Excellence in Mathematics CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
4Materials Science Research Center, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand

Received 17 December 2010; Revised 3 May 2011; Accepted 19 May 2011

Academic Editor: Martin D.Β Schechter

Copyright Β© 2011 Kanit Mukdasai and Piyapong Niamsup. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider Lyapunov stability theory of linear time-varying system and derive sufficient conditions for uniform stability, uniform exponential stability, πœ“-uniform stability, and h-stability for linear time-varying system with nonlinear perturbation on time scales. We construct appropriate Lyapunov functions and derive several stability conditions. Numerical examples are presented to illustrate the effectiveness of the theoretical results.

1. Introduction

In the past decades, stability analysis of dynamic systems has become an important topic both theoretically and practically because dynamic systems occur in many areas such as mechanics, physics, and economics. The theory of dynamic equations on time scales was first introduced by Hilger [1] with analysis of measure chains in order to unify continuous and discrete calculus on time scale. The generalized derivative or Hilger derivative 𝑓Δ(𝑑) of a function π‘“βˆΆπ•‹β†’β„, where 𝕋 is a so-called time scale (an arbitrary closed nonempty subset of ℝ) becomes the usual derivative when 𝕋=ℝ, namely, 𝑓Δ(𝑑)=π‘“ξ…ž(𝑑). On the other hand, if 𝕋=β„€, then 𝑓Δ(𝑑) reduces to the usual forward difference, namely, 𝑓Δ(𝑑)=Δ𝑓(𝑑). The development of theory on time scale calculus allows one to get some insight into and better understanding of the subtle differences between discrete and continuous systems [2, 3]. Therefore, the problem of stability analysis for dynamic equations (systems) on time scales has been investigated by many researchers, see [1–6], in which most results on stability of dynamic systems are obtained by the method of estimation of general solution of the systems. It seems that there are not many researches concerning with stability of dynamic systems on time scales by using Lyapunov functions on time scales.

There are various types of stability of dynamic systems on time scales such as uniform stability, uniform asymptotic stability [5], πœ“-uniform stability [6], and β„Ž-stability [4]. In [5], necessary and sufficient conditions for uniform stability and uniform asymptotic stability for dynamic systems on time scales are obtained. In [4, 6], the method presents in [5] are used to derive sufficient conditions for πœ“-uniformly stability [6] and β„Ž-stability [4] for dynamic systems on time scales.

In this paper, we shall develop Lyapunov stability theory for various types of stability for linear time-varying system with nonlinear perturbation on time scales. By using this Lyapunov stability theory, we derive several sufficient conditions for stabilities of dynamic systems on time scales.

2. Problem Formulation and Preliminaries

In this section, we introduce some notations, definitions, and preliminary results which will be used throughout the paper. ℝ+ denotes the set of all nonnegative real numbers; ℝ denotes the set of all real numbers; β„€+ denotes the set of all non-negative integers; β„€ denotes the set of all integers; ℝ𝑛 denotes the 𝑛-dimensional Euclidean space with the usual Euclidean norm β€–β‹…β€–; β€–π‘₯β€– denotes the Euclidean vector norm of π‘₯βˆˆβ„π‘›; β„π‘›Γ—π‘Ÿ denotes the set of π‘›Γ—π‘Ÿ real matrix; 𝐴𝑇 denotes the transpose of the matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴𝑇; 𝐼 denotes the identity matrix; πœ†(𝐴) denotes the set of all eigenvalues of 𝐴; πœ†max(𝐴)=max{Reπœ†βˆΆπœ†βˆˆπœ†(𝐴)}; πœ†min(𝐴)=min{Reπœ†βˆΆπœ†βˆˆπœ†(𝐴)}.

Definition 2.1. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ.

Definition 2.2. The mapping 𝜎,πœŒβˆΆπ•‹β†’π•‹ defined by 𝜎(𝑑)=inf{π‘ βˆˆπ•‹βˆΆπ‘ >𝑑}, and 𝜌(𝑑)=sup{π‘ βˆˆπ•‹βˆΆπ‘ <𝑑} are called the jump operators.

Definition 2.3. A nonmaximal element π‘‘βˆˆπ•‹ is said to be right-scattered (rs) if 𝜎(𝑑)>𝑑 and right-dense (rd) if 𝜎(𝑑)=𝑑. A nonminimal element π‘‘βˆˆπ•‹ is called left-scattered (ls) if 𝜌(𝑑)<𝑑 and left-dense (ld) if 𝜌(𝑑)=𝑑.

Definition 2.4. The mapping πœ‡βˆΆπ•‹β†’β„+ defined by πœ‡(𝑑)=𝜎(𝑑)βˆ’π‘‘ is called the graininess function.

Definition 2.5. (Delta derivative) assume π‘“βˆΆπ•‹β†’β„ is a function and let π‘‘βˆˆπ•‹. Then we define 𝑓Δ(𝑑) to be the number (provided it exists) with the property that given any πœ–>0, there is a neighborhood π‘ˆ of 𝑑 (i.e., π‘ˆ=(π‘‘βˆ’π›Ώ,𝑑+𝛿)βˆ©π•‹ for some 𝛿>0) such that |[𝑓(𝜎(𝑑))βˆ’π‘“(𝑠)]βˆ’π‘“Ξ”(𝑑)[𝜎(𝑑)βˆ’π‘ ]|β‰€πœ–|𝜎(𝑑)βˆ’π‘ | for all π‘ βˆˆπ‘ˆ.
The function 𝑓Δ(𝑑) is the delta derivative of 𝑓 at 𝑑.
In the case that 𝕋=ℝ, we have 𝑓Δ(𝑑)=π‘“ξ…ž(𝑑). In the case that 𝕋=β„€, we have 𝑓Δ(𝑑)=𝑓(𝑑+1)βˆ’π‘“(𝑑).
The following are some useful relationships regarding the delta derivative, see [2].

