Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 860506, 15 pages
doi:10.1155/2011/860506
Research Article
An LMI Approach to Stability for Linear
Time-Varying System with Nonlinear Perturbation
on Time Scales
Kanit Mukdasai1 and Piyapong Niamsup2, 3, 4
1
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
3
Center of Excellence in Mathematics CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
4
Materials Science Research Center, Faculty of Science, Chiang Mai University,
Chiangmai 50200, Thailand
2
Correspondence should be addressed to Kanit Mukdasai, kanit@kku.ac.th
Received 17 December 2010; Revised 3 May 2011; Accepted 19 May 2011
Academic Editor: Martin D. Schechter
Copyright q 2011 K. Mukdasai and P. 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.
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 f Δ t
of a function f : T → R, where T is a so-called time scale an arbitrary closed nonempty
subset of R becomes the usual derivative when T R, namely, f Δ t f t. On the other
hand, if T Z, then f Δ t reduces to the usual forward difference, namely, f Δ t Δft.
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
2
Abstract and Applied Analysis
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, ψ-uniform stability 6, and h-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 h-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; Z
denotes the set of all non-negative integers; Z denotes the set
of all integers; Rn denotes the n-dimensional Euclidean space with the usual Euclidean norm
· ; x denotes the Euclidean vector norm of x ∈ Rn ; Rn×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; λA denotes the set of all eigenvalues of A; λmax A max{Re λ : λ ∈ λA};
λmin A min{Re λ : λ ∈ λA}.
Definition 2.1. A time scale T is an arbitrary nonempty closed subset of the real numbers R.
Definition 2.2. The mapping σ, ρ : T → T defined by σt inf{s ∈ T : s > t}, and ρt sup{s ∈ T : s < t} are called the jump operators.
Definition 2.3. A nonmaximal element t ∈ T is said to be right-scattered rs if σt > t and
right-dense rd if σt t. A nonminimal element t ∈ T is called left-scattered ls if ρt < t
and left-dense ld if ρt t.
Definition 2.4. The mapping μ : T → R
defined by μt σt − t is called the graininess
function.
Definition 2.5. Delta derivative assume f : T → R is a function and let t ∈ T. Then we
define f Δ t to be the number provided it exists with the property that given any > 0,
there is a neighborhood U of t i.e., U t − δ, t δ ∩ T for some δ > 0 such that |fσt −
fs − f Δ tσt − s| ≤ |σt − s| for all s ∈ U.
The function f Δ t is the delta derivative of f at t.
In the case that T R, we have f Δ t f t. In the case that T Z, we have f Δ t ft 1 − ft.
Abstract and Applied Analysis
3
The following are some useful relationships regarding the delta derivative, see 2.
Theorem 2.6 see 2. Assume that f : T → Rn and let t ∈ T.
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
fσt − ft
.
σt − t
f Δ t 2.1
iii If f is differentiable at t and t is right dense, then
f Δ t lim
s→t
ft − fs
.
t−s
2.2
iv If f is differentiable at t, then
fσt ft μtf Δ t.
2.3
Theorem 2.7 see 2. Assume that f, g : T → Rn and let t ∈ T.
i The sum f, g : T → Rn are differentiable at t with
f g
Δ
Δ
Δ
t f t g t.
2.4
ii For any constant α, αf : T → Rn is differentiable at t with
αf
Δ
t αf Δ t.
2.5
iii The product fg : T → Rn is differentiable at t with
fg
Δ
t f Δ tgt fσtg Δ t ftg Δ t f Δ tgσt.
2.6
Definition 2.8. The function f : T → Rn is said to be rd-continuous denoted by f ∈
Crd T, Rn if the following conditions hold.
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Abstract and Applied Analysis
i f is continuous at every right-dense point t ∈ T.
ii lims → t− fs exists and is finite at every ld-point t ∈ T.
Definition 2.9. Let f ∈ Crd T, Rn . Then g : T → Rn is called the antiderivative of f on T if it
is differentiable on T and satisfies g Δ t ft for t ∈ T. In this case, we define
t
a
fsΔs gt − ga,
2.7
a ≤ t ∈ T.
