Abstract

We consider Lyapunov stability theory of linear time-varying system and derive sufficient conditions for uniform stability, uniform exponential stability, 𝜓-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(𝑡) of a function 𝑓𝕋, where 𝕋 is a so-called time scale (an arbitrary closed nonempty subset of ) becomes the usual derivative when 𝕋=, namely, 𝑓Δ(𝑡)=𝑓(𝑡). On the other hand, if 𝕋=, then 𝑓Δ(𝑡) reduces to the usual forward difference, namely, 𝑓Δ(𝑡)=Δ𝑓(𝑡). 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 [16], 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], 𝜓-uniform stability [6], and -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 𝜓-uniformly stability [6] and -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. + denotes the set of all nonnegative real numbers; denotes the set of all real numbers; + denotes the set of all non-negative integers; denotes the set of all integers; 𝑛 denotes the 𝑛-dimensional Euclidean space with the usual Euclidean norm ; 𝑥 denotes the Euclidean vector norm of 𝑥𝑛; 𝑛×𝑟 denotes the set of 𝑛×𝑟 real matrix; 𝐴𝑇 denotes the transpose of the matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆max(𝐴)=max{Re𝜆𝜆𝜆(𝐴)}; 𝜆min(𝐴)=min{Re𝜆𝜆𝜆(𝐴)}.

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

Definition 2.2. The mapping 𝜎,𝜌𝕋𝕋 defined by 𝜎(𝑡)=inf{𝑠𝕋𝑠>𝑡}, and 𝜌(𝑡)=sup{𝑠𝕋𝑠<𝑡} are called the jump operators.

Definition 2.3. A nonmaximal element 𝑡𝕋 is said to be right-scattered (rs) if 𝜎(𝑡)>𝑡 and right-dense (rd) if 𝜎(𝑡)=𝑡. A nonminimal element 𝑡𝕋 is called left-scattered (ls) if 𝜌(𝑡)<𝑡 and left-dense (ld) if 𝜌(𝑡)=𝑡.

Definition 2.4. The mapping 𝜇𝕋+ defined by 𝜇(𝑡)=𝜎(𝑡)𝑡 is called the graininess function.

Definition 2.5. (Delta derivative) assume 𝑓𝕋 is a function and let 𝑡𝕋. Then we define 𝑓Δ(𝑡) to be the number (provided it exists) with the property that given any 𝜖>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡𝛿,𝑡+𝛿)𝕋 for some 𝛿>0) such that |[𝑓(𝜎(𝑡))𝑓(𝑠)]𝑓Δ(𝑡)[𝜎(𝑡)𝑠]|𝜖|𝜎(𝑡)𝑠| for all 𝑠𝑈.
The function 𝑓Δ(𝑡) is the delta derivative of 𝑓 at 𝑡.
In the case that 𝕋=, we have 𝑓Δ(𝑡)=𝑓(𝑡). In the case that 𝕋=, we have 𝑓Δ(𝑡)=𝑓(𝑡+1)𝑓(𝑡).
The following are some useful relationships regarding the delta derivative, see [2].

Theorem 2.6 (see [2]). Assume that 𝑓𝕋𝑛 and let 𝑡𝕋. (i)If 𝑓 is differentiable at 𝑡, then 𝑓 is continuous at 𝑡. (ii)If 𝑓 is continuous at 𝑡 and 𝑡 is right scattered, then 𝑓 is differentiable at 𝑡 with 𝑓Δ(𝑡)=𝑓(𝜎(𝑡))𝑓(𝑡)𝜎(𝑡)𝑡.(2.1)(iii)If 𝑓 is differentiable at 𝑡 and 𝑡 is right dense, then 𝑓Δ(𝑡)=lim𝑠𝑡𝑓(𝑡)𝑓(𝑠)𝑡𝑠.(2.2)(iv)If 𝑓 is differentiable at 𝑡, then 𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓Δ(𝑡).(2.3)

Theorem 2.7 (see [2]). Assume that 𝑓,𝑔𝕋𝑛 and let 𝑡𝕋. (i)The sum 𝑓,𝑔𝕋𝑛 are differentiable at 𝑡 with (𝑓+𝑔)Δ(𝑡)=(𝑓)Δ(𝑡)+(𝑔)Δ(𝑡).(2.4)(ii) For any constant 𝛼,𝛼𝑓𝕋𝑛 is differentiable at 𝑡 with (𝛼𝑓)Δ(𝑡)=𝛼𝑓Δ(𝑡).(2.5)(iii)The product 𝑓𝑔𝕋𝑛 is differentiable at 𝑡 with (𝑓𝑔)Δ(𝑡)=𝑓Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔Δ(𝑡)=𝑓(𝑡)𝑔Δ(𝑡)+𝑓Δ(𝑡)𝑔(𝜎(𝑡)).(2.6)