Theorem 2.6 (see [2]). Assume that π‘“βˆΆπ•‹β†’β„π‘› and let π‘‘βˆˆπ•‹. (i)If 𝑓 is differentiable at 𝑑, then 𝑓 is continuous at 𝑑. (ii)If 𝑓 is continuous at 𝑑 and 𝑑 is right scattered, then 𝑓 is differentiable at 𝑑 with 𝑓Δ(𝑑)=𝑓(𝜎(𝑑))βˆ’π‘“(𝑑)𝜎(𝑑)βˆ’π‘‘.(2.1)(iii)If 𝑓 is differentiable at 𝑑 and 𝑑 is right dense, then 𝑓Δ(𝑑)=lim𝑠→𝑑𝑓(𝑑)βˆ’π‘“(𝑠)π‘‘βˆ’π‘ .(2.2)(iv)If 𝑓 is differentiable at 𝑑, then 𝑓(𝜎(𝑑))=𝑓(𝑑)+πœ‡(𝑑)𝑓Δ(𝑑).(2.3)

Theorem 2.7 (see [2]). Assume that 𝑓,π‘”βˆΆπ•‹β†’β„π‘› and let π‘‘βˆˆπ•‹. (i)The sum 𝑓,π‘”βˆΆπ•‹β†’β„π‘› are differentiable at 𝑑 with (𝑓+𝑔)Ξ”(𝑑)=(𝑓)Ξ”(𝑑)+(𝑔)Ξ”(𝑑).(2.4)(ii) For any constant 𝛼,π›Όπ‘“βˆΆπ•‹β†’β„π‘› is differentiable at 𝑑 with (𝛼𝑓)Ξ”(𝑑)=𝛼𝑓Δ(𝑑).(2.5)(iii)The product π‘“π‘”βˆΆπ•‹β†’β„π‘› is differentiable at 𝑑 with (𝑓𝑔)Ξ”(𝑑)=𝑓Δ(𝑑)𝑔(𝑑)+𝑓(𝜎(𝑑))𝑔Δ(𝑑)=𝑓(𝑑)𝑔Δ(𝑑)+𝑓Δ(𝑑)𝑔(𝜎(𝑑)).(2.6)

Definition 2.8. The function π‘“βˆΆπ•‹β†’β„π‘› is said to be rd-continuous (denoted by π‘“βˆˆπ’žrd(𝕋,ℝ𝑛)) if the following conditions hold. (i)𝑓 is continuous at every right-dense point π‘‘βˆˆπ•‹. (ii)limπ‘ β†’π‘‘βˆ’π‘“(𝑠) exists and is finite at every ld-point π‘‘βˆˆπ•‹.

Definition 2.9. Let π‘“βˆˆπ’žrd(𝕋,ℝ𝑛). Then π‘”βˆΆπ•‹β†’β„π‘› is called the antiderivative of 𝑓 on 𝕋 if it is differentiable on 𝕋 and satisfies 𝑔Δ(𝑑)=𝑓(𝑑) for π‘‘βˆˆπ•‹. In this case, we define ξ€œπ‘‘π‘Žπ‘“(𝑠)Δ𝑠=𝑔(𝑑)βˆ’π‘”(π‘Ž),π‘Žβ‰€π‘‘βˆˆπ•‹.(2.7) Consider the linear time-varying system with nonlinear perturbation on time scales (𝕋) of the form π‘₯Ξ”(𝑑)=𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯(𝑑)),π‘‘βˆˆπ•‹,(2.8) where π‘₯(𝑑)βˆˆβ„π‘›,π΄βˆΆπ•‹β†’β„π‘›Γ—π‘› is an 𝑛×𝑛 matrix-valued function and π‘“βˆΆπ•‹Γ—β„π‘›β†’β„π‘› is rd-continuous in the first argument with 𝑓(𝑑,0)=0. The uncertain perturbation is known to satisfy a bound of the form ‖𝑓(𝑑,π‘₯(𝑑))‖≀𝛾‖π‘₯(𝑑)β€–,(2.9) or equivalently, the perturbation is conically bounded. The solution of (2.8) through (𝑑0,π‘₯(𝑑0)) satisfies the variation of constants formula π‘₯(𝑑)=Φ𝐴𝑑,𝑑0ξ€Έπ‘₯𝑑0ξ€Έ+ξ€œπ‘‘π‘‘0Φ𝐴(𝑑,𝜎(𝑠))𝑓(𝑠,π‘₯(𝑠))Δ𝑠,𝑑β‰₯𝑑0.(2.10) When 𝑓(𝑑,π‘₯(𝑑))=0, (2.8) becomes the linear time-varying system π‘₯Ξ”(𝑑𝑑)=𝐴(𝑑)π‘₯(𝑑),π‘₯0ξ€Έ=π‘₯0,𝑑0βˆˆπ•‹.(2.11) For the case when 𝑓(𝑑,π‘₯(𝑑))=𝐡(𝑑)π‘₯(𝑑),𝐡(𝑑)βˆˆβ„π‘›Γ—π‘›, (2.8) becomes the linear time-varying system π‘₯Ξ”([]𝑑𝑑)=𝐴(𝑑)+𝐡(𝑑)π‘₯(𝑑),π‘₯0ξ€Έ=π‘₯0,𝑑0βˆˆπ•‹.(2.12) The norm of 𝑛×𝑛 matrix 𝐴 is defined as ‖𝐴‖=maxβ€–π‘₯β€–=1‖𝐴π‘₯β€–.(2.13) The Euclidean norm of 𝑛×1 vector π‘₯(𝑑) is defined by βˆšβ€–π‘₯(𝑑)β€–=π‘₯𝑇(𝑑)π‘₯(𝑑).(2.14)

Definition 2.10. A function πœ™βˆΆ[0,π‘Ÿ]β†’[0,+∞) is of class 𝒦 if it is well-defined, continuous, and strictly increasing on [0,π‘Ÿ] with πœ™(0)=0.

Definition 2.11. Assume π‘”βˆΆπ•‹β†’β„. Define and denote π‘”βˆˆπ’žrd(𝕋;ℝ) as right-dense continuous (rd-continuous) if 𝑔 is continuous at every right-dense point π‘‘βˆˆπ•‹ and limπ‘ β†’π‘‘βˆ’π‘”(𝑠) exists, and is finite, at every left-dense point π‘‘βˆˆπ•‹. Now define the so-called set of regressive functions, β„›, by ξ€½β„›=π‘βˆΆπ•‹β†’β„βˆ£π‘βˆˆπ’žrdξ€Ύ(𝕋;ℝ),1+𝑝(𝑑)πœ‡(𝑑)β‰ 0,π‘‘βˆˆπ•‹,(2.15) and define the set of positively regressive functions by β„›+={π‘βˆˆβ„›βˆ£1+𝑝(𝑑)πœ‡(𝑑)>0,π‘‘βˆˆπ•‹}.(2.16)

Definition 2.12. The zero solution of system (2.8) is called uniformly stable if there exists a finite constant 𝛾>0 such that β€–β€–π‘₯𝑑,π‘₯0,𝑑0ξ€Έβ€–β€–β€–β€–π‘₯≀𝛾0β€–β€–,(2.17) for all π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.

