# An LMI Approach to Stability for Linear Time-Varying System with Nonlinear Perturbation on Time Scales

Authors: Kanit Mukdasai; Piyapong Niamsup
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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 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, fΔ(t)=f′(t). On the other hand, if 𝕋=ℤ, 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 [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; λ(A) denotes the set of all eigenvalues of A; λmax(A)=max{Reλ:λ∈λ(A)}; λmin(A)=min{Reλ:λ∈λ(A)}.Definition 2.1.
A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ.Definition 2.2.
The mappingσ,ρ:𝕋→𝕋 defined by σ(t)=inf{s∈𝕋:s>t}, and ρ(t)=sup{s∈𝕋:s<t} are called the jump operators.Definition 2.3.
A nonmaximal elementt∈𝕋 is said to be right-scattered (rs) if σ(t)>t and right-dense (rd) if σ(t)=t. A nonminimal element t∈𝕋 is called left-scattered (ls) if ρ(t)<t and left-dense (ld) if ρ(t)=t.Definition 2.4.
The mappingμ:𝕋→ℝ+ defined by μ(t)=σ(t)-t is called the graininess function.Definition 2.5.
(Delta derivative) assumef:𝕋→ℝ is a function and let t∈𝕋. 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=(t-δ,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 s∈U.
The functionfΔ(t) is the delta derivative of f at t.
In the case that𝕋=ℝ, we have fΔ(t)=f′(t). In the case that 𝕋=ℤ, 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 thatf:𝕋→ℝn and let  t∈𝕋.   (i)
Iff is differentiable at t, then f is continuous at t.(ii)
Iff is continuous at t and t is right scattered, then f is differentiable at t with(2.1)fΔ(t)=f(σ(t))-f(t)σ(t)-t.(iii)
Iff is differentiable at t and t is right dense, then(2.2)fΔ(t)=lim⁡s→tf(t)-f(s)t-s.(iv)
Iff is differentiable at t, then(2.3)f(σ(t))=f(t)+μ(t)fΔ(t).Theorem 2.7 (see [2]).
Assume thatf,g:𝕋→ℝn and let  t∈𝕋.   (i)
The sumf,g:𝕋→ℝn are differentiable at t with(2.4)(f+g)Δ(t)=(f)Δ(t)+(g)Δ(t).(ii)
For any constantα,αf:𝕋→ℝn is differentiable at t with(2.5)(αf)Δ(t)=αfΔ(t).(iii)
The productfg:𝕋→ℝn is differentiable at t with(2.6)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).Definition 2.8.
The functionf:𝕋→ℝn is said to be rd-continuous (denoted by f∈𝒞rd(𝕋,ℝn)) if the following conditions hold.  (i)
f is continuous at every right-dense point t∈𝕋.(ii)
lims→t-f(s) exists and is finite at every ld-point t∈𝕋.Definition 2.9.
Letf∈𝒞rd(𝕋,ℝn). Then g:𝕋→ℝn is called the antiderivative of f on 𝕋 if it is differentiable on 𝕋 and satisfies gΔ(t)=f(t) for t∈𝕋. In this case, we define
(2.7)∫atf(s)Δs=g(t)-g(a),a≤t∈T.
Consider the linear time-varying system with nonlinear perturbation on time scales (𝕋) of the form
(2.8)xΔ(t)=A(t)x(t)+f(t,x(t)),t∈T,
where x(t)∈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 f(t,0)=0. The uncertain perturbation is known to satisfy a bound of the form
(2.9)‖f(t,x(t))‖≤γ‖x(t)‖,
or equivalently, the perturbation is conically bounded. The solution of (2.8) through (t0,x(t0)) satisfies the variation of constants formula
(2.10)x(t)=ΦA(t,t0)x(t0)+∫t0tΦA(t,σ(s))f(s,x(s))Δs,t≥t0.
When f(t,x(t))=0, (2.8) becomes the linear time-varying system
(2.11)xΔ(t)=A(t)x(t),x(t0)=x0,t0∈T.
