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Variationally stable dynamic systems on time scales
Advances in Difference Equations volume 2012, Article number: 129 (2012)
In this paper we give a Lyapunov functional characterization of h-stability for nonlinear dynamic systems on time scales under the condition of -similarity between their variational systems. Furthermore, we give some examples related to the notions of -similarity and h-stability.
MSC:34N05, 39A30, 34D23, 34K20, 26E70.
It is widely known that the various types of stability of nonlinear differential equations or difference equations can be characterized by using Lyapunov’s second method, the method of variation of parameters, and inequalities, etc. [1, 21, 22, 24, 38, 40].
Pinto  introduced the notion of h-stability for differential equations with the intention of obtaining results about stability for weakly stable differential systems under some perturbations. Also, Medina and Pinto  applied the h-stability to obtain a uniform treatment for the various stability notions in difference systems and extended the study of exponential stability to a variety of reasonable systems called h-systems. Pinto and Medina obtained the important properties about h-stability for the various differential systems and difference systems [26–28, 30, 33–35].
Choi et al.  investigated h-stability for the nonlinear differential systems by means of the notions of Lyapunov functions and -similarity introduced by Conti . Trench  introduced summable similarity as a discrete analog of Conti’s definition of -similarity and investigated the various stabilities of linear difference systems by using summable similarity. Choi and Koo  studied the variational stability for nonlinear difference systems by means of -similarity. Also, Choi et al. studied the asymptotic property and the h-stability of difference systems via discrete similarities and comparison principle [9–11]. For detailed results about the various stabilities including the notions of h-stability and strong stability of dynamic systems on time scales, see [12–18, 20, 25, 31].
In this paper we introduce the notion of -similarity which extends the continuous -similarity  and the discrete -summable similarity . Then we give a Lyapunov functional characterization of h-stability for nonlinear dynamic systems on time scales by assuming the condition of -similarity between its variational systems. Furthermore, we give some examples related to the notions of -similarity and h-stability on time scales.
2 Main results
We refer the reader to Ref.  for all the basic definitions and results on time scales necessary for this work (e.g., delta differentiability, rd-continuity, exponential function and its properties).
It is assumed throughout that a time scale will be unbounded above. If has a left-scattered maximum m, then . Otherwise, . Let be the n-dimensional real Euclidean space. denotes the set of all rd-continuous functions from to and .
We consider the dynamic system
where with , and is the delta derivative of with respect to . We assume that exists and is rd-continuous on . Let be the unique solution of (2.1) satisfying the initial condition . For the existence and uniqueness of solutions of nonlinear dynamic system (2.1), see .
Also, we consider its associated variational systems
where is invertible for all and I denotes the identity matrix.
To establish our main results we will use the following lemmas.
Lemma 2.1 , Theorem 2.6.4]
Assume that and are the solutions of system (2.1) through and respectively, which exist for each and are such that and belong to a convex subset D of . Then
where Φ is a fundamental matrix of (2.3) and .
This lemma can be proved in the same manner as that of Theorem 2.6.4 in , so we omit the detail.
Lemma 2.2 , Lemma 2.7.1]
Let , where D is an open convex set in . Suppose that exists and is rd-continuous. Then
In order to prove the variation of parameters formula on time scales, we need the following result on differentiability of solutions with respect to initial values.
Lemma 2.3 Assume that possesses partial derivatives on and is rd-continuous on . Let be the solution of (2.1), which exists for and let
exists and is the solution of
The proof of Lemma 2.3 follows simply by differentiating the solution identity
with respect to . It is a special case of , Satz 1.2.22].
Remark 2.1 , Theorem 2.7.1]
in Lemma 2.3 is given by
It follows from Lemma 2.3 that the fundamental matrix solution of (2.2) is given by
and the fundamental matrix solution of (2.3) is given by
Let be the set of all matrices over . The class of all rd-continuous functions is denoted by
Consider the quasilinear dynamic system
where and is rd-continuous in the first argument with .
We need the following result which is a slight modification of the variation of constants formula in , Theorem 4.6.1].
Lemma 2.4 
The solution of (2.9) satisfies the equation
where Φ is a transition matrix of the linear system
For the Lyapunov-like function , we recall the following definition.
Definition 2.2 , Definition 3.1.1]
We define the generalized derivative of relative to (2.1) as follows: given , there exists a neighborhood of such that
where is any solution of (2.1) and the upper right Dini derivative of is given by
Then it is well known that
In case is right dense, we have
In case is right scattered and is continuous at t, we have
In fact, if is a solution of (2.1), we have
by the chain rule of a differentiable function , Theorem 1].
We note that the total difference of the function V along the solutions x of (2.1) is given by
Choi et al.  investigated h-stability for nonlinear differential systems using the notions of -similarity and Lyapunov functions. Also, Choi et al.  introduced the notion of -summable similarity which is the corresponding -similarity for the discrete case and then characterized h-stability in variation and asymptotic equilibrium in variation for nonlinear difference system via -summable similarity and comparison principle.