Definition 2.13. The zero solution of system (2.8) is called uniformly exponentially stable if there exist finite constants 𝛾,πœ†>0 with βˆ’πœ†βˆˆβ„›+ such that β€–β€–π‘₯𝑑,π‘₯0,𝑑0ξ€Έβ€–β€–β€–β€–π‘₯≀𝛾0β€–β€–π‘’βˆ’πœ†ξ€·π‘‘,𝑑0ξ€Έ,(2.18) for all π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.

Definition 2.14. The zero solution of system (2.8) is called πœ“-uniformly stable if there exists a finite constant 𝛾>0 such that for any 𝑑0 and π‘₯(𝑑0), the corresponding solution satisfies β€–β€–πœ“ξ€·(𝑑)π‘₯𝑑,π‘₯0,𝑑0ξ€Έβ€–β€–β€–β€–πœ“ξ€·π‘‘β‰€π›Ύ0ξ€Έπ‘₯0β€–β€–,(2.19) for all π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.

Definition 2.15. System (2.8) is called an β„Ž-π‘ π‘¦π‘ π‘‘π‘’π‘š if there exist a positive function β„ŽβˆΆπ•‹β†’β„, a constant 𝑐β‰₯1 and 𝛿>0 such that β€–β€–π‘₯𝑑,π‘₯0,𝑑0ξ€Έβ€–β€–β€–β€–π‘₯≀𝑐0β€–β€–ξ€·π‘‘β„Ž(𝑑)β„Ž0ξ€Έβˆ’1,𝑑β‰₯𝑑0,(2.20) if β€–π‘₯0β€–<𝛿(hereβ„Ž(𝑑)βˆ’1=1/β„Ž(𝑑)). If β„Ž is bounded, then (2.8) is said to be β„Ž-stable.

Definition 2.16. A continuous function π‘ƒβˆΆπ•‹β†’β„ with 𝑃(0)=0 is called positive definite (negative definite) on 𝕋 if there exists a function πœ™βˆˆπ’¦ such that πœ™(𝑑)≀𝑃(𝑑) (πœ™(𝑑)β‰€βˆ’π‘ƒ(𝑑)) for all π‘‘βˆˆπ•‹.

Definition 2.17. A continuous function π‘ƒβˆΆπ•‹β†’β„ with 𝑃(0)=0 is called positive semidefinite (negative semi-definite) on 𝕋 if 𝑃(𝑑)β‰₯0 (𝑃(𝑑)≀0) for all π‘‘βˆˆπ•‹.

Definition 2.18. A continuous function π‘ƒβˆΆπ•‹Γ—β„π‘›β†’β„ with 𝑃(𝑑,0)=0 is called positive definite (negative definite) on 𝕋×ℝ𝑛 if there exists a function πœ™βˆˆπ’¦ such that πœ™(β€–π‘₯β€–)≀𝑃(𝑑,π‘₯) (πœ™(β€–π‘₯β€–)β‰€βˆ’π‘ƒ(𝑑,π‘₯)) for all π‘‘βˆˆπ•‹ and π‘₯βˆˆβ„π‘›.

Definition 2.19. A continuous function π‘ƒβˆΆπ•‹Γ—β„π‘›β†’β„ with 𝑃(𝑑,0)=0 is called positive semi-definite (negative semi-definite) on 𝕋×ℝ𝑛 if 0≀𝑃(𝑑,π‘₯) (0β‰₯𝑃(𝑑,π‘₯)) for all π‘‘βˆˆπ•‹ and π‘₯βˆˆβ„π‘›.

Lemma 2.20 ([7], Completing the square). assume that π‘†βˆˆπ‘€π‘›Γ—π‘› is a symmetric positive definite matrix. Then for every π‘„βˆˆπ‘€π‘›Γ—π‘›, we obtain 2π‘₯π‘‡π‘„π‘¦βˆ’π‘¦π‘‡π‘†π‘¦β‰€π‘₯π‘‡π‘„π‘†βˆ’1𝑄𝑇π‘₯,βˆ€π‘₯,π‘¦βˆˆπ‘…π‘›.(2.21)

3. Main Results

In this section, we first introduce Lyapunov stability theory of various types stability for linear time varying system with nonlinear perturbation on time scales. Then, we use this Lyapunov stability theory to obtain sufficient conditions for various types of stabilities of this system.

3.1. Lyapunov Stability Theory

Theorem 3.1. If there exist a continuously differentiable positive definite function 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’ž1rd(𝕋×ℝ𝑛,ℝ+), and π‘Ž,π‘βˆˆβ„+ such that (i)𝑉Δ(𝑑,π‘₯(𝑑))≀0,(ii)π‘Žβ€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))≀𝑏‖π‘₯(𝑑)β€–2, then the zero solution of system (2.8) is πœ“-uniformly stable if there exists πœ“(𝑑)βˆˆπ’ž1rd(𝕋,ℝ+) satisfying πœ“Ξ”(𝑑)≀0.