For the case when f(t,x(t))=B(t)x(t),B(t)∈ℝn×n, (2.8) becomes the linear time-varying system
(2.12)xΔ(t)=[A(t)+B(t)]x(t),x(t0)=x0,t0∈T.
The norm of n×n matrix A is defined as
(2.13)‖A‖=max⁡‖x‖=1‖Ax‖.
The Euclidean norm of n×1 vector x(t) is defined by
(2.14)‖x(t)‖=xT(t)x(t).Definition 2.10.
A functionϕ:[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.
Assumeg: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-g(s) exists, and is finite, at every left-dense point t∈𝕋. Now define the so-called set of regressive functions, ℛ, by
(2.15)R={p:T→R∣p∈Crd⁡(T;R),1+p(t)μ(t)≠0,t∈T},
and define the set of positively regressive functions by
(2.16)R+={p∈R∣1+p(t)μ(t)>0,t∈T}.Definition 2.12.
The zero solution of system (2.8) is called uniformly stable if there exists a finite constant γ>0 such that
(2.17)‖x(t,x0,t0)‖≤γ‖x0‖,
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
(2.18)‖x(t,x0,t0)‖≤γ‖x0‖e-λ(t,t0),
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 x(t0), the corresponding solution satisfies
(2.19)‖ψ(t)x(t,x0,t0)‖≤γ‖ψ(t0)x0‖,
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
(2.20)‖x(t,x0,t0)‖≤c‖x0‖h(t)h(t0)-1,t≥t0,
if ∥x0∥<δ(hereh(t)-1=1/h(t)). If h is bounded, then (2.8) is said to be h-stable.Definition 2.16.
A continuous functionP:𝕋→ℝ with P(0)=0 is called positive definite (negative definite) on 𝕋 if there exists a function ϕ∈𝒦 such that ϕ(t)≤P(t) (ϕ(t)≤-P(t)) for all t∈𝕋.Definition 2.17.
A continuous functionP: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 functionP: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 ϕ(∥x∥)≤P(t,x) (ϕ(∥x∥)≤-P(t,x)) for all t∈𝕋 and x∈ℝn.Definition 2.19.
A continuous functionP: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 thatS∈Mn×n is a symmetric positive definite matrix. Then for every Q∈Mn×n, we obtain
(2.21)2xTQy-yTSy≤xTQS-1QTx,∀x,y∈Rn.


## 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 functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+), and  a,b∈ℝ+ such that   (i)
VΔ(t,x(t))≤0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is ψ-uniformly stable if there exists ψ(t)∈𝒞rd1(𝕋,ℝ+) satisfying ψΔ(t)≤0.Proof.
Fort0∈𝕋, we let x(t0)=x0. Then, by (i), we have
(3.1)∫t0tVΔ(s,x(s))Δs=V(t,x(t))-V(t0,x(t0))≤0,∫t0tψΔ(s)Δs=ψ(t)-ψ(t0)≤0.
We obtain V(t,x(t))≤V(t0,x(t0)) and ψ(t)≤ψ(t0) for all t∈𝕋,t≥t0. By (ii), we get the estimation as follows:
(3.2)a‖ψ(t)‖2‖x(t)‖2≤‖ψ(t)‖2V(t,x(t))≤‖ψ(t0)‖2V(t0,x(t0))≤b‖ψ(t0)‖2‖x(t0)‖2.
We conclude that ∥ψ(t)x(t)∥≤γ∥ψ(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 functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+) and  a,b∈ℝ+ such that   (i)
VΔ(t,x(t))≤0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is uniformly stable.Theorem 3.3.
If there exist a continuously differentiable positive definite functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+) and  a,b,ϵ∈ℝ+ with  -ϵ/b∈ℛ+ satisfying   (i)
VΔ(t,x(t))≤-ϵ∥x(t)∥2,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2, then the zero solution of system (2.8) is uniformly exponentially stable.Proof.
