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Volume 2011, Article ID 979705, 27 pagesdoi:10.1155/2011/979705 Research Article Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales 1 Department of Mathematics,

Volume 2011, Article ID 979705, 27 pages

doi:10.1155/2011/979705

Research Article

Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales

1 Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam

2 Department of Mathematics, Tamkang University, 151 Ying Chuang Road, Tamsui, Taipei County

25317, Taiwan

Correspondence should be addressed to N H Du,dunh@vnu.edu.vn

Received 29 September 2010; Accepted 4 February 2011

Academic Editor: Stevo Stevic

Copyrightq 2011 N H Du et al This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

This paper studies the stability of the solution x≡ 0 for a class of quasilinear implicit dynamic

equations on time scales of the form A t xΔ  ft, x We deal with an index concept to study the

solvability and use Lyapunov functions as a tool to approach the stability problem

1 Introduction

The stability theory of quasilinear differential-algebraic equations DAEs for short

both theory and practice This problem can be seen in many real problems, such as in

g.

Together with the theory of DAEs, there has been a great interest in singular difference

This model appears in many practical areas, such as the Leontiev dynamic model ofmultisector economy, the Leslie population growth model, and singular discrete optimalcontrol problems On the other hand, SDEs occur in a natural way of using discretizationtechniques for solving DAEs and partial differential-algebraic equations, and so forth, which

the nonlinear perturbation gn, xn 1, xn is small and does not depend on xn 1.

Further, in recent years, to unify the presentation of continuous and discrete analysis,

a new theory was born and is more and more extensively concerned, that is, the theory of the

The purpose of this paper is to answer the question whether results of stability for

direct Lyapunov method, and the results of this paper can be considered as a generalization

of1.1 and 1.2

basic notions of the analysis on time scales and give the solvability of Cauchy problem forquasilinear implicit dynamic equations

with small perturbation ft, x and for quasilinear implicit dynamic equations of the style

with the assumption of differentiability for ft, x The main results of this paper are

because of the complicated structure of every time scale

2 Nonlinear Implicit Dynamic Equations on Time Scales

2.1 Some Basic Notations of the Theory of the Analysis on Time Scales

A time scale is a nonempty closed subset of the real numbers , and we usually denote it

inherited from the real numbers with the standard topology We define the forward jump

operator and the backward jump operator σ, ρ : → by σt  inf{s ∈ : s > t}

supplemented by inf ∅  sup and ρt  sup{s ∈  : s < t} supplemented by

is said to be right-dense if σt  t, right-scattered if σt > t, left-dense if ρt  t, left-scattered if

k

is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists

X is denoted by Crd , X  A matrix function f from  to m ×m is said to be regressive if

Theorem 2.1 see 9 11 Let t ∈and let A t be a rd-continuous m × m-matrix function and q t

rd-continuous function Then, for any t0∈

k , the initial value problem (IVP)

has a unique solution x · defined on tt0 Further, if A t is regressive, this solution exists on t∈.

We now recall the chain rule for multivariable functions on time scales, this result has

2.2 Linear Equations with Small Nonlinear Perturbation

where A . , B . ∈ Crd

t toobtain an ordinary dynamic equation

xΔ A−1

t B t x A−1

where the solutions of Cauchy problem may exist only on a submanifold or even they do notexist One of the ways to solve this equation is to impose some further assumptions statedunder the form of indices of the equation

t · · ·σ t r > 0 on its main diagonal Since A . ∈ Crd

Under these notations, we have the following Lemma

Lemma 2.2 The following assertions are equivalent

i kerA ρ t ∩ S t  {0};

ii the matrix G t  A t − B t T t Q t is nonsingular;

iii m  kerA ρ t ⊕ S t , for all t∈ .