Proof. For 𝑑0βˆˆπ•‹, we let π‘₯(𝑑0)=π‘₯0. Then, by (i), we have ξ€œπ‘‘π‘‘0𝑉Δ(𝑑𝑠,π‘₯(𝑠))Δ𝑠=𝑉(𝑑,π‘₯(𝑑))βˆ’π‘‰0𝑑,π‘₯0ξ€œξ€Έξ€Έβ‰€0,𝑑𝑑0πœ“Ξ”(𝑑𝑠)Δ𝑠=πœ“(𝑑)βˆ’πœ“0≀0.(3.1) We obtain 𝑉(𝑑,π‘₯(𝑑))≀𝑉(𝑑0,π‘₯(𝑑0)) and πœ“(𝑑)β‰€πœ“(𝑑0) for all π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0. By (ii), we get the estimation as follows: π‘Žβ€–πœ“(𝑑)β€–2β€–π‘₯(𝑑)β€–2β‰€β€–πœ“(𝑑)β€–2β€–β€–πœ“ξ€·π‘‘π‘‰(𝑑,π‘₯(𝑑))≀0ξ€Έβ€–β€–2𝑉𝑑0𝑑,π‘₯0β€–β€–πœ“ξ€·π‘‘ξ€Έξ€Έβ‰€π‘0ξ€Έβ€–β€–2β€–β€–π‘₯𝑑0ξ€Έβ€–β€–2.(3.2) We conclude that β€–πœ“(𝑑)π‘₯(𝑑)β€–β‰€π›Ύβ€–πœ“(𝑑0)π‘₯(𝑑0)β€– where βˆšπ›Ύ=𝑏/π‘Ž>0. Therefore, the zero solution of system (2.8) is πœ“-uniformly stable. The proof of the theorem is complete.

Corollary 3.2. If there exist a continuously differentiable positive definite function 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’ž1rd(𝕋×ℝ𝑛,ℝ+) and π‘Ž,π‘βˆˆβ„+ such that (i)𝑉Δ(𝑑,π‘₯(𝑑))≀0,(ii)π‘Žβ€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))≀𝑏‖π‘₯(𝑑)β€–2,then the zero solution of system (2.8) is uniformly stable.

Theorem 3.3. If there exist a continuously differentiable positive definite function 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’ž1rd(𝕋×ℝ𝑛,ℝ+) and π‘Ž,𝑏,πœ–βˆˆβ„+ with βˆ’πœ–/π‘βˆˆβ„›+ satisfying (i)𝑉Δ(𝑑,π‘₯(𝑑))β‰€βˆ’πœ–β€–π‘₯(𝑑)β€–2,(ii)π‘Žβ€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))≀𝑏‖π‘₯(𝑑)β€–2, then the zero solution of system (2.8) is uniformly exponentially stable.

Proof. For 𝑑0βˆˆπ•‹, we let π‘₯(𝑑0)=π‘₯0. We obtain, by (i) and (ii), that for all 𝑑β‰₯𝑑0, 𝑉Δ‖(𝑑,π‘₯(𝑑))β‰€βˆ’πœ–β€–π‘₯(𝑑)2πœ–β‰€βˆ’π‘π‘‰(𝑑,π‘₯(𝑑)).(3.3) Since βˆ’πœ–/π‘βˆˆβ„›+, it follows from Gronwall’s inequality for time scales [2] and (ii) that π‘Žβ€–π‘₯(𝑑)β€–2𝑑≀𝑉(𝑑,π‘₯(𝑑))≀𝑉0𝑑,π‘₯0π‘’ξ€Έξ€Έβˆ’πœ–/𝑏𝑑,𝑑0ξ€Έβ€–β€–π‘₯𝑑≀𝑏0ξ€Έβ€–β€–2π‘’βˆ’πœ–/𝑏𝑑,𝑑0ξ€Έ.(3.4) Hence, we get β€–β€–β€–π‘₯𝑑π‘₯(𝑑)‖≀𝛾0ξ€Έβ€–β€–ξ€Ίπ‘’βˆ’πœ–/𝑏(𝑑,𝑑0)ξ€»1/2,(3.5) where βˆšπ›Ύ=𝑏/π‘Ž for all 𝑑β‰₯𝑑0. Therefore, the zero solution of system (2.8) is uniformly exponentially stable. The proof of the theorem is complete.

Theorem 3.4. If there exist a continuously differentiable positive definite function 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’ž1rd(𝕋×ℝ𝑛,ℝ+), a bounded positive differentiable function β„ŽβˆΆπ•‹β†’β„ and π‘Ž,π‘βˆˆβ„+ such that (i)𝑉Δ(𝑑,π‘₯(𝑑))β‰€π›Ύβ„ŽΞ”(𝑑)/β„Ž(𝑑)β€–π‘₯(𝑑)β€–2⎧βŽͺ⎨βŽͺ⎩,𝛾=π‘Ž,β„ŽΞ”(𝑑)β‰₯0;𝑏,β„ŽΞ”(𝑑)<0,(3.6)(ii)π‘Žβ€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))≀𝑏‖π‘₯(𝑑)β€–2, then the zero solution of system (2.8) is β„Ž-stable.

Proof. Let 𝑑0βˆˆπ•‹, π‘₯(𝑑0)=π‘₯0 and π‘₯(𝑑,𝑑0,π‘₯0)=π‘₯(𝑑) be any solution of system (2.8). By (i), we have π‘‰Ξ”β„Ž(𝑑,π‘₯(𝑑))≀𝛾Δ(𝑑)β„Ž(𝑑)β€–π‘₯(𝑑)β€–2β‰€βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›Ύπ‘Žβ„ŽΞ”(𝑑)β„Ž(𝑑)𝑉(𝑑,π‘₯(𝑑)),β„ŽΞ”π›Ύ(𝑑)β‰₯0;π‘β„ŽΞ”(𝑑)β„Ž(𝑑)𝑉(𝑑,π‘₯(𝑑)),β„ŽΞ”β‰€β„Ž(𝑑)<0,Ξ”(𝑑)π‘‰β„Ž(𝑑)(𝑑,π‘₯(𝑑)).(3.7) From Gronwall’s inequality for time scales [2], (ii) and Lemma 2.15 [4], we obtain π‘Žβ€–π‘₯(𝑑)β€–2𝑑≀𝑉(𝑑,π‘₯(𝑑))≀𝑉0𝑑,π‘₯0π‘’ξ€Έξ€Έβ„ŽΞ”(𝑑)/β„Ž(𝑑)𝑑,𝑑0ξ€Έβ€–β€–π‘₯𝑑≀𝑏0ξ€Έβ€–β€–2π‘’β„ŽΞ”(𝑑)/β„Ž(𝑑)𝑑,𝑑0ξ€Έ,β€–β€–π‘₯𝑑≀𝑏0ξ€Έβ€–β€–2β„Ž(𝑑)β„Žξ€·π‘‘0ξ€Έ.(3.8) Thus, β€–β€–β€–π‘₯𝑑π‘₯(𝑑)‖≀𝛾0‖‖𝑑𝐻(𝑑)𝐻0ξ€Έβˆ’1,𝑑β‰₯𝑑0,(3.9) where βˆšπ›Ύ=𝑏/π‘Ž and √𝐻(𝑑)=β„Ž(𝑑). Therefore, zero solution of (2.8) is β„Ž-stable.