Fort0∈𝕋, we let x(t0)=x0. We obtain, by (i) and (ii), that for all t≥t0,
(3.3)VΔ(t,x(t))≤-ϵ‖x(t)‖2≤-ϵbV(t,x(t)).
Since -ϵ/b∈ℛ+, it follows from Gronwall’s inequality for time scales [2] and (ii) that
(3.4)a‖x(t)‖2≤V(t,x(t))≤V(t0,x(t0))e-ϵ/b(t,t0)≤b‖x(t0)‖2e-ϵ/b(t,t0).
Hence, we get
(3.5)‖x(t)‖≤γ‖x(t0)‖[e-ϵ/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 functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+), a bounded positive differentiable function  h:𝕋→ℝ and  a,b∈ℝ+ such that   (i)
(3.6)VΔ(t,x(t))≤γhΔ(t)/h(t)‖x(t)‖2,γ={a,hΔ(t)≥0;b,hΔ(t)<0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is h-stable.Proof.
Lett0∈𝕋, x(t0)=x0 and x(t,t0,x0)=x(t) be any solution of system (2.8). By (i), we have
(3.7)VΔ(t,x(t))≤γhΔ(t)h(t)‖x(t)‖2≤{γahΔ(t)h(t)V(t,x(t)),hΔ(t)≥0;γbhΔ(t)h(t)V(t,x(t)),hΔ(t)<0,≤hΔ(t)h(t)V(t,x(t)).
From Gronwall’s inequality for time scales [2], (ii) and Lemma 2.15 [4], we obtain
(3.8)a‖x(t)‖2≤V(t,x(t))≤V(t0,x(t0))ehΔ(t)/h(t)(t,t0)≤b‖x(t0)‖2ehΔ(t)/h(t)(t,t0),≤b‖x(t0)‖2h(t)h(t0).
Thus,
(3.9)‖x(t)‖≤γ‖x(t0)‖H(t)H(t0)-1,t≥t0,
where γ=b/a and H(t)=h(t). Therefore, zero solution of (2.8) is h-stable.


### 3.2. Stability Conditions

We introduce the following notation for later use:(3.10)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)+ϵ2-1γ2μ(t)2I+ϵ3AT(t)P(t)P(t)A(t)+ϵ3-1γ2μ(t)2I+ϵ4AT(t)PΔ(t)PΔ(t)A(t)+ϵ4-1γ2μ(t)4I+ϵ1-1γ2I+η2γ2μ(t)I+ρ2γ2μ(t)2I.Theorem 3.5.
The system (2.11) is uniformly stable if there exist a positive definite symmetric matrix function  P(t)∈𝒞rd1(𝕋,ℝn×n) and  η,ρ∈ℝ+ such that   (i)
ηI≤P(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 Theorem3.5 (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
(3.11)VΔ(t)=xT(t)[AT(t)P(t)+(I+μ(t)AT(t))(PΔ(t)+P(t)A(t)+μ(t)PΔ(t)A(t))]x(t).Theorem 3.7.
The system (2.8) is uniformly stable if there exist a positive definite symmetric matrix function  P(t)∈𝒞rd1(𝕋,ℝn×n) and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ such that   (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤0.Proof.
We consider the following Lyapunov function for system (2.8). 									(3.12)V(t,x(t))=xT(t)P(t)x(t).
By (i), it is easy to see that
(3.13)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.14)VΔ(t)=[xT(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(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(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).
By (i), (ii), and Lemma 2.20, we have the following estimate:
(3.15)xT(t)P(t)x(t)≤η2xT(t)x(t),xT(t)PΔ(t)x(t)≤ρ2xT(t)x(t),2xT(t)P(t)f(t,x)≤ϵ1xT(t)P(t)P(t)x(t)+ϵ1-1fT(t,x)f(t,x),2μ(t)xT(t)PΔ(t)f(t,x)≤ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1μ(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)+ϵ3-1μ(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)+ϵ4-1μ(t)4fT(t,x)f(t,x).