Definition 2.8. The function 𝑓𝕋𝑛 is said to be rd-continuous (denoted by 𝑓𝒞rd(𝕋,𝑛)) if the following conditions hold. (i)𝑓 is continuous at every right-dense point 𝑡𝕋. (ii)lim𝑠𝑡𝑓(𝑠) exists and is finite at every ld-point 𝑡𝕋.

Definition 2.9. Let 𝑓𝒞rd(𝕋,𝑛). Then 𝑔𝕋𝑛 is called the antiderivative of 𝑓 on 𝕋 if it is differentiable on 𝕋 and satisfies 𝑔Δ(𝑡)=𝑓(𝑡) for 𝑡𝕋. In this case, we define 𝑡𝑎𝑓(𝑠)Δ𝑠=𝑔(𝑡)𝑔(𝑎),𝑎𝑡𝕋.(2.7) Consider the linear time-varying system with nonlinear perturbation on time scales (𝕋) of the form 𝑥Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),𝑡𝕋,(2.8) where 𝑥(𝑡)𝑛,𝐴𝕋𝑛×𝑛 is an 𝑛×𝑛 matrix-valued function and 𝑓𝕋×𝑛𝑛 is rd-continuous in the first argument with 𝑓(𝑡,0)=0. The uncertain perturbation is known to satisfy a bound of the form 𝑓(𝑡,𝑥(𝑡))𝛾𝑥(𝑡),(2.9) or equivalently, the perturbation is conically bounded. The solution of (2.8) through (𝑡0,𝑥(𝑡0)) satisfies the variation of constants formula 𝑥(𝑡)=Φ𝐴𝑡,𝑡0𝑥𝑡0+𝑡𝑡0Φ𝐴(𝑡,𝜎(𝑠))𝑓(𝑠,𝑥(𝑠))Δ𝑠,𝑡𝑡0.(2.10) When 𝑓(𝑡,𝑥(𝑡))=0, (2.8) becomes the linear time-varying system 𝑥Δ(𝑡𝑡)=𝐴(𝑡)𝑥(𝑡),𝑥0=𝑥0,𝑡0𝕋.(2.11) For the case when 𝑓(𝑡,𝑥(𝑡))=𝐵(𝑡)𝑥(𝑡),𝐵(𝑡)𝑛×𝑛, (2.8) becomes the linear time-varying system 𝑥Δ([]𝑡𝑡)=𝐴(𝑡)+𝐵(𝑡)𝑥(𝑡),𝑥0=𝑥0,𝑡0𝕋.(2.12) The norm of 𝑛×𝑛 matrix 𝐴 is defined as 𝐴=max𝑥=1𝐴𝑥.(2.13) The Euclidean norm of 𝑛×1 vector 𝑥(𝑡) is defined by 𝑥(𝑡)=𝑥𝑇(𝑡)𝑥(𝑡).(2.14)

Definition 2.10. A function 𝜙[0,𝑟][0,+) is of class 𝒦 if it is well-defined, continuous, and strictly increasing on [0,𝑟] with 𝜙(0)=0.

Definition 2.11. Assume 𝑔𝕋. Define and denote 𝑔𝒞rd(𝕋;) as right-dense continuous (rd-continuous) if 𝑔 is continuous at every right-dense point 𝑡𝕋 and lim𝑠𝑡𝑔(𝑠) exists, and is finite, at every left-dense point 𝑡𝕋. Now define the so-called set of regressive functions, , by =𝑝𝕋𝑝𝒞rd(𝕋;),1+𝑝(𝑡)𝜇(𝑡)0,𝑡𝕋,(2.15) and define the set of positively regressive functions by +={𝑝1+𝑝(𝑡)𝜇(𝑡)>0,𝑡𝕋}.(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 𝑥𝑡,𝑥0,𝑡0𝑥𝛾0,(2.17) for all 𝑡𝕋,𝑡𝑡0.