3.2. Stability Conditions

We introduce the following notation for later use:𝑍(𝑑)∢=𝑃Δ(𝑑)+𝐴𝑇(𝑑)𝑃(𝑑)+𝑃(𝑑)𝐴(𝑑)+πœ‡(𝑑)𝑃Δ(𝑑)𝐴(𝑑)+πœ‡(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)+πœ–1𝑃(𝑑)𝑃(𝑑)+πœ‡(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)+πœ‡2(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)+πœ–2𝑃Δ(𝑑)𝑃Δ(𝑑)+πœ–2βˆ’1𝛾2πœ‡(𝑑)2𝐼+πœ–3𝐴𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)𝐴(𝑑)+πœ–3βˆ’1𝛾2πœ‡(𝑑)2𝐼+πœ–4𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)𝐴(𝑑)+πœ–4βˆ’1𝛾2πœ‡(𝑑)4𝐼+πœ–1βˆ’1𝛾2𝐼+πœ‚2𝛾2πœ‡(𝑑)𝐼+𝜌2𝛾2πœ‡(𝑑)2𝐼.(3.10)

Theorem 3.5. The system (2.11) is uniformly stable if there exist a positive definite symmetric matrix function 𝑃(𝑑)βˆˆπ’ž1rd(𝕋,ℝ𝑛×𝑛) and πœ‚,πœŒβˆˆβ„+ such that (i)πœ‚πΌβ‰€π‘ƒ(𝑑)β‰€πœŒπΌ,(ii)𝐴𝑇(𝑑)𝑃(𝑑)+(𝐼+πœ‡(𝑑)𝐴𝑇(𝑑))(𝑃Δ(𝑑)+𝑃(𝑑)𝐴(𝑑)+πœ‡(𝑑)𝑃Δ(𝑑)𝐴(𝑑))≀0.

Remark 3.6. We can prove Theorem 3.5 (see Theorem 3.1 in [5] DaCunha) by using the same approach as in Theorem 3.1 by choosing 𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑). In this case, we obtain 𝑉Δ(𝑑)=π‘₯𝑇(𝐴𝑑)𝑇(𝑑)𝑃(𝑑)+𝐼+πœ‡(𝑑)𝐴𝑇(𝑃𝑑)ξ€Έξ€·Ξ”(𝑑)+𝑃(𝑑)𝐴(𝑑)+πœ‡(𝑑)𝑃Δ(𝑑)𝐴(𝑑)ξ€Έξ€»π‘₯(𝑑).(3.11)

Theorem 3.7. The system (2.8) is uniformly stable if there exist a positive definite symmetric matrix function 𝑃(𝑑)βˆˆπ’ž1rd(𝕋,ℝ𝑛×𝑛) and πœ‚1,πœ‚2,𝛾,πœ–1,πœ–2,πœ–3,πœ–4βˆˆβ„+, 𝜌1,𝜌2βˆˆβ„ such that (i)πœ‚1𝐼≀𝑃(𝑑)β‰€πœ‚2𝐼,(ii)𝜌1𝐼≀𝑃Δ(𝑑)β‰€πœŒ2𝐼,(iii)𝑍(𝑑)≀0.

Proof. We consider the following Lyapunov function for system (2.8). 𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑).(3.12) By (i), it is easy to see that πœ‚1β€–β€–π‘₯(𝑑)2≀𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)β‰€πœ‚2β€–β€–π‘₯(𝑑)2.(3.13) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δπ‘₯(𝑑)=𝑇(𝑑)𝑃(𝑑)Ξ”π‘₯(𝑑)+π‘₯𝑇(𝜎(𝑑))𝑃(𝜎(𝑑))π‘₯Ξ”(𝑑)=π‘₯𝑇(𝑑)Δ𝑃(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝜎(𝑑))𝑃Δ(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝜎(𝑑))𝑃(𝜎(𝑑))π‘₯Ξ”=ξ€Ίπ‘₯(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇π‘₯(𝑑,π‘₯)𝑃(𝑑)π‘₯(𝑑)+𝑇π‘₯(𝑑)+πœ‡(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇×𝑃(𝑑,π‘₯)ξ€Έξ€»Ξ”(𝑑)π‘₯(𝑑)+𝑃(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))+πœ‡(𝑑)𝑃Δ(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))=π‘₯𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)2π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)𝑓𝑇(𝑑,π‘₯)𝑃(𝑑)𝑓(𝑑,π‘₯)+πœ‡(𝑑)2𝑓𝑇(𝑑,π‘₯)𝑃Δ(𝑑)𝑓(𝑑,π‘₯)+𝑓𝑇(𝑑,π‘₯)𝑃(𝑑)π‘₯(𝑑)+πœ‡(𝑑)𝑓𝑇(𝑑,π‘₯)𝑃Δ(𝑑)π‘₯(𝑑)+πœ‡(𝑑)𝑓𝑇(𝑑,π‘₯)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)2𝑓𝑇(𝑑,π‘₯)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃(𝑑)𝑓(𝑑,π‘₯)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝑓(𝑑,π‘₯)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝑓(𝑑,π‘₯)+πœ‡(𝑑)2π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑓(𝑑,π‘₯).(3.14) By (i), (ii), and Lemma 2.20, we have the following estimate: π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)β‰€πœ‚2π‘₯𝑇π‘₯(𝑑)π‘₯(𝑑),𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)β‰€πœŒ2π‘₯𝑇(𝑑)π‘₯(𝑑),2π‘₯𝑇(𝑑)𝑃(𝑑)𝑓(𝑑,π‘₯)β‰€πœ–1π‘₯𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)π‘₯(𝑑)+πœ–1βˆ’1𝑓𝑇(𝑑,π‘₯)𝑓(𝑑,π‘₯),2πœ‡(𝑑)π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝑓(𝑑,π‘₯)β‰€πœ–2π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ–2βˆ’1πœ‡(𝑑)2𝑓𝑇(𝑑,π‘₯)𝑓(𝑑,π‘₯),2πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝑓(𝑑,π‘₯)β‰€πœ–3π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–3βˆ’1πœ‡(𝑑)2𝑓𝑇(𝑑,π‘₯)𝑓(𝑑,π‘₯),2πœ‡(𝑑)2π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑓(𝑑,π‘₯)β‰€πœ–4π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–4βˆ’1πœ‡(𝑑)4𝑓𝑇(𝑑,π‘₯)𝑓(𝑑,π‘₯).(3.15) From the above inequalities and ‖𝑓(𝑑,π‘₯)‖≀𝛾‖π‘₯(𝑑)β€–, we obtain 𝑉Δ(𝑑)≀π‘₯𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)2π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–1π‘₯𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)π‘₯(𝑑)+πœ–1βˆ’1𝛾2β€–π‘₯(𝑑)β€–2+πœ–2π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ–2βˆ’1𝛾2πœ‡(𝑑)2β€–β€–π‘₯(𝑑)2+πœ–3π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–3βˆ’1𝛾2πœ‡(𝑑)2β€–π‘₯(𝑑)β€–2+πœ–4βˆ’1𝛾2πœ‡(𝑑)4β€–π‘₯(𝑑)β€–2+πœ–4π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‚2𝛾2πœ‡(𝑑)β€–π‘₯(𝑑)β€–2+𝜌2𝛾2πœ‡(𝑑)2β€–π‘₯(𝑑)β€–2=π‘₯𝑇(𝑑)𝑍(𝑑)π‘₯(𝑑).(3.16) By (iii), we conclude that 𝑉Δ(𝑑)≀0. Therefore, the zero solution of (2.8) is uniformly stable by Corollary 3.2.