Now, we define -similarity on time scales in order to unify (continuous ) -similarity and (discrete) -similarity for matrix-valued functions.
Let be the set of all invertible matrices over , and be the set of all rd-continuous differentiable functions S from to such that S and are bounded on .
Definition 2.3 A function is called regressive if for each the matrix is invertible.
The class of all rd-continuous and regressive functions from to is denoted by
Definition 2.4 
Let and . A function A is -similar to a function B if there exists an absolutely integrable function , i.e., , such that
for some .
Remark 2.5 If , then -similarity becomes -similarity and if , then -similarity becomes -similarity. Also if A and B are -similar with defined on , then they are kinematically similar .
Let , where is a nonnegative integer and denote the set of all invertible matrix-valued functions defined on .
Remark 2.6 , Definition 2.5]
A matrix function is -summably similar to a matrix function if there exists an absolutely summable matrix over , that is,
for some .
For the example of -summable similarity, see .
Remark 2.7 We can easily show that the -summable similarity is an equivalence relation in the similar manner of Trench in . Also, if A and B are -summably similar with , then we say that they are kinematically similar.
Pinto  introduced the notion of h-stability which is an extension of the notions of exponential stability and uniform stability of the solutions of differential equations. The symbol will be used to denote any convenient vector norm in . We recall the notions of h-stability for dynamic systems on time scales in .
Definition 2.8 System (2.1) is called an h-system if there exist a positive rd-continuous function , a constant and such that
for (here ).
Moreover, system (2.1) is said to be
(h S) h-stable if h is a bounded function in the definition of h-system,
(Gh S) globally h-stable if system (2.1) is h S for every , where is a region which includes the origin,
(h SV) h-stable in variation if system (2.3) is h S,
(Gh SV) globally h-stable in variation if system (2.3) is Gh S.
For the various definitions of stability, we refer to  and we obtain the following possible implications for system (2.1) among the various types of stability:
We consider two linear dynamic systems
We say that if A and B are -similar, then systems (2.15) and (2.16) are -similar.
Lemma 2.5 , Lemma 2.3]
System (2.15) is an h-system if and only if there exist a positive rd-continuous function h defined on and a constant such that
where is a fundamental matrix solution of (2.15).
We obtain the following result from Lemma 2.3 in .
Lemma 2.6 Assume that A and B are -similar. Then
where and are the matrix exponential functions of (2.15) and (2.16) respectively.
Medina and Pinto , Theorem 3] showed that h SV implies h S. Also, they proved the converse when the condition
for holds , Theorem 14].
In order to establish our main results, we will introduce the following condition.
(H): and are -similar for and for some constant and with the positive rd-continuous function defined on .
Lemma 2.7 , Theorem 3.4]
Assume that condition (H) is satisfied. Then variational system (2.2) is also an h-system if and only if variational system (2.3) is an h-system.
We can obtain the same result about Lemma 2.7 by assuming that and are -quasisimilar for instead of the condition (H) in Lemma 2.7 , Theorem 3.3].
For nonlinear dynamic system (2.1), we can show that
by using the concept of -similarity.
We study the relation of h-stability between two systems (2.1) and (2.3) by assuming the condition (H) is satisfied.
Theorem 2.8 , Theorem 2]
Suppose that condition (H) is satisfied. If of (2.1) is h-stable, then of (2.2) is h-stable.
We obtain the following result from (2.8).
Theorem 2.9 If of (2.3) is h-stable, then of (2.1) is h-stable.
We can obtain the following result by using Lemma 2.7 and Theorem 2.8.
Theorem 2.10 Assume that condition (H) is satisfied. If of (2.1) is h-stable, then of (2.1) is h-stable in variation.
Remark 2.9 For nonlinear dynamic system (2.1), we show that two concepts of h-stability and h-stability in variation are equivalent under the condition that two variational systems (2.2) and (2.3) are -similar.
Choi et al. investigated Massera type converse theorems for the nonlinear difference system via -similarity in , Theorem 5] and , Theorem 2.1]. Furthermore, they characterized h-stability in variation for the nonlinear difference system by using the notion of -summable similarity in .
We need the following lemma to prove our main theorem.
Lemma 2.11 , Korollar 2.1.13]
If the delta differentiable function is positive, then is positively regressive, and satisfies
We can obtain the following result that characterizes h-stability for nonlinear dynamic system (2.1) via the notions of Lyapunov functions and -similarity. It is adapted from Theorem 3.6.1 in  and Theorem 3.1 in .
Theorem 2.12 Assume that condition (H) is satisfied. Suppose further that exists and is rd-continuous on . Then system (2.1) is GhS if and only if there exists a function defined on such that the following properties hold:
for and a constant ;
for and ;
is continuous on ;
Proof Necessity: Suppose that system (2.1) is Gh S. Then system (2.1) is Gh SV by Theorem 2.10, i.e., there exist a constant and a positive rd-continuous bounded function h defined on such that for each
where Φ is a fundamental matrix solution of (2.3).
Fix . Let . Then we note that is nonempty from .