Consider the linear time-varying system with nonlinear perturbation on time scales T of
the form
xΔ t Atxt ft, xt,
2.8
t ∈ T,
where xt ∈ Rn , A : T → Rn×n is an n × n matrix-valued function and f : T × Rn → Rn is
rd-continuous in the first argument with ft, 0 0. The uncertain perturbation is known to
satisfy a bound of the form
ft, xt ≤ γxt,
2.9
or equivalently, the perturbation is conically bounded. The solution of 2.8 through t0 , xt0 satisfies the variation of constants formula
xt ΦA t, t0 xt0 t
t0
ΦA t, σsfs, xsΔs,
t ≥ t0 .
2.10
When ft, xt 0, 2.8 becomes the linear time-varying system
xΔ t Atxt,
xt0 x0 ,
t0 ∈ T.
2.11
For the case when ft, xt Btxt, Bt ∈ Rn×n , 2.8 becomes the linear time-varying
system
xΔ t At Btxt,
xt0 x0 ,
t0 ∈ T.
2.12
The norm of n × n matrix A is defined as
A maxAx.
x1
2.13
The Euclidean norm of n × 1 vector xt is defined by
xt xT txt.
2.14
Abstract and Applied Analysis
5
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 φ0 0.
Definition 2.11. Assume g : T → R. Define and denote g ∈ Crd T; R as right-dense
continuous rd-continuous if g is continuous at every right-dense point t ∈ T and
lims → t− gs exists, and is finite, at every left-dense point t ∈ T. Now define the so-called
set of regressive functions, R, by
0, t ∈ T ,
R p : T → R | p ∈ Crd T; R, 1 ptμt /
2.15
and define the set of positively regressive functions by
R
p ∈ R | 1 ptμt > 0, t ∈ T .
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 , t0 ≤ γx0 ,
2.17
for all t ∈ T, t ≥ t0 .
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 ∈ R
such that
xt, x0 , t0 ≤ γx0 e−λ t, t0 ,
2.18
for all t ∈ T, t ≥ t0 .
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 xt0 , the corresponding solution satisfies
ψtxt, x0 , t0 ≤ γ ψt0 x0 ,
2.19
for all t ∈ T, t ≥ t0 .
Definition 2.15. System 2.8 is called an h-system if there exist a positive function h : T → R,
a constant c ≥ 1 and δ > 0 such that
xt, x0 , t0 ≤ cx0 htht0 −1 ,
t ≥ t0 ,
2.20
if x0 < δ here ht−1 1/ht. If h is bounded, then 2.8 is said to be h-stable.
Definition 2.16. A continuous function P : T → R with P 0 0 is called positive definite
negative definite on T if there exists a function φ ∈ K such that φt ≤ P t φt ≤ −P t
for all t ∈ T.
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Abstract and Applied Analysis
Definition 2.17. A continuous function P : T → R with P 0 0 is called positive semidefinite
negative semi-definite on T if P t ≥ 0 P t ≤ 0 for all t ∈ T.
Definition 2.18. A continuous function P : T × Rn → R with P t, 0 0 is called positive
definite negative definite on T × Rn if there exists a function φ ∈ K such that φx ≤ P t, x
φx ≤ −P t, x for all t ∈ T and x ∈ Rn .
Definition 2.19. A continuous function P : T × Rn → R with P t, 0 0 is called positive
semi-definite negative semi-definite on T × Rn if 0 ≤ P t, x 0 ≥ P t, x for all t ∈ T and
x ∈ Rn .
Lemma 2.20 7, Completing the square. assume that S ∈ Mn×n is a symmetric positive definite
matrix. Then for every Q ∈ Mn×n , we obtain
2xT Qy − yT Sy ≤ xT QS−1 QT x,
∀x, y ∈ Rn .
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, xt ∈
C1rd T × Rn , R
, and a, b ∈ R
such that
i V Δ t, xt ≤ 0,
ii axt2 ≤ V t, xt ≤ bxt2 ,
then the zero solution of system 2.8 is ψ-uniformly stable if there exists ψt ∈ C1rd T, R
satisfying
ψ Δ t ≤ 0.