From the above inequalities and ∥f(t,x)∥≤γ∥x(t)∥, we obtain
(3.16)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)+ϵ1-1γ2‖x(t)‖2+ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1γ2μ(t)2‖x(t)‖2+ϵ3xT(t)AT(t)P(t)P(t)A(t)x(t)+ϵ3-1γ2μ(t)2‖x(t)‖2+ϵ4-1γ2μ(t)4‖x(t)‖2+ϵ4xT(t)AT(t)PΔ(t)PΔ(t)A(t)x(t)+η2γ2μ(t)‖x(t)‖2+ρ2γ2μ(t)2‖x(t)‖2=xT(t)Z(t)x(t).
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(3.17)xΔ(t)=A(t)x(t)+f(t,x(t)),(3.18)A(t)=[-a(t)-11-a(t)],f(t,x(t))=[-0.125sin(t)[x2(t)]0.125cos⁡(t)[x1(t)]],‖f(t,x(t))‖≤0.125‖x(t)‖,
where a(t)=-e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈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. By assuming that 0≤μ(t)≤0.25 for all t∈𝕋, we can find solution P(t) satisfying conditions (i)–(iii) of Theorem 3.7 as P(t)=[(1/8)e⊖8(t,0)+(1/8)00(1/8)e⊖8(t,0)+(1/8)]. Observe that,
(3.19)PΔ(t)=[-e⊖8(t,0)00-e⊖8(t,0)].
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)∈𝒞rd1(𝕋,ℝn×n) and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4,ϵ5∈ℝ+,  ρ1,ρ2∈ℝ such that(i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤-ϵ5I.Proof.
Consider a Lyapunov function for system (2.8) of the form
(3.20)V(t,x(t))=xT(t)P(t)x(t).
It is easy to see that (i) yields
(3.21)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.22)VΔ(t)=[xT(t)P(t)]Δx(t)+xT(σ(t))P(σ(t))xΔ(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(t)x(t)+P(t)(A(t)x(t)+f(t,x))+μ(t)PΔ(t)(A(t)x(t)+f(t,x))].
From Theorem 3.7, we obtain
(3.23)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)+ϵ1-1γ2‖x(t)‖2+ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1γ2μ(t)2‖x(t)‖2+ϵ3xT(t)AT(t)P(t)P(t)A(t)x(t)+ϵ3-1γ2μ(t)2‖x(t)‖2+ϵ4-1γ2μ(t)4‖x(t)‖2+ϵ4xT(t)AT(t)PΔ(t)PΔ(t)A(t)x(t)+η2γ2μ(t)‖x(t)‖2+ρ2γ2μ(t)2‖x(t)‖2.
By (iii), we conclude that VΔ(t)≤-ϵ5∥x(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(3.24)xΔ(t)=A(t)x(t)+f(t,x(t)),
where
(3.25)A(t)=[-a(t)-11-a(t)],f(t,x(t))=[-0.125cos⁡(t)[x2(t)]0.125sin(t)[x1(t)]].a(t)=e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. Then, 1≤a(t)≤2 and ∥f(t,x(t))∥≤0.125∥x(t)∥ for all t∈𝕋. 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 0≤μ(t)≤0.25, for all t∈𝕋, we can find a solution P(t) satisfying (i)–(iii) of Theorem 3.9 as
(3.26)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18].
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)∈𝒞rd1(𝕋,ℝn×n),ψ(t)∈𝒞rd1(𝕋,ℝ+), and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ such that   (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤0,(iv)
ψΔ(t)≤0.Proof.
We consider the following Lyapunov function for system (2.8)
(3.27)V(t,x(t))=xT(t)P(t)x(t).
By (i), it is easy to see that
(3.28)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
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(3.29)xΔ(t)=[-a(t)-11-a(t)]x(t)+f(t,x(t)),a(t)=|sin(t)|+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. We let ψ(t)=-t and
(3.30)f(t,x(t))=[0.125sin(t)[x1(t)]-0.125cos⁡(t)[x2(t)]].