Definition 2.13. The zero solution of system (2.8) is called uniformly exponentially stable if there exist finite constants 𝛾,𝜆>0 with 𝜆+ such that 𝑥𝑡,𝑥0,𝑡0𝑥𝛾0𝑒𝜆𝑡,𝑡0,(2.18) for all 𝑡𝕋,𝑡𝑡0.

Definition 2.14. The zero solution of system (2.8) is called 𝜓-uniformly stable if there exists a finite constant 𝛾>0 such that for any 𝑡0 and 𝑥(𝑡0), the corresponding solution satisfies 𝜓(𝑡)𝑥𝑡,𝑥0,𝑡0𝜓𝑡𝛾0𝑥0,(2.19) for all 𝑡𝕋,𝑡𝑡0.

Definition 2.15. System (2.8) is called an -𝑠𝑦𝑠𝑡𝑒𝑚 if there exist a positive function 𝕋, a constant 𝑐1 and 𝛿>0 such that 𝑥𝑡,𝑥0,𝑡0𝑥𝑐0𝑡(𝑡)01,𝑡𝑡0,(2.20) if 𝑥0<𝛿(here(𝑡)1=1/(𝑡)). If is bounded, then (2.8) is said to be -stable.

Definition 2.16. A continuous function 𝑃𝕋 with 𝑃(0)=0 is called positive definite (negative definite) on 𝕋 if there exists a function 𝜙𝒦 such that 𝜙(𝑡)𝑃(𝑡) (𝜙(𝑡)𝑃(𝑡)) for all 𝑡𝕋.

Definition 2.17. A continuous function 𝑃𝕋 with 𝑃(0)=0 is called positive semidefinite (negative semi-definite) on 𝕋 if 𝑃(𝑡)0 (𝑃(𝑡)0) for all 𝑡𝕋.

Definition 2.18. A continuous function 𝑃𝕋×𝑛 with 𝑃(𝑡,0)=0 is called positive definite (negative definite) on 𝕋×𝑛 if there exists a function 𝜙𝒦 such that 𝜙(𝑥)𝑃(𝑡,𝑥) (𝜙(𝑥)𝑃(𝑡,𝑥)) for all 𝑡𝕋 and 𝑥𝑛.

Definition 2.19. A continuous function 𝑃𝕋×𝑛 with 𝑃(𝑡,0)=0 is called positive semi-definite (negative semi-definite) on 𝕋×𝑛 if 0𝑃(𝑡,𝑥) (0𝑃(𝑡,𝑥)) for all 𝑡𝕋 and 𝑥𝑛.

Lemma 2.20 ([7], Completing the square). assume that 𝑆𝑀𝑛×𝑛 is a symmetric positive definite matrix. Then for every 𝑄𝑀𝑛×𝑛, we obtain 2𝑥𝑇𝑄𝑦𝑦𝑇𝑆𝑦𝑥𝑇𝑄𝑆1𝑄𝑇𝑥,𝑥,𝑦𝑅𝑛.(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(𝑡,𝑥(𝑡))𝒞1rd(𝕋×𝑛,+), and 𝑎,𝑏+ such that (i)𝑉Δ(𝑡,𝑥(𝑡))0,(ii)𝑎𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝑏𝑥(𝑡)2, then the zero solution of system (2.8) is 𝜓-uniformly stable if there exists 𝜓(𝑡)𝒞1rd(𝕋,+) satisfying 𝜓Δ(𝑡)0.

Proof. For 𝑡0𝕋, we let 𝑥(𝑡0)=𝑥0. Then, by (i), we have 𝑡𝑡0𝑉Δ(𝑡𝑠,𝑥(𝑠))Δ𝑠=𝑉(𝑡,𝑥(𝑡))𝑉0𝑡,𝑥00,𝑡𝑡0𝜓Δ(𝑡𝑠)Δ𝑠=𝜓(𝑡)𝜓00.(3.1) We obtain 𝑉(𝑡,𝑥(𝑡))𝑉(𝑡0,𝑥(𝑡0)) and 𝜓(𝑡)𝜓(𝑡0) for all 𝑡𝕋,𝑡𝑡0. By (ii), we get the estimation as follows: 𝑎𝜓(𝑡)2𝑥(𝑡)2𝜓(𝑡)2𝜓𝑡𝑉(𝑡,𝑥(𝑡))02𝑉𝑡0𝑡,𝑥0𝜓𝑡𝑏02𝑥𝑡02.(3.2) We conclude that 𝜓(𝑡)𝑥(𝑡)𝛾𝜓(𝑡0)𝑥(𝑡0) where 𝛾=𝑏/𝑎>0. Therefore, the zero solution of system (2.8) is 𝜓-uniformly stable. The proof of the theorem is complete.