Proof. For t0 ∈ T, we let xt0 x0 . Then, by i, we have
t
t0
V Δ s, xsΔs V t, xt − V t0 , xt0 ≤ 0,
t
t0
3.1
ψ Δ sΔs ψt − ψt0 ≤ 0.
We obtain V t, xt ≤ V t0 , xt0 and ψt ≤ ψt0 for all t ∈ T, t ≥ t0 . By ii, we get the
estimation as follows:
2
2
2
2
aψt xt2 ≤ ψt V t, xt ≤ ψt0 V t0 , xt0 ≤ bψt0 xt0 2 .
3.2
We conclude that ψtxt ≤ γψt0 xt0 where γ b/a > 0. Therefore, the zero solution
of system 2.8 is ψ-uniformly stable. The proof of the theorem is complete.
Abstract and Applied Analysis
7
Corollary 3.2. If there exist a continuously differentiable positive definite function V t, xt ∈
C1rd T × Rn , R
and a, b ∈ R
such that
i V Δ t, xt ≤ 0,
ii axt2 ≤ V t, xt ≤ bxt2 ,
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, xt ∈
C1rd T × Rn , R
and a, b, ∈ R
with −/b ∈ R
satisfying
i V Δ t, xt ≤ −xt2 ,
ii axt2 ≤ V t, xt ≤ bxt2 ,
then the zero solution of system 2.8 is uniformly exponentially stable.
Proof. For t0 ∈ T, we let xt0 x0 . We obtain, by i and ii, that for all t ≥ t0 ,
V Δ t, xt ≤ −xt2 ≤ − V t, xt.
b
3.3
Since −/b ∈ R
, it follows from Gronwall’s inequality for time scales 2 and ii that
axt2 ≤ V t, xt ≤ V t0 , xt0 e−/b t, t0 ≤ bxt0 2 e−/b t, t0 .
3.4
Hence, we get
3.5
xt ≤ γxt0 e−/b t, t0 1/2 ,
where γ b/a for all t ≥ t0 . 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, xt ∈
C1rd T × Rn , R
, a bounded positive differentiable function h : T → R and a, b ∈ R
such that
i
V Δ t, xt ≤ γhΔ t/ht xt2 ,
ii axt2 ≤ V t, xt ≤ bxt2 ,
then the zero solution of system 2.8 is h-stable.
γ
⎧
⎨a,
hΔ t ≥ 0;
⎩b,
hΔ t < 0,
3.6
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Abstract and Applied Analysis
Proof. Let t0 ∈ T, xt0 x0 and xt, t0 , x0 xt be any solution of system 2.8. By i, we
have
hΔ t
xt2
ht
⎧
γ hΔ t
⎪
⎪
⎪
⎨ a ht V t, xt,
≤
⎪
γ hΔ t
⎪
⎪
⎩
V t, xt,
b ht
V Δ t, xt ≤ γ
≤
hΔ t ≥ 0;
3.7
Δ
h t < 0,
hΔ t
V t, xt.
ht
From Gronwall’s inequality for time scales 2, ii and Lemma 2.15 4, we obtain
axt2 ≤ V t, xt ≤ V t0 , xt0 ehΔ t/ht t, t0 ≤ bxt0 2 ehΔ t/ht t, t0 ,
≤ bxt0 2
3.8
ht
.
ht0 Thus,
xt ≤ γxt0 HtHt0 −1 ,
where γ t ≥ t0 ,
3.9
b/a and Ht ht. Therefore, zero solution of 2.8 is h-stable.
3.2. Stability Conditions
We introduce the following notation for later use:
Zt : P Δ t AT tP t P tAt μtP Δ tAt μtAT tP Δ t 1 P tP t
μtAT tP tAt μ2 tAT tP Δ tAt 2 P Δ tP Δ t 2−1 γ 2 μt2 I
3 AT tP tP tAt 3−1 γ 2 μt2 I 4 AT tP Δ tP Δ tAt 4−1 γ 2 μt4 I
3.10
1−1 γ 2 I η2 γ 2 μtI ρ2 γ 2 μt2 I.
