#### Abstract

We consider Lyapunov stability theory of linear time-varying system and derive sufficient conditions for uniform stability, uniform exponential stability, -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 of a function , where is a so-called time scale (an arbitrary closed nonempty subset of ) becomes the usual derivative when , namely, . On the other hand, if , then reduces to the usual forward difference, namely, . 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], -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 -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; denotes the -dimensional Euclidean space with the usual Euclidean norm ; denotes the Euclidean vector norm of ; denotes the set of real matrix; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; .

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

*Definition 2.2. *The mapping defined by , and are called the jump operators.

*Definition 2.3. *A nonmaximal element is said to be right-scattered (rs) if and right-dense (rd) if . A nonminimal element is called left-scattered (ls) if and left-dense (ld) if .

*Definition 2.4. *The mapping defined by is called the graininess function.

*Definition 2.5. * (Delta derivative) assume is a function and let . Then we define to be the number (provided it exists) with the property that given any , there is a neighborhood of (i.e., for some ) such that for all .

The function is the delta derivative of at .

In the case that , we have . In the case that , we have .

The following are some useful relationships regarding the delta derivative, see [2].

Theorem 2.6 (see [2]). * Assume that and let . *(i)*If is differentiable at , then is continuous at . *(ii)*If is continuous at and is right scattered, then is differentiable at with *(iii)*If is differentiable at and is right dense, then *(iv)*If is differentiable at , then *

Theorem 2.7 (see [2]). * Assume that and let . *(i)*The sum are differentiable at with *(ii)* For any constant is differentiable at with *(iii)*The product is differentiable at with *

*Definition 2.8. *The function is said to be rd-continuous (denoted by if the following conditions hold. (i) is continuous at every right-dense point . (ii) exists and is finite at every ld-point .

*Definition 2.9. *Let . Then is called the antiderivative of on if it is differentiable on and satisfies for . In this case, we define
Consider the linear time-varying system with nonlinear perturbation on time scales of the form
where is an matrix-valued function and is rd-continuous in the first argument with . The uncertain perturbation is known to satisfy a bound of the form
or equivalently, the perturbation is conically bounded. The solution of (2.8) through satisfies the variation of constants formula
When , (2.8) becomes the linear time-varying system
For the case when , (2.8) becomes the linear time-varying system
The norm of matrix is defined as
The Euclidean norm of vector is defined by

*Definition 2.10. *A function is of class if it is well-defined, continuous, and strictly increasing on with .

*Definition 2.11. *Assume . Define and denote as right-dense continuous (rd-continuous) if is continuous at every right-dense point and exists, and is finite, at every left-dense point . Now define the so-called set of regressive functions, , by
and define the set of positively regressive functions by

*Definition 2.12. *The zero solution of system (2.8) is called uniformly stable if there exists a finite constant such that
for all .

*Definition 2.13. *The zero solution of system (2.8) is called uniformly exponentially stable if there exist finite constants with such that
for all .

*Definition 2.14. *The zero solution of system (2.8) is called -uniformly stable if there exists a finite constant such that for any and , the corresponding solution satisfies
for all .

*Definition 2.15. *System (2.8) is called an if there exist a positive function , a constant and such that
if . If is bounded, then (2.8) is said to be -stable.

*Definition 2.16. *A continuous function with is called positive definite (negative definite) on if there exists a function such that () for all .

*Definition 2.17. *A continuous function with is called positive semidefinite (negative semi-definite) on if () for all .

*Definition 2.18. *A continuous function with is called positive definite (negative definite) on if there exists a function such that () for all and .

*Definition 2.19. *A continuous function with is called positive semi-definite (negative semi-definite) on if () for all and .

Lemma 2.20 ([7], Completing the square). *assume that is a symmetric positive definite matrix. Then for every , we obtain
*

#### 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 , and such that *(i)*,*(ii)*, **then the zero solution of system (2.8) is -uniformly stable if there exists satisfying .*

*Proof. *For , we let . Then, by (i), we have
We obtain and for all . By (ii), we get the estimation as follows:
We conclude that where . Therefore, the zero solution of system (2.8) is -uniformly stable. The proof of the theorem is complete.