Corollary 3.2. If there exist a continuously differentiable positive definite function 𝑉(𝑡,𝑥(𝑡))𝒞1rd(𝕋×𝑛,+) and 𝑎,𝑏+ such that (i)𝑉Δ(𝑡,𝑥(𝑡))0,(ii)𝑎𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝑏𝑥(𝑡)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(𝑡,𝑥(𝑡))𝒞1rd(𝕋×𝑛,+) and 𝑎,𝑏,𝜖+ with 𝜖/𝑏+ satisfying (i)𝑉Δ(𝑡,𝑥(𝑡))𝜖𝑥(𝑡)2,(ii)𝑎𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝑏𝑥(𝑡)2, then the zero solution of system (2.8) is uniformly exponentially stable.

Proof. For 𝑡0𝕋, we let 𝑥(𝑡0)=𝑥0. We obtain, by (i) and (ii), that for all 𝑡𝑡0, 𝑉Δ(𝑡,𝑥(𝑡))𝜖𝑥(𝑡)2𝜖𝑏𝑉(𝑡,𝑥(𝑡)).(3.3) Since 𝜖/𝑏+, it follows from Gronwall’s inequality for time scales [2] and (ii) that 𝑎𝑥(𝑡)2𝑡𝑉(𝑡,𝑥(𝑡))𝑉0𝑡,𝑥0𝑒𝜖/𝑏𝑡,𝑡0𝑥𝑡𝑏02𝑒𝜖/𝑏𝑡,𝑡0.(3.4) Hence, we get 𝑥𝑡𝑥(𝑡)𝛾0𝑒𝜖/𝑏(𝑡,𝑡0)1/2,(3.5) where 𝛾=𝑏/𝑎 for all 𝑡𝑡0. 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(𝑡,𝑥(𝑡))𝒞1rd(𝕋×𝑛,+), a bounded positive differentiable function 𝕋 and 𝑎,𝑏+ such that (i)𝑉Δ(𝑡,𝑥(𝑡))𝛾Δ(𝑡)/(𝑡)𝑥(𝑡)2,𝛾=𝑎,Δ(𝑡)0;𝑏,Δ(𝑡)<0,(3.6)(ii)𝑎𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝑏𝑥(𝑡)2, then the zero solution of system (2.8) is -stable.

Proof. Let 𝑡0𝕋, 𝑥(𝑡0)=𝑥0 and 𝑥(𝑡,𝑡0,𝑥0)=𝑥(𝑡) be any solution of system (2.8). By (i), we have 𝑉Δ(𝑡,𝑥(𝑡))𝛾Δ(𝑡)(𝑡)𝑥(𝑡)2𝛾𝑎Δ(𝑡)(𝑡)𝑉(𝑡,𝑥(𝑡)),Δ𝛾(𝑡)0;𝑏Δ(𝑡)(𝑡)𝑉(𝑡,𝑥(𝑡)),Δ(𝑡)<0,Δ(𝑡)𝑉(𝑡)(𝑡,𝑥(𝑡)).(3.7) From Gronwall’s inequality for time scales [2], (ii) and Lemma 2.15 [4], we obtain 𝑎𝑥(𝑡)2𝑡𝑉(𝑡,𝑥(𝑡))𝑉0𝑡,𝑥0𝑒Δ(𝑡)/(𝑡)𝑡,𝑡0𝑥𝑡𝑏02𝑒Δ(𝑡)/(𝑡)𝑡,𝑡0,𝑥𝑡𝑏02(𝑡)𝑡0.(3.8) Thus, 𝑥𝑡𝑥(𝑡)𝛾0𝑡𝐻(𝑡)𝐻01,𝑡𝑡0,(3.9) where 𝛾=𝑏/𝑎 and 𝐻(𝑡)=(𝑡). Therefore, zero solution of (2.8) is -stable.