Proof i⇒ii Let t ∈ and x ∈ m such thatA t − B t T t Q t x  0 ⇔ B t T t Q t x   Ax This equation implies T t Q t x ∈ S t Since ker A ρ t ∩ S t  {0} and T t Q t x ∈ ker A ρ t, it follows that

0, that is, the matrix G t  A t − B t T t Q tis nonsingular

ii⇒iii It is obvious that x  I T t Q t G−1t B t x−T t Q t G−1t B t x We see that T t Q t G−1t B t x∈

ker A ρ t and B t I T t Q t G−1t B t x  B t x − A t − B t T t Q t G−1

t B t x A t G−1t B t x  A t G−1t B t x ∈ imA t.Thus,I T t Q t G−1t B t x ∈ S tand we have m  S t ker A ρ t

Let x ∈ ker A ρ t ∩ S t , that is, x ∈ S t and x ∈ ker A ρ t Since x ∈ S t , there is a z∈ m

such that B t x  A t z  A t P t z and since x ∈ ker A ρ t , T t−1x ∈ ker A t Therefore, T t−1x  Q t T t−1x.

Hence,A t − B t T t Q t T−1

t x  −A t − B t T t Q t P t z which follows that T t−1x  −P t z Thus, T t−1x 0

Proof 1 Noting that G t P t  A t − B t T t Q t P t  A t P t  A t, we get2.8

2 From B t T t Q t  A t − G t , it follows G−1t B t T t Q t  P t − I  −Q t Thus, we have2.9

3 Q2t  T t Q t G−1t B t T t Q t G−1t B t 2.9  −T t Q t Q t G−1t B t  −T t Q t G−1t B t  Q t and A ρ t Q t 

−A ρ t T t Q t G−1t B t  0 This means that Q t is a projector onto ker A ρ t From the proof ofiii,

Lemma 2.2, we see that Q t is the projector onto ker A ρ t along S t

5 Let T

B t T tQt It is easy to see that

t Qt Gt−1 The proof ofLemma 2.3is complete

Definition 2.4 The LIDE2.4 is said to be index-1 if for all t ∈ , the following conditionshold:

ii kerA ρ t ∩ S t {0}

Now, we add the following assumptions

Hypothesis 2.5. 1 The homogeneous LIDE 2.4 is of index-1

2 ft, x is rd-continuous and satisfies the Lipschitz condition,

P t P t  P t and P t P t  P t are true for each two projectors P t and P t along the space ker A t

We now describe shortly the decomposition technique for2.3 as follows.

k Multiplying2.3 by P t G−1t and Q t G−1t , respectively, it yields

Thus, g t is Lipschitz continuous with the Lipschitz constant δ t : γt 1 − γ t−1L−1t L t B t .

condition ut0  P ρ t0 x0has a unique solution ut  ut; t0 , x0, t t0.

Thus, we get the following theorem

Theorem 2.7 Let Hypothesis 2.5 and the assumptions on the projector Q t be satisfied Then,2.3

with the initial condition

has a unique solution This solution is expressed by

x t  xt; t0 , x0  ut; t0, x0 gt ut; t0 , x0, t t0, t∈

where u t  ut; t0 , x0 is the solution of 2.31 with ut0  P ρ t0 x0.

ii If ft, 0  0 for all t ∈thenΩt ∩ ker A ρ t  {0}.

Proof i Let y ∈ Ł t , that is, Q ρ t y  T t Q t G−1t B t P ρ t y T t Q t G−1t f t, y We have

ii Let y ∈ Ω t ∩ ker A ρ t Then y ∈ Ωt and P ρ t y  0 Since Ωt Łt , we have y ∈ Łt.

This means that Q ρ t y  T t Q t G−1t B t P ρ t y T t Q t G−1t f t, y  T t Q t G−1t f t, Q ρ t y From the

assumption f t, 0  0, it follows that Q ρ t y L t T t Q t G−1t Q ρ t y  γ t Q ρ t y The fact

γ t < 1 implies that Q ρ t y  0 Thus y  P ρ t y Q ρ t y 0 The lemma is proved

Remark 2.9. 1 By virtue ofLemma 2.8, we find out that the solution spaceŁtis independent

ρ t0 A ρ t0  P ρ t0  and A ρ t0P ρ t0  A ρ t0 , the initial condition 2.33 is

equivalent to the condition A ρ t0 x t0   A ρ t0 x0 This implies that the initial condition isnot also dependent on choice of projectors

2.3 satisfying the initial condition P ρ t xt; t, x0 − x0  0 We see that xt; t, x0  P ρ t x

g t P ρ t x   P ρ t x0 g t P ρ t x0  x0 This means that there exists a solution of2.3 passing

x0∈ Łt

2.3 Quasilinear Implicit Dynamic Equations

Now we consider a quasilinear implicit dynamic equation of the form

one of the following conditions:

Lemma 2.10 Suppose that the bounded linear operator triplet: : X → Y, : Y → Z,  : Z →

X is given, where X, Y, Z are Banach spaces Then the operator I−  is invertible if and only if

I− is invertible.

