Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales

Consider the linear dynamic equation on time scales ( ) ( ) ( ) ( ) ( ) [ ) 0 0 0 , ; , , , T x t A t x t f t x x t x t t ∆ = + = ∈ ∞ (1) where ( ) . n x R ∈ , ( ) [ ) ( ) ( ) 0 . , , rd n T A C t M R ∈ ∞ , [ ) 0 : , n n T f t R R ∞ × → is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.


T x t A t x t f t x x t x t t
where ( ) , , → is a rd-continuous function, T

Introduction
Let n R be a n-dimension Euclidean space, T be a time scales (a nonempty closed subset of R).We denote For convenience, we shall use the notions which appear in the book by Bohner and Peterson (see [1] [2]).The notions related to the Lyapunov function that we use follow the results of B. Kaymakcalan (see [3]).For necessary, we recall them in this process.
We consider a dynamic equation ,0 0 F t = .We suppose that F satisfies all conditions such that (2) has a unique solution ( ) 0 0 , , x t t x with ( ) 0 0 x t x = .In this paper, we define the stable notions of the trivial solution ( ) 0 x t = of (2) as the followings: Definition 1.The trivial solution ( ) 0 x t = of (2) is stable on 0 t T + forall 0 ε > , there exists ( ) Definition 2. The trivial solution ( ) 0 x t = of (2) is asymptotically stable if it is stable and there exists ( ) In these definitions, if the numbers δ and 0 δ do not depend on 0 t , we say that the trivial solution of ( 2) is uniformly stable (uniformly asymptotically stable).

Definition 3. The trivial solution ( )
In the simple case (see [2]), consider the dynamic equation The solution of ( 3) is exponential function

∫
. We recall some properties of the exponential function which are used later.Assume S T is set of exponential stability of ( ) , e t s λ (see [4]).Theory of stability of dynamic equation on time scales is an area of mathematics that has recently received a lot of attention (see [1] [2] [4]- [7]).And almost of the results which involve the methods of Lyapunov to investigate the stability, have been developed and obtained the interesting results to expand for dynamic equation on time scales.Besides that the criterions and sufficient conditions were given, there were short of some particular examples.We know that the calculus for functions on general time scales is complex and difficult to implement.In order to overcome obstacles, in some cases we can combine the different methods of Lyapunov to investigate the stability of the solution.The content of this paper contains two parts: the first part presents the sufficient conditions following the first approximate method for the exponential stability of the solution of the linear dynamic Equation ( 1) on time scales.The second one gives some specific examples for applications.Besides the part two we add a theorem about the stability of the solution following the second method of Lyapunov.This theorem can be seen as a corollary of the stable criterion which was presented in [3].

The Stability of Linear Dynamic Equation under Perturbation on Time Scales
Consider the dynamic equation . , In proportion to the system (4), we consider where ( ) , We assume that ( ) . We denote ( ) ( ) = is exponential matrix of ( 5) with ( ) 0 0 x t x = .We easily verify that ( ) ( ) We assume that the trivial solution of (5) is exponentially stable, there exists 0 K > , ( ) then the trivial solution of ( 4) is exponentially stable if one of these conditions is satisfied i) ii) There exists a function Proof.We assume that ( ) x t is the solution of ( 4) with ( ) ( ) With 0 q < , we can choose 0 ε > , which is sufficiently small and 0 q εγ + < .So that the trivial solution of ( 4) is exponentially stable on 0 t T + .For ii), by argument similarly as in i), the proof is completed.

The Stability of Scalar Dynamic Equation on Time Scales
For convenience, the first we consider the scalar dynamic equation where Theorem 5. We assume that ( ) Then the trivial solution of ( 6) is exponentially stable if one of these conditions is satisfied i) ii) There exists a function where is the solution of ( 6) with ( ) 0 0 x t x = , we have ( x t e t t x e t s f s x s s By argument similarly as the proof in theorem 4, we obtain results. In the next part, for convenience to investigate the stability in specific examples, we represent a theorem about the sufficient condition for the exponential stability of the trivial solution of system (2).This result can be seen as a corollary of the stable criterion B. Kaymakcalan (see [3]).
We assume 0 : the solution of ( 2) with ( ) 0 0 x t x = .Then derivative of ( ) , V t x following the trajectory of ( ) x t is defined by .
x with above properties is a Lyapunov function.Theorem 6.We assume that there exists function 0 : × → is a Lyapunov function which satisfies the following conditions , where 1 0 λ > and 1 α ≥ are positive real numbers, ( ) is exponentially stable then the trivial solution of ( 2) is also exponentially stable.
Proof.By the assumption the trivial solution of ( 7) is exponentially stable, then the maximal solution ( ) r t of ( 7) with ( ) .By theorem 2.1 (see [3]) we obtain ( ) ( ) Using the assumption, we have ( ) Therefore By the assumption 1 α ≥ implies the trivial solution of ( 2) is exponentially stable.

Applications
In this part, we represent some examples of applications.Example 1. Assume that , α β are positive constants.These functions ( ) satisfy one of the conditions i) or ii) of theorem 4. Consider system , , , , We assume that ( ) ∀ ∈ in order that system (8) has the trivial solution.We consider ( ) ( ) In order to investigate the stability of (9), we choose Lyapunov function ( Therefore the derivative of right-hand side of (9) is  , , V x y z x y z = + + , we obtain system (10), we investigate the stability of the trivial solution of system ) , By using the results of theorem 6, the trivial solution of (9) is exponentially stable.Therefore following theorem 4, the trivial solution of (8) is exponentially stable.