Theorem 3.5. The system 2.11 is uniformly stable if there exist a positive definite symmetric matrix
function P t ∈ C1rd T, Rn×n and η, ρ ∈ R
such that
i ηI ≤ P t ≤ ρI,
ii AT tP t I μtAT tP Δ t P tAt μtP Δ tAt ≤ 0.
Abstract and Applied Analysis
9
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, xt xT tP txt. In this case, we obtain
V Δ t xT t AT tP t I μtAT t P Δ t P tAt μtP Δ tAt xt. 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, Rn×n and η1 , η2 , γ, 1 , 2 , 3 , 4 ∈ R
, ρ1 , ρ2 ∈ R such that
i η1 I ≤ P t ≤ η2 I,
ii ρ1 I ≤ P Δ t ≤ ρ2 I,
iii Zt ≤ 0.
Proof. We consider the following Lyapunov function for system 2.8.
V t, xt xT tP txt.
3.12
η1 xt2 ≤ V t, xt xT tP txt ≤ η2 xt2 .
3.13
By i, it is easy to see that
The delta derivative of V along the trajectories of system 2.8 is given by
Δ
V Δ t xT tP t xt xT σtP σtxΔ t
xT tΔ P txt xT σtP Δ txt xT σtP σtxΔ t
xT tAT t f T t, x P txt xT t μt xT tAT t f T t, x
× P Δ txt P t Atxt ft, x μtP Δ t Atxt ft, x
xT tP Δ txt xT tAT tP txt xT tP tAtxt
μtxT tP Δ tAtxt
μtxT tAT tP Δ txt μtxT tAT tP tAtxt
μt2 xT tAT tP Δ tAtxt
μtf T t, xP tft, x μt2 f T t, xP Δ tft, x f T t, xP txt
μtf T t, xP Δ txt μtf T t, xP tAtxt
μt2 f T t, xP Δ tAtxt
xT tP tft, x μtxT tP Δ tft, x μtxT tAT tP tft, x
μt2 xT tAT tP Δ tft, x.
3.14
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Abstract and Applied Analysis
By i, ii, and Lemma 2.20, we have the following estimate:
xT tP txt ≤ η2 xT txt,
xT tP Δ txt ≤ ρ2 xT txt,
2xT tP tft, x ≤ 1 xT tP tP txt 1−1 f T t, xft, x,
2μtxT tP Δ tft, x ≤ 2 xT tP Δ tP Δ txt 2−1 μt2 f T t, xft, x,
2μtxT tAT tP tft, x ≤ 3 xT tAT tP tP tAtxt 3−1 μt2 f T t, xft, x,
2μt2 xT tAT tP Δ tft, x ≤ 4 xT tAT tP Δ tP Δ tAtxt 4−1 μt4 f T t, xft, x.
3.15
From the above inequalities and ft, x ≤ γxt, we obtain
V Δ t ≤ xT tP Δ txt xT tAT tP txt xT tP tAtxt
μtxT tP Δ tAtxt
μtxT tAT tP Δ txt μtxT tAT tP tAtxt
μt2 xT tAT tP Δ tAtxt
1 xT tP tP txt 1−1 γ 2 xt2 2 xT tP Δ tP Δ txt
3.16
2−1 γ 2 μt2 xt2
3 xT tAT tP tP tAtxt 3−1 γ 2 μt2 xt2 4−1 γ 2 μt4 xt2
4 xT tAT tP Δ tP Δ tAtxt η2 γ 2 μtxt2 ρ2 γ 2 μt2 xt2
xT tZtxt.