3.2. Stability Conditions

We introduce the following notation for later use:𝑍(𝑡)=𝑃Δ(𝑡)+𝐴𝑇(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)+𝜇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)+𝜇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)+𝜖1𝑃(𝑡)𝑃(𝑡)+𝜇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)+𝜇2(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)+𝜖2𝑃Δ(𝑡)𝑃Δ(𝑡)+𝜖21𝛾2𝜇(𝑡)2𝐼+𝜖3𝐴𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝐴(𝑡)+𝜖31𝛾2𝜇(𝑡)2𝐼+𝜖4𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝐴(𝑡)+𝜖41𝛾2𝜇(𝑡)4𝐼+𝜖11𝛾2𝐼+𝜂2𝛾2𝜇(𝑡)𝐼+𝜌2𝛾2𝜇(𝑡)2𝐼.(3.10)

Theorem 3.5. The system (2.11) is uniformly stable if there exist a positive definite symmetric matrix function 𝑃(𝑡)𝒞1rd(𝕋,𝑛×𝑛) and 𝜂,𝜌+ such that (i)𝜂𝐼𝑃(𝑡)𝜌𝐼,(ii)𝐴𝑇(𝑡)𝑃(𝑡)+(𝐼+𝜇(𝑡)𝐴𝑇(𝑡))(𝑃Δ(𝑡)+𝑃(𝑡)𝐴(𝑡)+𝜇(𝑡)𝑃Δ(𝑡)𝐴(𝑡))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(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡). In this case, we obtain 𝑉Δ(𝑡)=𝑥𝑇(𝐴𝑡)𝑇(𝑡)𝑃(𝑡)+𝐼+𝜇(𝑡)𝐴𝑇(𝑃𝑡)Δ(𝑡)+𝑃(𝑡)𝐴(𝑡)+𝜇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡).(3.11)

Theorem 3.7. The system (2.8) is uniformly stable if there exist a positive definite symmetric matrix function 𝑃(𝑡)𝒞1rd(𝕋,𝑛×𝑛) and 𝜂1,𝜂2,𝛾,𝜖1,𝜖2,𝜖3,𝜖4+, 𝜌1,𝜌2 such that (i)𝜂1𝐼𝑃(𝑡)𝜂2𝐼,(ii)𝜌1𝐼𝑃Δ(𝑡)𝜌2𝐼,(iii)𝑍(𝑡)0.