Proof See17, Lemma 1

I T t Q t G−1t 

B t − f

respectively, and putting u  P ρ t x, v  Q ρ t x, we obtain

Let h ∈ Q ρ t m be a vector satisfying T t Q t G−1t f xt, u vh  0 Paying attention to

T t Q t G−1t B t h  −h, we have

−T t Q t G−1t f xt, u vh I T t Q t G−1t 

B t − f

2.33 can be expressed by xt; t0 , x0  ut; t0, x0 gt ut; t0 , x0.



T t Q t G−1t f xt, x| Q ρ tÊm

−1

Therefore, we have the following theorem

Theorem 2.11 Given an index-1 quasilinear implicit dynamic equation 2.43, then there holds the

following.

(1) Equation2.43 is locally solvable, that is, for any t0∈k, x0∈ m , there exists a unique solution x t; t0 , x0 of 2.43, defined on t0 , b  with some b ∈ , b > t0, satisfying the initial condition2.33.

(2) Moreover, if f t, x satisfies the Lipschitz condition in x and we can find a matrix B t such that



T t Q t G−1t f xt, x| Q m

−1

is bounded, then this solution is defined on t0 , sup and we have the expression

where u t; t0 , x0 is the solution of 2.56 with ut0   P ρ t0 x0.

Remark 2.12 1 We note that the expression T t Q t G−1t B t depends only on choosing the

matrix B t

2 The assumption that T t Q t G−1t f xt, x| Q ρ tÊm−1T t Q t G−1t f xt, x| P ρ tÊmis bounded for

solvability via Lyapunov functions

3 If x0∈ Ωt , there exists z∈ m satisfying A t z  ft, x0 Hence, T t Q t G−1t f t, x0  0.

is a unique solution passing through x0.

3 Stability Theorems of Implicit Dynamic Equations

Consider an implicit dynamic equation of the form

N

k

τ , m

Let ft, 0  0 for all t ∈τ, which follows that3.1 has the trivial solution x ≡ 0.

is the solution of3.1 and 3.2 then xt ∈ Ω t for all t∈t0

Definition 3.1 The trivial solution x≡ 0 of 3.1 is said to be

k

δ t0 ,   such that A ρ t0 x0 < δ resp., Pρ t0 x0 < δ implies xt; t0, x0 <  for all

tt0,

2 A-uniformly resp., P-uniformly stable if it is A-stable resp., P-stable and the



k

asymptotically stable,

4 A-uniformly globally asymptotically resp., P-uniformly globally asymptotically stable if for any δ0 > 0 there exist functions δ ·, T· such that A ρ t0 x0 < δ

resp., P ρ t0 x0 < δ implies xt; t0, x0 <  for all t t0and if A ρ t0 x0 < δ0

resp., P ρ t0 x0 < δ0 then xt; t0, x0 <  for all t t0 T,

k

condition P ρ t0 xt0 − x0  0 satisfies xt; t0 , x0 N P ρ t0 x0 e−α t, t0, t 

called P -uniformly exponentially stable.

Remark 3.2 From G−1t A t  P t and A t  A t P t , the notions of stable and P -stable as well as asymptotically stable and P -asymptotically stable are equivalent Therefore, in the following theorems we will omit the prefixes A and P when talking about stability and asymptotical stability However, the concept of A-uniform stability implies P -uniform stability if the

Denote

Proposition 3.3 The trivial solution x ≡ 0 of 3.1 is A-uniformly (resp., P-uniformly) stable if and

only if there exists a function ϕ∈such that for each t0 ∈

holds, provided A ρ t0 x0 ∈ ϕ (resp., Pρ t0 x0 ∈ ϕ).