Define the function by
where is a unique solution of system (2.1) for with the initial value . From Gh S of (2.1) we have
Furthermore, we obtain
Thus V satisfies property (i).
Let . Then we have
It follows from Lemma 2.2 that for each and in a convex subset D of
In view of (2.18), (2.19) and (2.20), we have
This implies that is globally Lipschitzian in x for fixed .
Next, we will prove property (iii). Let be a unique solution of system (2.1) for each initial point . We will consider two cases, and , in the proof.
Suppose that and let . By the uniqueness of solutions of (2.1) and the definition of h S, we have
Suppose that . Then it follows from the definition of that
Since the solution of (2.1) is unique, we have the following derivative:
Thus property (iii) was satisfied for two cases.
The continuity of can be proved in a similar manner of Theorem 3.6.1 in  and Theorem 3.1 in . It remains to show that V is continuous in the sense of (iv): let , be fixed and choose arbitrary. Then and must be found such that
holds for all
and all , where is an open ball centered on x of radius .
If is right scattered, then we can always choose a suitable such that is the only point satisfying condition (2.21) (see , Theorem 3.6.1]). Thus is continuous in since V is globally Lipschitz continuous in x for fixed .
Suppose that is right dense and let for with . Then we have
Since is Lipschitzian in x and is continuous in ν, the first two terms (2.22) and (2.23) on the right-hand side of the preceding inequality are small when and ν are small. That is, we have
for all when and
since it follows from that there exists a such that
for all with .
Let us consider the third term in (2.24). We note that
Thus we have
where for .
We have for all with . Furthermore, is a nonincreasing function in ν with
Hence, there exists a such that
Now, choose . For with where and , combining all of the above estimates of the terms in (2.22)-(2.24) gives
which proves the continuity of .
Sufficiency: Assume that satisfies the properties (i)-(iv). Let be any solution of system (2.1). Then it follows from condition (iii) of that
where . From (2.25) and property (i) of , we have
for each and . Hence the zero solution of (2.1) is Gh S. This completes the proof of the theorem. □
Remark 2.10 Assume that condition (H) is satisfied for . Furthermore, suppose that exists and is continuous on . Then we can obtain Theorems 2.4 and 2.6 in  as a continuous version of Theorem 2.12.
Also, we can obtain the following result as a discrete version of Theorem 2.12.
Corollary 2.13 , Theorem 3.7]
Assume that is -summably similar to for and every with . Then system (2.1) is GhS if and only if there exists a function defined on such that the following properties hold:
is continuous on ;
for , ;
for with .
Remark 2.11 Choi et al. [8, 9] introduced the notion of -similarity which is slightly different from -summable similarity and studied a general variational stability for a nonlinear difference system via -similarity and Lyapunov functions. We can obtain the discrete analogues , Theorem 5, Corollary 8] and , Theorem 2.1] as a discrete version of Theorem 2.12.
Remark 2.12 Choi et al. , Theorem 2.16] studied h-stability for linear dynamic equations on time scales by using the unified time scale quadratic Lyapunov functions. Also, Mukdasai and Niamsup , Theorem 3.13] derived a sufficient condition for h-stability for a linear time-varying system with nonlinear perturbation on time scales by constructing appropriate Lyapunov functions.
We can obtain the following Massera type converse theorem for the uniform exponential asymptotic stability of linear dynamic equations on time scales as a special case of Theorem 2.12.
Corollary 2.14 , Theorem 3.1]
Assume that is linear, where . If system (2.1) is hS with on time scales for a nonnegative constant λ, then there exists a function such that
for all .
for any fixed and all .
The upper right Dini derivative of exists and the estimates hold for all for a positive constants K and λ. Here the function is given by
V is continuous from the right in , that is,
In this section we give some examples which illustrate some results from the previous section.
To illustrate the notion of -similarity, we will give an example for scalar functions defined on time scales.
Example 3.1 Let be scalar functions given by
where α is a negative regressive constant and for fixed . If we put for each , then and are bounded and nonzero functions. Moreover, we have
Thus we have
This implies that a and b are -similar on .
Example 3.2 To illustrate Lemma 2.5, we consider the linear dynamic system
where . If for , then (3.1) is h-stable.
Proof A matrix exponential function of (3.1) is given by
where and . Here the cylinder transformation is given by
It follows that
for each with , where M is a positive constant. Thus we obtain
where is a positive bounded rd-continuous function for a fixed point , and is a positive constant. Hence system (3.1) is h-stable by Lemma 2.5. □
To illustrate that the converse of Theorem 2.8 does not hold in general, we give the following example.
Example 3.3 , Example 5.2]
Let be the unbounded above time scales with for each . We consider the nonlinear dynamic equation
and its variational dynamic equation
where . Then of (3.3) is h-stable, but of (3.2) is not h-stable.
Proof Since the fundamental solution is for each , Eq. (3.3) is h-stable with a positive bounded function for a fixed point . But (3.2) is not h-stable because there exists an unbounded solution of (3.2) such that
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This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2010-0008835). The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.