Then ψΔ(t)=-1≤0 and ∥f(t,x(t))∥≤0.125∥x(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
(3.31)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18]ψΔ(t)=-1≤0.
Therefore, by Theorem3.11, 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)∈𝒞rd1(𝕋,ℝn×n), a bounded positive differentiable function h:𝕋→ℝ, and η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ satisfying  (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
(3.32)Z(t)≤γ1hΔ(t)h(t)I,γ1={η1,hΔ(t)≥0;η2,hΔ(t)<0.Proof.
Lett0∈𝕋, 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
(3.33)V(t,x(t))=xT(t)P(t)x(t).
By (i), we get
(3.34)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.35)VΔ(t)=[xT(t)P(t)]Δx(t)+xT(σ(t))P(σ(t))xΔ(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(t)x(t)+P(t)(A(t)x(t)+f(t,x))+μ(t)PΔ(t)(A(t)x(t)+f(t,x))].
By using (i), (ii), (iii), and Lemma 2.20, we obtain
(3.36)VΔ(t,x(t))=[xT(t)P(t)x(t)]Δ≤γ1hΔ(t)h(t)‖x(t)‖2≤{γ1η1hΔ(t)h(t)V(t,x(t)),hΔ(t)≥0,γ1η2hΔ(t)h(t)V(t,x(t)),hΔ(t)<0,≤hΔ(t)h(t)[xT(t)P(t)x(t)].
From Gronwall’s inequality for time scales [3], (i) and Lemma 2.15 in [2], we obtain
(3.37)η1‖x(t)‖2≤xT(t)P(t)x(t)≤[xT(t0)P(t0)x(t0)]ehΔ(t)/h(t)(t,t0),≤η2‖x(t0)‖2ehΔ(t)/h(t)(t,t0)≤η2‖x(t0)‖2h(t)h(t0).
Hence, we get
(3.38)‖x(t)‖≤ω‖x(t0)‖H(t)H(t0)-1,t≥t0,
where ω=η2/η1 and H(t)=h(t). Therefore, the zero solution of (2.8) is h-stable.Example 3.14.
We consider the linear time-varying dynamic system of the form(3.39)xΔ(t)=[-a(t)-11-a(t)]x(t)+f(t,x(t)),
where a(t)=e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. Let h(t)=5 and
(3.40)f(t,x(t))=[0.125cos⁡(t)[x2(t)]-0.125sin(t)[x1(t)]].
ThenhΔ(t)=0 and ∥f(t,x(t))∥≤0.125∥x(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 as(3.41)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18].
Therefore, by Theorem 3.13, the system (3.39) is 5-stable.


## 3.1. Lyapunov Stability Theory

Theorem 3.1.
If there exist a continuously differentiable positive definite functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+), and  a,b∈ℝ+ such that   (i)
VΔ(t,x(t))≤0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is ψ-uniformly stable if there exists ψ(t)∈𝒞rd1(𝕋,ℝ+) satisfying ψΔ(t)≤0.Proof.
Fort0∈𝕋, we let x(t0)=x0. Then, by (i), we have
(3.1)∫t0tVΔ(s,x(s))Δs=V(t,x(t))-V(t0,x(t0))≤0,∫t0tψΔ(s)Δs=ψ(t)-ψ(t0)≤0.
We obtain V(t,x(t))≤V(t0,x(t0)) and ψ(t)≤ψ(t0) for all t∈𝕋,t≥t0. By (ii), we get the estimation as follows:
(3.2)a‖ψ(t)‖2‖x(t)‖2≤‖ψ(t)‖2V(t,x(t))≤‖ψ(t0)‖2V(t0,x(t0))≤b‖ψ(t0)‖2‖x(t0)‖2.
We conclude that ∥ψ(t)x(t)∥≤γ∥ψ(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 functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+) and  a,b∈ℝ+ such that   (i)
VΔ(t,x(t))≤0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is uniformly stable.Theorem 3.3.