By iii, we conclude that V Δ 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
At −at
−1
1
−at
,
xΔ t Atxt ft, xt,
−0.125 sintx2 t
,
ft, xt 0.125 costx1 t
3.17
ft, xt ≤ 0.125xt,
3.18
where at −e8 t, 0
1 and ft, xt are rd-continuous in the first argument with ft, 0 0
for all t ∈ T. Let γ 1/8, 1 1, 2 1/16, 3 1/2, 4 1/16, η1 1/8, η2 1/4, ρ1 −1,
Abstract and Applied Analysis
11
and ρ2 0. By assuming that 0 ≤ μt ≤ 0.25 for all t ∈ T, we can find solution
P t satisfying
1/8e8 t,0
1/8
0
conditions i–iii of Theorem 3.7 as P t 0
1/8e8 t,0
1/8 . Observe that,
Δ
P t −e8 t, 0
0
0
−e8 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, Rn×n and η1 , η2 , γ, 1 , 2 , 3 , 4 , 5 ∈ R
, ρ1 , ρ2 ∈ R such
that
i η1 I ≤ P t ≤ η2 I,
ii ρ1 I ≤ P Δ t ≤ ρ2 I,
iii Zt ≤ −5 I.
Proof. Consider a Lyapunov function for system 2.8 of the form
V t, xt xT tP txt.
3.20
η1 xt2 ≤ V t, xt xT tP txt ≤ η2 xt2 .
3.21
It is easy to see that i yields
The delta derivative of V along the trajectories of system 2.8 is given by
Δ
V Δ t xT tP t xt xT σtP σtxΔ t
xT tAT t f T t, x P txt xT t μt xT tAT t f T t, x
3.22
× P Δ txt P t Atxt ft, x μtP Δ t Atxt ft, x .
From Theorem 3.7, we obtain
V Δ t ≤ xT tP Δ txt xT tAT tP txt xT tP tAtxt μtxT tP Δ tAtxt
μtxT tAtP Δ txt μtxT tAtP tAtxt μt2 xT tAtP Δ tAtxt
1 xT tP tP txt 1−1 γ 2 xt2 2 xT tP Δ tP Δ txt 2−1 γ 2 μt2 xt2
3 xT tAT tP tP tAtxt 3−1 γ 2 μt2 xt2 4−1 γ 2 μt4 xt2
4 xT tAT tP Δ tP Δ tAtxt η2 γ 2 μtxt2 ρ2 γ 2 μt2 xt2 .
3.23
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Abstract and Applied Analysis
By iii, we conclude that V Δ t ≤ −5 xt2 . 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 Atxt ft, xt,
3.24
where
At −at
−1
1
−at
−0.125 costx2 t
ft, xt .
0.125 sintx1 t
,
3.25
at e8 t, 0 1 and ft, xt are rd-continuous in the first argument with ft, 0 0 for all
t ∈ T. Then, 1 ≤ at ≤ 2 and ft, xt ≤ 0.125xt for all t ∈ T. Let γ 1/8, 1 1, 2 1/16, 3 1/2, 4 5 1/16, η1 1/8, η2 1/4, ρ1 −1, and ρ2 0. By assuming that
0 ≤ μt ≤ 0.25, for all t ∈ T, we can find a solution P t satisfying i–iii of Theorem 3.9 as
⎡1
⎢
P t ⎣ 8
e8 t, 0 0
1
8
⎤
0
⎥
.
1
1⎦
e8 t, 0 8
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, Rn×n , ψt ∈ C1rd T, R
, and η1 , η2 , γ, 1 , 2 , 3 , 4 ∈ R
, ρ1 , ρ2 ∈ R
such that
i η1 I ≤ P t ≤ η2 I,
ii ρ1 I ≤ P Δ t ≤ ρ2 I,
iii Zt ≤ 0,
iv ψ Δ t ≤ 0.
Proof. We consider the following Lyapunov function for system 2.8
V t, xt xT tP txt.
3.27
η1 xt2 ≤ V t, xt xT tP txt ≤ η2 xt2 .
3.28
By i, it is easy to see that
By the same argument as in the proof of Theorem 3.7, we obtain V Δ t ≤ 0. By iv and
Theorem 3.1, the zero solution of 2.8 is ψ-uniformly stable.
Abstract and Applied Analysis
13
Example 3.12. We consider the linear time-varying dynamic system of the form
Δ
x t −at
−1
1
−at
xt ft, xt,
3.29
at | sint| 1 and ft, xt are rd-continuous in the first argument with ft, 0 0 for all
t ∈ T. We let ψt −t and
ft, xt 0.125 sintx1 t
−0.125 costx2 t
.