Example 3.8. We consider the time-varying dynamic system of the form π‘₯Ξ”(π‘₯𝑑)=𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯(𝑑)),(3.17)𝐴(𝑑)=βˆ’π‘Ž(𝑑)βˆ’11βˆ’π‘Ž(𝑑),𝑓(𝑑,π‘₯(𝑑))=βˆ’0.125sin(𝑑)2ξ€»ξ€Ίπ‘₯(𝑑)0.125cos(𝑑)1ξ€»ξƒ­(𝑑),‖𝑓(𝑑,π‘₯(𝑑))‖≀0.125β€–π‘₯(𝑑)β€–,(3.18) where π‘Ž(𝑑)=βˆ’π‘’βŠ–8(𝑑,0)+1 and 𝑓(𝑑,π‘₯(𝑑)) are rd-continuous in the first argument with 𝑓(𝑑,0)=0 for all π‘‘βˆˆπ•‹. Let 𝛾=1/8,πœ–1=1,πœ–2=1/16,πœ–3=1/2,πœ–4=1/16,πœ‚1=1/8,πœ‚2=1/4, 𝜌1=βˆ’1, and 𝜌2=0. By assuming that 0β‰€πœ‡(𝑑)≀0.25 for all π‘‘βˆˆπ•‹, we can find solution 𝑃(𝑑) satisfying conditions (i)–(iii) of Theorem 3.7 as 𝑃(𝑑)=(1/8)π‘’βŠ–8(𝑑,0)+(1/8)00(1/8)π‘’βŠ–8(𝑑,0)+(1/8)ξ‚„. Observe that, 𝑃Δ(𝑑)=βˆ’π‘’βŠ–8(𝑑,0)00βˆ’π‘’βŠ–8ξƒ­(𝑑,0).(3.19) Therefore, by Theorem 3.7, the system (3.17) is uniformly stable.

Theorem 3.9. The system (2.8) is uniformly exponentially stable if there exist positive definite symmetric matrix function 𝑃(𝑑)βˆˆπ’ž1rd(𝕋,ℝ𝑛×𝑛) and πœ‚1,πœ‚2,𝛾,πœ–1,πœ–2,πœ–3,πœ–4,πœ–5βˆˆβ„+, 𝜌1,𝜌2βˆˆβ„ such that (i)πœ‚1𝐼≀𝑃(𝑑)β‰€πœ‚2𝐼,(ii)𝜌1𝐼≀𝑃Δ(𝑑)β‰€πœŒ2𝐼,(iii)𝑍(𝑑)β‰€βˆ’πœ–5𝐼.

Proof. Consider a Lyapunov function for system (2.8) of the form 𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑).(3.20) It is easy to see that (i) yields πœ‚1β€–β€–π‘₯(𝑑)2≀𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)β‰€πœ‚2β€–β€–π‘₯(𝑑)2.(3.21) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δπ‘₯(𝑑)=𝑇(𝑑)𝑃(𝑑)Ξ”π‘₯(𝑑)+π‘₯𝑇(𝜎(𝑑))𝑃(𝜎(𝑑))π‘₯Ξ”=ξ€Ίπ‘₯(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇π‘₯(𝑑,π‘₯)𝑃(𝑑)π‘₯(𝑑)+𝑇π‘₯(𝑑)+πœ‡(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇×𝑃(𝑑,π‘₯)ξ€Έξ€»Ξ”(𝑑)π‘₯(𝑑)+𝑃(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))+πœ‡(𝑑)𝑃Δ.(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))(3.22) From Theorem 3.7, we obtain 𝑉Δ(𝑑)≀π‘₯𝑇(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ‡(𝑑)π‘₯𝑇(𝑑)𝐴(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‡(𝑑)2π‘₯𝑇(𝑑)𝐴(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–1π‘₯𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)π‘₯(𝑑)+πœ–1βˆ’1𝛾2β€–π‘₯(𝑑)β€–2+πœ–2π‘₯𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)π‘₯(𝑑)+πœ–2βˆ’1𝛾2πœ‡(𝑑)2β€–π‘₯(𝑑)β€–2+πœ–3π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃(𝑑)𝑃(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ–3βˆ’1𝛾2πœ‡(𝑑)2(β€–π‘₯𝑑)β€–2+πœ–4βˆ’1𝛾2πœ‡(𝑑)4(β€–π‘₯𝑑)β€–2+πœ–4π‘₯𝑇(𝑑)𝐴𝑇(𝑑)𝑃Δ(𝑑)𝑃Δ(𝑑)𝐴(𝑑)π‘₯(𝑑)+πœ‚2𝛾2πœ‡(𝑑)β€–π‘₯(𝑑)β€–2+𝜌2𝛾2πœ‡(𝑑)2(β€–π‘₯𝑑)β€–2.(3.23) By (iii), we conclude that 𝑉Δ(𝑑)β‰€βˆ’πœ–5β€–π‘₯(𝑑)β€–2. By Theorem 3.3, the zero solution of (2.8) is uniformly exponentially stable.