If there exist a continuously differentiable positive definite functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+) and  a,b,ϵ∈ℝ+ with  -ϵ/b∈ℛ+ satisfying   (i)
VΔ(t,x(t))≤-ϵ∥x(t)∥2,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2, then the zero solution of system (2.8) is uniformly exponentially stable.Proof.
Fort0∈𝕋, we let x(t0)=x0. We obtain, by (i) and (ii), that for all t≥t0,
(3.3)VΔ(t,x(t))≤-ϵ‖x(t)‖2≤-ϵbV(t,x(t)).
Since -ϵ/b∈ℛ+, it follows from Gronwall’s inequality for time scales [2] and (ii) that
(3.4)a‖x(t)‖2≤V(t,x(t))≤V(t0,x(t0))e-ϵ/b(t,t0)≤b‖x(t0)‖2e-ϵ/b(t,t0).
Hence, we get
(3.5)‖x(t)‖≤γ‖x(t0)‖[e-ϵ/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 functionV(t,x(t))∈𝒞rd1(𝕋×ℝn,ℝ+), a bounded positive differentiable function  h:𝕋→ℝ and  a,b∈ℝ+ such that   (i)
(3.6)VΔ(t,x(t))≤γhΔ(t)/h(t)‖x(t)‖2,γ={a,hΔ(t)≥0;b,hΔ(t)<0,(ii)
a∥x(t)∥2≤V(t,x(t))≤b∥x(t)∥2,then the zero solution of system (2.8) is h-stable.Proof.
Lett0∈𝕋, x(t0)=x0 and x(t,t0,x0)=x(t) be any solution of system (2.8). By (i), we have
(3.7)VΔ(t,x(t))≤γhΔ(t)h(t)‖x(t)‖2≤{γahΔ(t)h(t)V(t,x(t)),hΔ(t)≥0;γbhΔ(t)h(t)V(t,x(t)),hΔ(t)<0,≤hΔ(t)h(t)V(t,x(t)).
From Gronwall’s inequality for time scales [2], (ii) and Lemma 2.15 [4], we obtain
(3.8)a‖x(t)‖2≤V(t,x(t))≤V(t0,x(t0))ehΔ(t)/h(t)(t,t0)≤b‖x(t0)‖2ehΔ(t)/h(t)(t,t0),≤b‖x(t0)‖2h(t)h(t0).
Thus,
(3.9)‖x(t)‖≤γ‖x(t0)‖H(t)H(t0)-1,t≥t0,
where γ=b/a and H(t)=h(t). Therefore, zero solution of (2.8) is h-stable.


## 3.2. Stability Conditions

We introduce the following notation for later use:(3.10)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)+ϵ2-1γ2μ(t)2I+ϵ3AT(t)P(t)P(t)A(t)+ϵ3-1γ2μ(t)2I+ϵ4AT(t)PΔ(t)PΔ(t)A(t)+ϵ4-1γ2μ(t)4I+ϵ1-1γ2I+η2γ2μ(t)I+ρ2γ2μ(t)2I.Theorem 3.5.
The system (2.11) is uniformly stable if there exist a positive definite symmetric matrix function  P(t)∈𝒞rd1(𝕋,ℝn×n) and  η,ρ∈ℝ+ such that   (i)
ηI≤P(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 Theorem3.5 (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
(3.11)VΔ(t)=xT(t)[AT(t)P(t)+(I+μ(t)AT(t))(PΔ(t)+P(t)A(t)+μ(t)PΔ(t)A(t))]x(t).Theorem 3.7.
The system (2.8) is uniformly stable if there exist a positive definite symmetric matrix function  P(t)∈𝒞rd1(𝕋,ℝn×n) and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ such that   (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤0.Proof.
We consider the following Lyapunov function for system (2.8). 									(3.12)V(t,x(t))=xT(t)P(t)x(t).
By (i), it is easy to see that
(3.13)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.14)VΔ(t)=[xT(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(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(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).