3.30
Then ψ Δ t −1 ≤ 0 and ft, xt ≤ 0.125xt. Let γ 1/8, 1 1, 2 1/16, 3 1/2, 4 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
⎡1
⎢
P t ⎣ 8
e8 t, 0 1
8
0
⎤
0
⎥
1
1⎦
e8 t, 0 8
8
3.31
ψ Δ t −1 ≤ 0.
Therefore, by Theorem 3.11, the system 3.29 is ψ-uniformly stable.
Theorem 3.13. The system 2.8 is h-stable if there exist a positive definite symmetric matrix
function P t ∈ C1rd T, Rn×n , a bounded positive differentiable function h : T → R, and η1 , η2 ,
γ, 1 , 2 , 3 , 4 ∈ R
, ρ1 , ρ2 ∈ R satisfying
i η1 I ≤ P t ≤ η2 I,
ii ρ1 I ≤ P Δ t ≤ ρ2 I,
iii
hΔ t
I,
Zt ≤ γ1
ht
γ1 ⎧
⎨η1 , hΔ t ≥ 0;
⎩η , hΔ t < 0.
2
3.32
Proof. Let t0 ∈ T, xt0 x0 and xt, t0 , x0 xt be any solution of system 2.8. We consider
a Lyapunov function for system 2.8 of the form
V t, xt xT tP txt.
3.33
η1 xt2 ≤ V t, xt xT tP txt ≤ η2 xt2 .
3.34
By i, we get
14
Abstract and Applied Analysis
The delta derivative of V along the trajectories of system 2.8 is given by
Δ
V Δ t xT tP t xt xT σtP σtxΔ t
xT tAT t f T t, x P txt xT t μt xT tAT t f T t, x
3.35
× P Δ txt P t Atxt ft, x μtP Δ t Atxt ft, x .
By using i, ii, iii, and Lemma 2.20, we obtain
Δ
hΔ t
V Δ t, xt xT tP txt ≤ γ1
xt2
ht
⎧
γ1 hΔ t
⎪
⎪
V t, xt, hΔ t ≥ 0,
⎨
η1 ht
≤
γ hΔ t
⎪
⎪
⎩ 1
V t, xt, hΔ t < 0,
η2 ht
≤
3.36
hΔ t T
x tP txt .
ht
From Gronwall’s inequality for time scales 3, i and Lemma 2.15 in 2, we obtain
η1 xt2 ≤ xT tP txt ≤ xT t0 P t0 xt0 ehΔ t/ht t, t0 ,
≤ η2 xt0 2 ehΔ t/ht t, t0 ≤ η2 xt0 2
ht
.
ht0 3.37
Hence, we get
xt ≤ ωxt0 HtHt0 −1 ,
where ω η2 /η1 and Ht t ≥ t0 ,
3.38
ht. Therefore, the zero solution of 2.8 is h-stable.
Example 3.14. We consider the linear time-varying dynamic system of the form
Δ
x t −at
−1
1
−at
xt ft, xt,
3.39
where at e8 t, 0 1 and ft, xt are rd-continuous in the first argument with ft, 0 0
for all t ∈ T. Let ht 5 and
ft, xt 0.125 costx2 t
−0.125 sintx1 t
.
3.40
Abstract and Applied Analysis
15
Then hΔ t 0 and ft, xt ≤ 0.125xt. Let γ 1/8, 1 1, 2 1/16, 3 1/2, 4 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 as
⎡1
⎢
P t ⎣ 8
e8 t, 0 0
1
8
⎤
0
⎥
.
1
1⎦
e8 t, 0 8
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, ψuniform stability and h-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.
References
1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
2 M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.
3 M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass,
USA, 2003.
4 S. K. Choi, N. J. Koo, and D. M. Im, “h—stability for linear dynamic equations on time scales,” Journal
of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 707–720, 2006.
5 J. J. DaCunha, “Stability for time varying linear dynamic systems on time scales,” Journal of
Computational and Applied Mathematics, vol. 176, no. 2, pp. 381–410, 2005.
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International Mathematical Forum, vol. 2, no. 17–20, pp. 963–972, 2007.
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