Example 3.10. We consider the linear time-varying system with nonlinear perturbation of the form π‘₯Ξ”(𝑑)=𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯(𝑑)),(3.24) where π‘₯𝐴(𝑑)=βˆ’π‘Ž(𝑑)βˆ’11βˆ’π‘Ž(𝑑),𝑓(𝑑,π‘₯(𝑑))=βˆ’0.125cos(𝑑)2ξ€»ξ€Ίπ‘₯(𝑑)0.125sin(𝑑)1ξ€»ξƒ­(𝑑).(3.25)π‘Ž(𝑑)=π‘’βŠ–8(𝑑,0)+1 and 𝑓(𝑑,π‘₯(𝑑)) are rd-continuous in the first argument with 𝑓(𝑑,0)=0 for all π‘‘βˆˆπ•‹. Then, 1β‰€π‘Ž(𝑑)≀2 and ‖𝑓(𝑑,π‘₯(𝑑))‖≀0.125β€–π‘₯(𝑑)β€– for all π‘‘βˆˆπ•‹. Let 𝛾=1/8,πœ–1=1,πœ–2=1/16,πœ–3=1/2,πœ–4=πœ–5=1/16,πœ‚1=1/8,πœ‚2=1/4,𝜌1=βˆ’1, and 𝜌2=0. By assuming that 0β‰€πœ‡(𝑑)≀0.25, for all π‘‘βˆˆπ•‹, we can find a solution 𝑃(𝑑) satisfying (i)–(iii) of Theorem 3.9 as ⎑⎒⎒⎒⎣1𝑃(𝑑)=8π‘’βŠ–81(𝑑,0)+80018π‘’βŠ–81(𝑑,0)+8⎀βŽ₯βŽ₯βŽ₯⎦.(3.26) Therefore, by Theorem 3.9, the system (3.24) is uniformly exponentially stable.

Theorem 3.11. The system (2.8) is πœ“-uniformly stable if there exist positive definite symmetric matrix function 𝑃(𝑑)βˆˆπ’ž1rd(𝕋,ℝ𝑛×𝑛),πœ“(𝑑)βˆˆπ’ž1rd(𝕋,ℝ+), and πœ‚1,πœ‚2,𝛾,πœ–1,πœ–2,πœ–3,πœ–4βˆˆβ„+, 𝜌1,𝜌2βˆˆβ„ such that (i)πœ‚1𝐼≀𝑃(𝑑)β‰€πœ‚2𝐼,(ii)𝜌1𝐼≀𝑃Δ(𝑑)β‰€πœŒ2𝐼,(iii)𝑍(𝑑)≀0, (iv)πœ“Ξ”(𝑑)≀0.

Proof. We consider the following Lyapunov function for system (2.8) 𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑).(3.27) By (i), it is easy to see that πœ‚1β€–β€–π‘₯(𝑑)2≀𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)β‰€πœ‚2β€–β€–π‘₯(𝑑)2.(3.28) By the same argument as in the proof of Theorem 3.7, we obtain 𝑉Δ(𝑑)≀0. By (iv) and Theorem 3.1, the zero solution of (2.8) is πœ“-uniformly stable.

Example 3.12. We consider the linear time-varying dynamic system of the form π‘₯Δ(𝑑)=βˆ’π‘Ž(𝑑)βˆ’11βˆ’π‘Ž(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯(𝑑)),(3.29)π‘Ž(𝑑)=|sin(𝑑)|+1 and 𝑓(𝑑,π‘₯(𝑑)) are rd-continuous in the first argument with 𝑓(𝑑,0)=0 for all π‘‘βˆˆπ•‹. We let πœ“(𝑑)=βˆ’π‘‘ and π‘₯𝑓(𝑑,π‘₯(𝑑))=0.125sin(𝑑)1ξ€»ξ€Ίπ‘₯(𝑑)βˆ’0.125cos(𝑑)2ξ€»ξƒ­(𝑑).(3.30) Then πœ“Ξ”(𝑑)=βˆ’1≀0 and ‖𝑓(𝑑,π‘₯(𝑑))‖≀0.125β€–π‘₯(𝑑)β€–. Let 𝛾=1/8,πœ–1=1,πœ–2=1/16,πœ–3=1/2,πœ–4=1/16,πœ‚1=1/8,πœ‚2=1/4,𝜌1=βˆ’1, and 𝜌2=0. We can find a solution 𝑃(𝑑) satisfying (i)–(iv) of Theorem 3.11 as ⎑⎒⎒⎒⎣1𝑃(𝑑)=8π‘’βŠ–81(𝑑,0)+80018π‘’βŠ–81(𝑑,0)+8⎀βŽ₯βŽ₯βŽ₯⎦(3.31)πœ“Ξ”(𝑑)=βˆ’1≀0.
Therefore, by Theorem 3.11, the system (3.29) is πœ“-uniformly stable.