By (i), (ii), and Lemma 2.20, we have the following estimate:
(3.15)xT(t)P(t)x(t)≤η2xT(t)x(t),xT(t)PΔ(t)x(t)≤ρ2xT(t)x(t),2xT(t)P(t)f(t,x)≤ϵ1xT(t)P(t)P(t)x(t)+ϵ1-1fT(t,x)f(t,x),2μ(t)xT(t)PΔ(t)f(t,x)≤ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1μ(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)+ϵ3-1μ(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)+ϵ4-1μ(t)4fT(t,x)f(t,x).
From the above inequalities and ∥f(t,x)∥≤γ∥x(t)∥, we obtain
(3.16)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)+ϵ1-1γ2‖x(t)‖2+ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1γ2μ(t)2‖x(t)‖2+ϵ3xT(t)AT(t)P(t)P(t)A(t)x(t)+ϵ3-1γ2μ(t)2‖x(t)‖2+ϵ4-1γ2μ(t)4‖x(t)‖2+ϵ4xT(t)AT(t)PΔ(t)PΔ(t)A(t)x(t)+η2γ2μ(t)‖x(t)‖2+ρ2γ2μ(t)2‖x(t)‖2=xT(t)Z(t)x(t).
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(3.17)xΔ(t)=A(t)x(t)+f(t,x(t)),(3.18)A(t)=[-a(t)-11-a(t)],f(t,x(t))=[-0.125sin(t)[x2(t)]0.125cos⁡(t)[x1(t)]],‖f(t,x(t))‖≤0.125‖x(t)‖,
where a(t)=-e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈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. By assuming that 0≤μ(t)≤0.25 for all t∈𝕋, we can find solution P(t) satisfying conditions (i)–(iii) of Theorem 3.7 as P(t)=[(1/8)e⊖8(t,0)+(1/8)00(1/8)e⊖8(t,0)+(1/8)]. Observe that,
(3.19)PΔ(t)=[-e⊖8(t,0)00-e⊖8(t,0)].
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)∈𝒞rd1(𝕋,ℝn×n) and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4,ϵ5∈ℝ+,  ρ1,ρ2∈ℝ such that(i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤-ϵ5I.Proof.
Consider a Lyapunov function for system (2.8) of the form
(3.20)V(t,x(t))=xT(t)P(t)x(t).
It is easy to see that (i) yields
(3.21)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.22)VΔ(t)=[xT(t)P(t)]Δx(t)+xT(σ(t))P(σ(t))xΔ(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(t)x(t)+P(t)(A(t)x(t)+f(t,x))+μ(t)PΔ(t)(A(t)x(t)+f(t,x))].
From Theorem 3.7, we obtain
(3.23)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)+ϵ1-1γ2‖x(t)‖2+ϵ2xT(t)PΔ(t)PΔ(t)x(t)+ϵ2-1γ2μ(t)2‖x(t)‖2+ϵ3xT(t)AT(t)P(t)P(t)A(t)x(t)+ϵ3-1γ2μ(t)2‖x(t)‖2+ϵ4-1γ2μ(t)4‖x(t)‖2+ϵ4xT(t)AT(t)PΔ(t)PΔ(t)A(t)x(t)+η2γ2μ(t)‖x(t)‖2+ρ2γ2μ(t)2‖x(t)‖2.
By (iii), we conclude that VΔ(t)≤-ϵ5∥x(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(3.24)xΔ(t)=A(t)x(t)+f(t,x(t)),
where
(3.25)A(t)=[-a(t)-11-a(t)],f(t,x(t))=[-0.125cos⁡(t)[x2(t)]0.125sin(t)[x1(t)]].a(t)=e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. Then, 1≤a(t)≤2 and ∥f(t,x(t))∥≤0.125∥x(t)∥ for all t∈𝕋. 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 0≤μ(t)≤0.25, for all t∈𝕋, we can find a solution P(t) satisfying (i)–(iii) of Theorem 3.9 as
(3.26)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18].
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)∈𝒞rd1(𝕋,ℝn×n),ψ(t)∈𝒞rd1(𝕋,ℝ+), and  η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ such that   (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
Z(t)≤0,(iv)
ψΔ(t)≤0.Proof.