Proof. We consider the following Lyapunov function for system (2.8). 𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡).(3.12) By (i), it is easy to see that 𝜂1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)𝜂2𝑥(𝑡)2.(3.13) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δ𝑥(𝑡)=𝑇(𝑡)𝑃(𝑡)Δ𝑥(𝑡)+𝑥𝑇(𝜎(𝑡))𝑃(𝜎(𝑡))𝑥Δ(𝑡)=𝑥𝑇(𝑡)Δ𝑃(𝑡)𝑥(𝑡)+𝑥𝑇(𝜎(𝑡))𝑃Δ(𝑡)𝑥(𝑡)+𝑥𝑇(𝜎(𝑡))𝑃(𝜎(𝑡))𝑥Δ=𝑥(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇𝑥(𝑡,𝑥)𝑃(𝑡)𝑥(𝑡)+𝑇𝑥(𝑡)+𝜇(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇×𝑃(𝑡,𝑥)Δ(𝑡)𝑥(𝑡)+𝑃(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))+𝜇(𝑡)𝑃Δ(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))=𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)2𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑓𝑇(𝑡,𝑥)𝑃(𝑡)𝑓(𝑡,𝑥)+𝜇(𝑡)2𝑓𝑇(𝑡,𝑥)𝑃Δ(𝑡)𝑓(𝑡,𝑥)+𝑓𝑇(𝑡,𝑥)𝑃(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑓𝑇(𝑡,𝑥)𝑃Δ(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑓𝑇(𝑡,𝑥)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)2𝑓𝑇(𝑡,𝑥)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝑃(𝑡)𝑓(𝑡,𝑥)+𝜇(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑓(𝑡,𝑥)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑓(𝑡,𝑥)+𝜇(𝑡)2𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑓(𝑡,𝑥).(3.14) By (i), (ii), and Lemma 2.20, we have the following estimate: 𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)𝜂2𝑥𝑇𝑥(𝑡)𝑥(𝑡),𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)𝜌2𝑥𝑇(𝑡)𝑥(𝑡),2𝑥𝑇(𝑡)𝑃(𝑡)𝑓(𝑡,𝑥)𝜖1𝑥𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝑥(𝑡)+𝜖11𝑓𝑇(𝑡,𝑥)𝑓(𝑡,𝑥),2𝜇(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑓(𝑡,𝑥)𝜖2𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜖21𝜇(𝑡)2𝑓𝑇(𝑡,𝑥)𝑓(𝑡,𝑥),2𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑓(𝑡,𝑥)𝜖3𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖31𝜇(𝑡)2𝑓𝑇(𝑡,𝑥)𝑓(𝑡,𝑥),2𝜇(𝑡)2𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑓(𝑡,𝑥)𝜖4𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖41𝜇(𝑡)4𝑓𝑇(𝑡,𝑥)𝑓(𝑡,𝑥).(3.15) From the above inequalities and 𝑓(𝑡,𝑥)𝛾𝑥(𝑡), we obtain 𝑉Δ(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)2𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖1𝑥𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝑥(𝑡)+𝜖11𝛾2𝑥(𝑡)2+𝜖2𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜖21𝛾2𝜇(𝑡)2𝑥(𝑡)2+𝜖3𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖31𝛾2𝜇(𝑡)2𝑥(𝑡)2+𝜖41𝛾2𝜇(𝑡)4𝑥(𝑡)2+𝜖4𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜂2𝛾2𝜇(𝑡)𝑥(𝑡)2+𝜌2𝛾2𝜇(𝑡)2𝑥(𝑡)2=𝑥𝑇(𝑡)𝑍(𝑡)𝑥(𝑡).(3.16) By (iii), we conclude that 𝑉Δ(𝑡)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 𝑥Δ(𝑥𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),(3.17)𝐴(𝑡)=𝑎(𝑡)11𝑎(𝑡),𝑓(𝑡,𝑥(𝑡))=0.125sin(𝑡)2𝑥(𝑡)0.125cos(𝑡)1(𝑡),𝑓(𝑡,𝑥(𝑡))0.125𝑥(𝑡),(3.18) where 𝑎(𝑡)=𝑒8(𝑡,0)+1 and 𝑓(𝑡,𝑥(𝑡)) are rd-continuous in the first argument with 𝑓(𝑡,0)=0 for all 𝑡𝕋. Let 𝛾=1/8,𝜖1=1,𝜖2=1/16,𝜖3=1/2,𝜖4=1/16,𝜂1=1/8,𝜂2=1/4, 𝜌1=1, and 𝜌2=0. By assuming that 0𝜇(𝑡)0.25 for all 𝑡𝕋, we can find solution 𝑃(𝑡) satisfying conditions (i)–(iii) of Theorem 3.7 as 𝑃(𝑡)=(1/8)𝑒8(𝑡,0)+(1/8)00(1/8)𝑒8(𝑡,0)+(1/8). Observe that, 𝑃Δ(𝑡)=𝑒8(𝑡,0)00𝑒8(𝑡,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(𝑡)𝒞1rd(𝕋,𝑛×𝑛) and 𝜂1,𝜂2,𝛾,𝜖1,𝜖2,𝜖3,𝜖4,𝜖5+, 𝜌1,𝜌2 such that (i)𝜂1𝐼𝑃(𝑡)𝜂2𝐼,(ii)𝜌1𝐼𝑃Δ(𝑡)𝜌2𝐼,(iii)𝑍(𝑡)𝜖5𝐼.