Proof We only need to prove the proposition for the A-uniformly stable case.

take δ  δ > 0 such that ϕδ < , that is, ϕ−1 > δ If xt; t0 , x0 is an arbitrary solution of

3.1 and A ρ t0 x0 < δ, then xt; t0, x0 ϕ  A ρ t0 x0  < ϕδ < , for all t t0

for each  > 0 there exists δ  δ > 0 such that for each t0∈k

τthe inequality A ρ t0 x0 < δ

δ  <  Denote

there holds

A ρ tx0 < γ  then xt; t0 , x0 <  ∀t t0. 3.6

By putting

β  : 1



0

it is seen that

ϕ  A ρ t0 x0  ∀t t0by ϕ∈remember that xt; t0 , x0  0 does not imply that x·; t0, x0 ≡0 Consider the case where t > 0 If A ρ t0 x0 < βt, then by the relations 3.6 and 3.8

we have xs; t0 , x0 < t , ∀s t0 In particular, xt; t0 , x0 < twhich is a contradiction.Thus A ρ t0 x0  β  t , this implies xt; t0 , x0  t ϕ  A ρ t0 x0 , ∀t  t0, provided

sup β > A ρ t0 x0

The proposition is proved

Similarly, we have the following proposition

Proposition 3.4 The trivial solution x ≡ 0 of 3.1 is A-stable (resp., P-stable) if and only if for each

10

Remark 3.5 Note that when the function V is independent of t and even if the vector field

3.10may

depend on t.

Theorem 3.6 Assume that there exist a constant c > 0, −c ∈ R and a function V :τ× m →

being rd-continuous and a function ψ∈, ψ defined on 0, ∞ satisfying

1 ψ x  V t, A ρ t x  for all x ∈ Ω t and t∈τ ,

3.10 t, A ρ t x c/1 − cμtV t, A ρ t x , for any x ∈ Ω t and t∈ k

τ

Assume further that3.1 is locally solvable Then, 3.1 is globally solvable, that is, every solution

with the initial condition3.2 is defined ont0.

Theorem 3.7 Assume that there exist a function V : τ × m → being rd-continuous and

a function ψ∈, ψ defined on 0, ∞ satisfying the conditions

Proof By virtue ofTheorem 3.6and the conditions2 and 3, it follows that 3.1 is globally

A ρ tx t0 < δ and xt1; t0 , x t0 0for some t1 t0 Put 1 ψ0.

By the assumption that V t0 , 0   0 and V t, x is rd-continuous, we can find δ0 > 0

such that if y < δ0 then V t0 , y  < 1 With given δ0 > 0, let x t be a solution of 3.1 suchthat A ρ t0 x t0 < δ0and xt1 ; t0, x t0 0for some t1t0

t1

t0

V 3.10Δ 

t, A ρ t x tΔt  Vt1, A ρ t1 x t1− Vt0, A ρ t0 x t0 0. 3.16

Therefore, V t1 , A ρ t1 x t1 V t0 , A ρ t0 x t0 < 1 Further, xt1 ∈ Ω t1and by the condition

Proof Also fromTheorem 3.6and the conditions2 and 3, it implies that 3.1 is globallysolvable

lim inft→ ∞V t, A ρ t x t  0 Assume on the contrary that inf t∈ Ì

t0 V t, A ρ t x t > 0 From

the condition1, it follows that inft∈ Ì

t0 A ρ t x t : r > 0 By the condition 3, we have

This means that V t, A ρ t x t is a decreasing function Consequently,

Theorem 3.9 Suppose that there exist a function a ∈ , a defined on 0, ∞, and a function V ∈

Proof The proof is similar to the one ofTheorem 3.7with a remark that since limx→ 0V t, x 

0 uniformly in t∈τ , we can find δ0 > 0 such that if y < δ0then supt∈Ìτ V t, y < 1

The proof is complete

Remark 3.10 The conclusion ofTheorem 3.9 is still true if the condition1 is replaced by

that a  x  V t, A ρ t x b  A ρ t x  for all x ∈ Ω t and t∈τ”

We present a theorem of uniform global asymptotical stability

Theorem 3.11 If there exist functions a, b, c ∈, a defined on 0, ∞, and a function V ∈ Crdτ×