We consider the following Lyapunov function for system (2.8)
(3.27)V(t,x(t))=xT(t)P(t)x(t).
By (i), it is easy to see that
(3.28)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
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(3.29)xΔ(t)=[-a(t)-11-a(t)]x(t)+f(t,x(t)),a(t)=|sin(t)|+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. We let ψ(t)=-t and
(3.30)f(t,x(t))=[0.125sin(t)[x1(t)]-0.125cos⁡(t)[x2(t)]].
Then ψΔ(t)=-1≤0 and ∥f(t,x(t))∥≤0.125∥x(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
(3.31)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18]ψΔ(t)=-1≤0.
Therefore, by Theorem3.11, 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)∈𝒞rd1(𝕋,ℝn×n), a bounded positive differentiable function h:𝕋→ℝ, and η1,η2,γ,ϵ1,ϵ2,ϵ3,ϵ4∈ℝ+,  ρ1,ρ2∈ℝ satisfying  (i)
η1I≤P(t)≤η2I,(ii)
ρ1I≤PΔ(t)≤ρ2I,(iii)
(3.32)Z(t)≤γ1hΔ(t)h(t)I,γ1={η1,hΔ(t)≥0;η2,hΔ(t)<0.Proof.
Lett0∈𝕋, 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
(3.33)V(t,x(t))=xT(t)P(t)x(t).
By (i), we get
(3.34)η1‖x(t)‖2≤V(t,x(t))=xT(t)P(t)x(t)≤η2‖x(t)‖2.
The delta derivative of V along the trajectories of system (2.8) is given by
(3.35)VΔ(t)=[xT(t)P(t)]Δx(t)+xT(σ(t))P(σ(t))xΔ(t)=[xT(t)AT(t)+fT(t,x)]P(t)x(t)+[xT(t)+μ(t)(xT(t)AT(t)+fT(t,x))]×[PΔ(t)x(t)+P(t)(A(t)x(t)+f(t,x))+μ(t)PΔ(t)(A(t)x(t)+f(t,x))].
By using (i), (ii), (iii), and Lemma 2.20, we obtain
(3.36)VΔ(t,x(t))=[xT(t)P(t)x(t)]Δ≤γ1hΔ(t)h(t)‖x(t)‖2≤{γ1η1hΔ(t)h(t)V(t,x(t)),hΔ(t)≥0,γ1η2hΔ(t)h(t)V(t,x(t)),hΔ(t)<0,≤hΔ(t)h(t)[xT(t)P(t)x(t)].
From Gronwall’s inequality for time scales [3], (i) and Lemma 2.15 in [2], we obtain
(3.37)η1‖x(t)‖2≤xT(t)P(t)x(t)≤[xT(t0)P(t0)x(t0)]ehΔ(t)/h(t)(t,t0),≤η2‖x(t0)‖2ehΔ(t)/h(t)(t,t0)≤η2‖x(t0)‖2h(t)h(t0).
Hence, we get
(3.38)‖x(t)‖≤ω‖x(t0)‖H(t)H(t0)-1,t≥t0,
where ω=η2/η1 and H(t)=h(t). Therefore, the zero solution of (2.8) is h-stable.Example 3.14.
We consider the linear time-varying dynamic system of the form(3.39)xΔ(t)=[-a(t)-11-a(t)]x(t)+f(t,x(t)),
where a(t)=e⊖8(t,0)+1 and f(t,x(t)) are rd-continuous in the first argument with f(t,0)=0 for all t∈𝕋. Let h(t)=5 and
(3.40)f(t,x(t))=[0.125cos⁡(t)[x2(t)]-0.125sin(t)[x1(t)]].
ThenhΔ(t)=0 and ∥f(t,x(t))∥≤0.125∥x(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 as(3.41)P(t)=[18e⊖8(t,0)+180018e⊖8(t,0)+18].
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 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.



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*Source: 860506-2011-07-19.xml*