Proof. Consider a Lyapunov function for system (2.8) of the form 𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡).(3.20) It is easy to see that (i) yields 𝜂1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)𝜂2𝑥(𝑡)2.(3.21) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δ𝑥(𝑡)=𝑇(𝑡)𝑃(𝑡)Δ𝑥(𝑡)+𝑥𝑇(𝜎(𝑡))𝑃(𝜎(𝑡))𝑥Δ=𝑥(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇𝑥(𝑡,𝑥)𝑃(𝑡)𝑥(𝑡)+𝑇𝑥(𝑡)+𝜇(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇×𝑃(𝑡,𝑥)Δ(𝑡)𝑥(𝑡)+𝑃(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))+𝜇(𝑡)𝑃Δ.(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))(3.22) From Theorem 3.7, we obtain 𝑉Δ(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)+𝑥𝑇(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑥𝑇(𝑡)𝐴(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜇(𝑡)2𝑥𝑇(𝑡)𝐴(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖1𝑥𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝑥(𝑡)+𝜖11𝛾2𝑥(𝑡)2+𝜖2𝑥𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝑥(𝑡)+𝜖21𝛾2𝜇(𝑡)2𝑥(𝑡)2+𝜖3𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃(𝑡)𝑃(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜖31𝛾2𝜇(𝑡)2(𝑥𝑡)2+𝜖41𝛾2𝜇(𝑡)4(𝑥𝑡)2+𝜖4𝑥𝑇(𝑡)𝐴𝑇(𝑡)𝑃Δ(𝑡)𝑃Δ(𝑡)𝐴(𝑡)𝑥(𝑡)+𝜂2𝛾2𝜇(𝑡)𝑥(𝑡)2+𝜌2𝛾2𝜇(𝑡)2(𝑥𝑡)2.(3.23) By (iii), we conclude that 𝑉Δ(𝑡)𝜖5𝑥(𝑡)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 𝑥Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),(3.24) where 𝑥𝐴(𝑡)=𝑎(𝑡)11𝑎(𝑡),𝑓(𝑡,𝑥(𝑡))=0.125cos(𝑡)2𝑥(𝑡)0.125sin(𝑡)1(𝑡).(3.25)𝑎(𝑡)=𝑒8(𝑡,0)+1 and 𝑓(𝑡,𝑥(𝑡)) are rd-continuous in the first argument with 𝑓(𝑡,0)=0 for all 𝑡𝕋. Then, 1𝑎(𝑡)2 and 𝑓(𝑡,𝑥(𝑡))0.125𝑥(𝑡) for all 𝑡𝕋. Let 𝛾=1/8,𝜖1=1,𝜖2=1/16,𝜖3=1/2,𝜖4=𝜖5=1/16,𝜂1=1/8,𝜂2=1/4,𝜌1=1, and 𝜌2=0. By assuming that 0𝜇(𝑡)0.25, for all 𝑡𝕋, we can find a solution 𝑃(𝑡) satisfying (i)–(iii) of Theorem 3.9 as 1𝑃(𝑡)=8𝑒81(𝑡,0)+80018𝑒81(𝑡,0)+8.(3.26) Therefore, by Theorem 3.9, the system (3.24) is uniformly exponentially stable.

Theorem 3.11. The system (2.8) is 𝜓-uniformly stable if there exist positive definite symmetric matrix function 𝑃(𝑡)𝒞1rd(𝕋,𝑛×𝑛),𝜓(𝑡)𝒞1rd(𝕋,+), and 𝜂1,𝜂2,𝛾,𝜖1,𝜖2,𝜖3,𝜖4+, 𝜌1,𝜌2 such that (i)𝜂1𝐼𝑃(𝑡)𝜂2𝐼,(ii)𝜌1𝐼𝑃Δ(𝑡)𝜌2𝐼,(iii)𝑍(𝑡)0, (iv)𝜓Δ(𝑡)0.

Proof. We consider the following Lyapunov function for system (2.8) 𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡).(3.27) By (i), it is easy to see that 𝜂1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)𝜂2𝑥(𝑡)2.(3.28) By the same argument as in the proof of Theorem 3.7, we obtain 𝑉Δ(𝑡)0. By (iv) and Theorem 3.1, the zero solution of (2.8) is 𝜓-uniformly stable.

Example 3.12. We consider the linear time-varying dynamic system of the form 𝑥Δ(𝑡)=𝑎(𝑡)11𝑎(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),(3.29)𝑎(𝑡)=|sin(𝑡)|+1 and 𝑓(𝑡,𝑥(𝑡)) are rd-continuous in the first argument with 𝑓(𝑡,0)=0 for all 𝑡𝕋. We let 𝜓(𝑡)=𝑡 and 𝑥𝑓(𝑡,𝑥(𝑡))=0.125sin(𝑡)1𝑥(𝑡)0.125cos(𝑡)2(𝑡).(3.30) Then 𝜓Δ(𝑡)=10 and 𝑓(𝑡,𝑥(𝑡))0.125𝑥(𝑡). Let 𝛾=1/8,𝜖1=1,𝜖2=1/16,𝜖3=1/2,𝜖4=1/16,𝜂1=1/8,𝜂2=1/4,𝜌1=1, and 𝜌2=0. We can find a solution 𝑃(𝑡) satisfying (i)–(iv) of Theorem 3.11 as 1𝑃(𝑡)=8𝑒81(𝑡,0)+80018𝑒81(𝑡,0)+8(3.31)𝜓Δ(𝑡)=10.
Therefore, by Theorem 3.11, the system (3.29) is 𝜓-uniformly stable.