Corollary 3.2. * If there exist a continuously differentiable positive definite function and such that *(i)*,*(ii)*,**then the zero solution of system (2.8) is uniformly stable.*

Theorem 3.3. * If there exist a continuously differentiable positive definite function and with satisfying *(i)*,*(ii)*, ** then the zero solution of system (2.8) is uniformly exponentially stable.*

*Proof. * For , we let . We obtain, by (i) and (ii), that for all ,
Since , it follows from Gronwall’s inequality for time scales [2] and (ii) that
Hence, we get
where for all . 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 , a bounded positive differentiable function and such that *(i)*(ii)**, **then the zero solution of system (2.8) is -stable.*

*Proof. *Let , and be any solution of system (2.8). By (i), we have
From Gronwall’s inequality for time scales [2], (ii) and Lemma 2.15 [4], we obtain
Thus,
where and . Therefore, zero solution of (2.8) is -stable.

##### 3.2. Stability Conditions

We introduce the following notation for later use:

Theorem 3.5. * The system (2.11) is uniformly stable if there exist a positive definite symmetric matrix function and such that *(i)*,*(ii)*.*

*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 . In this case, we obtain

Theorem 3.7. * The system (2.8) is uniformly stable if there exist a positive definite symmetric matrix function and , such that *(i)*,*(ii)*,*(iii)*. *

*Proof. * We consider the following Lyapunov function for system (2.8).
By (i), it is easy to see that
The delta derivative of along the trajectories of system (2.8) is given by
By (i), (ii), and Lemma 2.20, we have the following estimate:
From the above inequalities and , we obtain
By (iii), we conclude that . 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
where and are rd-continuous in the first argument with for all . Let , , and . By assuming that for all , we can find solution satisfying conditions (i)–(iii) of Theorem 3.7 as . Observe that,
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 and , such that *(i)*,*(ii)*,*(iii)*. *

*Proof. * Consider a Lyapunov function for system (2.8) of the form
It is easy to see that (i) yields
The delta derivative of along the trajectories of system (2.8) is given by
From Theorem 3.7, we obtain
By (iii), we conclude that . 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
where
and are rd-continuous in the first argument with for all . Then, and for all . Let , and . By assuming that , for all , we can find a solution satisfying (i)–(iii) of Theorem 3.9 as
Therefore, by Theorem 3.9, the system (3.24) is uniformly exponentially stable.

Theorem 3.11. * The system (2.8) is -uniformly stable if there exist positive definite symmetric matrix function , and , such that *(i)*,*(ii)*,*(iii)*,
*(iv)*. *

*Proof. * We consider the following Lyapunov function for system (2.8)
By (i), it is easy to see that
By the same argument as in the proof of Theorem 3.7, we obtain . By (iv) and Theorem 3.1, the zero solution of (2.8) is -uniformly stable.

*Example 3.12. *We consider the linear time-varying dynamic system of the form
and are rd-continuous in the first argument with for all . We let and
Then and . Let , and . We can find a solution satisfying (i)–(iv) of Theorem 3.11 as
.

Therefore, by Theorem 3.11, the system (3.29) is -uniformly stable.

Theorem 3.13. * The system (2.8) is -stable if there exist a positive definite symmetric matrix function , a bounded positive differentiable function , and , satisfying *(i)*,*(ii)*,*(iii)

*Proof. * Let , and be any solution of system (2.8). We consider a Lyapunov function for system (2.8) of the form
By (i), we get
The delta derivative of along the trajectories of system (2.8) is given by
By using (i), (ii), (iii), and Lemma 2.20, we obtain
From Gronwall’s inequality for time scales [3], (i) and Lemma 2.15 in [2], we obtain
Hence, we get
where and . Therefore, the zero solution of (2.8) is -stable.

*Example 3.14. *We consider the linear time-varying dynamic system of the form
where and are rd-continuous in the first argument with for all . Let and

Then and . Let , and . We can find a solution satisfying (i)–(iii) of Theorem 3.13 as
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, -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.