Theorem 3.13. The system (2.8) is β„Ž-stable if there exist a positive definite symmetric matrix function 𝑃(𝑑)βˆˆπ’ž1rd(𝕋,ℝ𝑛×𝑛), a bounded positive differentiable function β„ŽβˆΆπ•‹β†’β„, and πœ‚1,πœ‚2,𝛾,πœ–1,πœ–2,πœ–3,πœ–4βˆˆβ„+, 𝜌1,𝜌2βˆˆβ„ satisfying (i)πœ‚1𝐼≀𝑃(𝑑)β‰€πœ‚2𝐼,(ii)𝜌1𝐼≀𝑃Δ(𝑑)β‰€πœŒ2𝐼,(iii)𝑍(𝑑)≀𝛾1β„ŽΞ”(𝑑)β„Ž(𝑑)𝐼,𝛾1=⎧βŽͺ⎨βŽͺβŽ©πœ‚1,β„ŽΞ”πœ‚(𝑑)β‰₯0;2,β„ŽΞ”(𝑑)<0.(3.32)

Proof. Let 𝑑0βˆˆπ•‹, π‘₯(𝑑0)=π‘₯0 and π‘₯(𝑑,𝑑0,π‘₯0)=π‘₯(𝑑) be any solution of system (2.8). We consider a Lyapunov function for system (2.8) of the form 𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑).(3.33) By (i), we get πœ‚1β€–β€–π‘₯(𝑑)2≀𝑉(𝑑,π‘₯(𝑑))=π‘₯𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)β‰€πœ‚2β€–β€–π‘₯(𝑑)2.(3.34) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δπ‘₯(𝑑)=𝑇(𝑑)𝑃(𝑑)Ξ”π‘₯(𝑑)+π‘₯𝑇(𝜎(𝑑))𝑃(𝜎(𝑑))π‘₯Ξ”=ξ€Ίπ‘₯(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇π‘₯(𝑑,π‘₯)𝑃(𝑑)π‘₯(𝑑)+𝑇π‘₯(𝑑)+πœ‡(𝑑)𝑇(𝑑)𝐴𝑇(𝑑)+𝑓𝑇×𝑃(𝑑,π‘₯)ξ€Έξ€»Ξ”(𝑑)π‘₯(𝑑)+𝑃(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))+πœ‡(𝑑)𝑃Δ.(𝑑)(𝐴(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯))(3.35) By using (i), (ii), (iii), and Lemma 2.20, we obtain 𝑉Δπ‘₯(𝑑,π‘₯(𝑑))=𝑇(𝑑)𝑃(𝑑)π‘₯(𝑑)Δ≀𝛾1β„ŽΞ”(𝑑)β„Ž(𝑑)β€–π‘₯(𝑑)β€–2β‰€βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›Ύ1πœ‚1β„ŽΞ”(𝑑)β„Ž(𝑑)𝑉(𝑑,π‘₯(𝑑)),β„ŽΞ”π›Ύ(𝑑)β‰₯0,1πœ‚2β„ŽΞ”(𝑑)β„Ž(𝑑)𝑉(𝑑,π‘₯(𝑑)),β„ŽΞ”β‰€β„Ž(𝑑)<0,Ξ”(𝑑)ξ€Ίπ‘₯β„Ž(𝑑)𝑇.(𝑑)𝑃(𝑑)π‘₯(𝑑)(3.36) From Gronwall’s inequality for time scales [3], (i) and Lemma 2.15 in [2], we obtain πœ‚1β€–β€–π‘₯(𝑑)2≀π‘₯𝑇π‘₯(𝑑)𝑃(𝑑)π‘₯(𝑑)≀𝑇𝑑0𝑃𝑑0ξ€Έπ‘₯𝑑0π‘’ξ€Έξ€»β„ŽΞ”(𝑑)/β„Ž(𝑑)𝑑,𝑑0ξ€Έ,β‰€πœ‚2β€–β€–π‘₯𝑑0ξ€Έβ€–β€–2π‘’β„ŽΞ”(𝑑)/β„Ž(𝑑)𝑑,𝑑0ξ€Έβ‰€πœ‚2β€–β€–π‘₯𝑑0ξ€Έβ€–β€–2β„Ž(𝑑)β„Žξ€·π‘‘0ξ€Έ.(3.37) Hence, we get β€–β€–β€–π‘₯𝑑π‘₯(𝑑)β€–β‰€πœ”0‖‖𝑑𝐻(𝑑)𝐻0ξ€Έβˆ’1,𝑑β‰₯𝑑0,(3.38) where βˆšπœ”=πœ‚2/πœ‚1 and √𝐻(𝑑)=β„Ž(𝑑). Therefore, the zero solution of (2.8) is β„Ž-stable.

Example 3.14. We consider the linear time-varying dynamic system of the form π‘₯Δ(𝑑)=βˆ’π‘Ž(𝑑)βˆ’11βˆ’π‘Ž(𝑑)π‘₯(𝑑)+𝑓(𝑑,π‘₯(𝑑)),(3.39) where π‘Ž(𝑑)=π‘’βŠ–8(𝑑,0)+1 and 𝑓(𝑑,π‘₯(𝑑)) are rd-continuous in the first argument with 𝑓(𝑑,0)=0 for all π‘‘βˆˆπ•‹. Let β„Ž(𝑑)=5 and π‘₯𝑓(𝑑,π‘₯(𝑑))=0.125cos(𝑑)2ξ€»ξ€Ίπ‘₯(𝑑)βˆ’0.125sin(𝑑)1ξ€»ξƒ­(𝑑).(3.40)
Then β„ŽΞ”(𝑑)=0 and ‖𝑓(𝑑,π‘₯(𝑑))‖≀0.125β€–π‘₯(𝑑)β€–. Let 𝛾=1/8,πœ–1=1,πœ–2=1/16,πœ–3=1/2,πœ–4=1/16,πœ‚1=1/8,πœ‚2=1/4,𝜌1=βˆ’1, and 𝜌2=0. We can find a solution 𝑃(𝑑) satisfying (i)–(iii) of Theorem 3.13 as⎑⎒⎒⎒⎣1𝑃(𝑑)=8π‘’βŠ–81(𝑑,0)+80018π‘’βŠ–81(𝑑,0)+8⎀βŽ₯βŽ₯βŽ₯⎦.(3.41) Therefore, by Theorem 3.13, the system (3.39) is 5-stable.

4. Conclusion

In this paper, we have considered Lyapunov stability theory of linear time-varying system and derived sufficient conditions for uniform stability, uniform exponential stability, πœ“-uniform stability and β„Ž-stability for linear time-varying system with nonlinear perturbation on time scales. By construction of appropriate Lyapunov functions, we have derived several stability conditions. Numerical examples are presented to illustrate the effectiveness of the theoretical results.

Acknowledgments

The first author is supported by Khon Kaen University Research Fund and the Development and the Promotion of Science and Technology Talents Project (DPST). The second author is supported by the Center of Excellence in Mathematics, CHE, Thailand. He also wish to thank the National Research University Project under Thailand's Office of the Higher Education Commission for financial support.

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