Theorem 3.13. The system (2.8) is -stable if there exist a positive definite symmetric matrix function 𝑃(𝑡)𝒞1rd(𝕋,𝑛×𝑛), a bounded positive differentiable function 𝕋, and 𝜂1,𝜂2,𝛾,𝜖1,𝜖2,𝜖3,𝜖4+, 𝜌1,𝜌2 satisfying (i)𝜂1𝐼𝑃(𝑡)𝜂2𝐼,(ii)𝜌1𝐼𝑃Δ(𝑡)𝜌2𝐼,(iii)𝑍(𝑡)𝛾1Δ(𝑡)(𝑡)𝐼,𝛾1=𝜂1,Δ𝜂(𝑡)0;2,Δ(𝑡)<0.(3.32)

Proof. Let 𝑡0𝕋, 𝑥(𝑡0)=𝑥0 and 𝑥(𝑡,𝑡0,𝑥0)=𝑥(𝑡) be any solution of system (2.8). We consider a Lyapunov function for system (2.8) of the form 𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡).(3.33) By (i), we get 𝜂1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))=𝑥𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)𝜂2𝑥(𝑡)2.(3.34) The delta derivative of 𝑉 along the trajectories of system (2.8) is given by 𝑉Δ𝑥(𝑡)=𝑇(𝑡)𝑃(𝑡)Δ𝑥(𝑡)+𝑥𝑇(𝜎(𝑡))𝑃(𝜎(𝑡))𝑥Δ=𝑥(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇𝑥(𝑡,𝑥)𝑃(𝑡)𝑥(𝑡)+𝑇𝑥(𝑡)+𝜇(𝑡)𝑇(𝑡)𝐴𝑇(𝑡)+𝑓𝑇×𝑃(𝑡,𝑥)Δ(𝑡)𝑥(𝑡)+𝑃(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))+𝜇(𝑡)𝑃Δ.(𝑡)(𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥))(3.35) By using (i), (ii), (iii), and Lemma 2.20, we obtain 𝑉Δ𝑥(𝑡,𝑥(𝑡))=𝑇(𝑡)𝑃(𝑡)𝑥(𝑡)Δ𝛾1Δ(𝑡)(𝑡)𝑥(𝑡)2𝛾1𝜂1Δ(𝑡)(𝑡)𝑉(𝑡,𝑥(𝑡)),Δ𝛾(𝑡)0,1𝜂2Δ(𝑡)(𝑡)𝑉(𝑡,𝑥(𝑡)),Δ(𝑡)<0,Δ(𝑡)𝑥(𝑡)𝑇.(𝑡)𝑃(𝑡)𝑥(𝑡)(3.36) From Gronwall’s inequality for time scales [3], (i) and Lemma 2.15 in [2], we obtain 𝜂1𝑥(𝑡)2𝑥𝑇𝑥(𝑡)𝑃(𝑡)𝑥(𝑡)𝑇𝑡0𝑃𝑡0𝑥𝑡0𝑒Δ(𝑡)/(𝑡)𝑡,𝑡0,𝜂2𝑥𝑡02𝑒Δ(𝑡)/(𝑡)𝑡,𝑡0𝜂2𝑥𝑡02(𝑡)𝑡0.(3.37) Hence, we get 𝑥𝑡𝑥(𝑡)𝜔0𝑡𝐻(𝑡)𝐻01,𝑡𝑡0,(3.38) where 𝜔=𝜂2/𝜂1 and 𝐻(𝑡)=(𝑡). Therefore, the zero solution of (2.8) is -stable.

Example 3.14. We consider the linear time-varying dynamic system of the form 𝑥Δ(𝑡)=𝑎(𝑡)11𝑎(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),(3.39) where 𝑎(𝑡)=𝑒8(𝑡,0)+1 and 𝑓(𝑡,𝑥(𝑡)) are rd-continuous in the first argument with 𝑓(𝑡,0)=0 for all 𝑡𝕋. Let (𝑡)=5 and 𝑥𝑓(𝑡,𝑥(𝑡))=0.125cos(𝑡)2𝑥(𝑡)0.125sin(𝑡)1(𝑡).(3.40)
Then Δ(𝑡)=0 and 𝑓(𝑡,𝑥(𝑡))0.125𝑥(𝑡). Let 𝛾=1/8,𝜖1=1,𝜖2=1/16,𝜖3=1/2,𝜖4=1/16,𝜂1=1/8,𝜂2=1/4,𝜌1=1, and 𝜌2=0. We can find a solution 𝑃(𝑡) satisfying (i)–(iii) of Theorem 3.13 as1𝑃(𝑡)=8𝑒81(𝑡,0)+80018𝑒81(𝑡,0)+8.(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, 𝜓-uniform stability and -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.