# Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales

- NH Du
^{1}Email author, - NC Liem
^{1}, - CJ Chyan
^{2}and - SW Lin
^{2}

**2011**:979705

https://doi.org/10.1155/2011/979705

© N. H. Du et al. 2011

**Received: **29 September 2010

**Accepted: **4 February 2011

**Published: **27 February 2011

## Abstract

## Keywords

## 1. Introduction

with being a given -matrix function, has been an intensively discussed field in both theory and practice. This problem can be seen in many real problems, such as in electric circuits, chemical reactions, and vehicle systems. März in [1] has dealt with the question whether the zero-solution of (1.1) is asymptotically stable in the Lyapunov sense with , with being constant and small perturbation .

This model appears in many practical areas, such as the Leontiev dynamic model of multisector economy, the Leslie population growth model, and singular discrete optimal control problems. On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and partial differential-algebraic equations, and so forth, which have already attracted much attention from researchers (cf. [2–4]). When , in [5], the authors considered the solvability of Cauchy problem for (1.2); the question of stability of the zero-solution of (1.2) has been considered in [6] where the nonlinear perturbation is small and does not depend on .

with in time scale and being the derivative operator on . When , (1.3) is (1.1); if , we have a similar equation to (1.2) if it is rewritten under the form .

The purpose of this paper is to answer the question whether results of stability for (1.1) and (1.2) can be extended and unified for the implicit dynamic equations of the form (1.3). The main tool to study the stability of this implicit dynamic equation is a generalized direct Lyapunov method, and the results of this paper can be considered as a generalization of (1.1) and (1.2).

with the assumption of differentiability for . The main results of this paper are established in Section 3 where we deal with the stability of (1.5). The technique we use in this section is somewhat similar to the one in [6–8]. However, we need some improvements 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 by the symbol
. We assume throughout that a time scale
is endowed with the topology inherited from the real numbers with the standard topology. We define the *forward jump operator* and the *backward jump operator*
by
(supplemented by
) and
(supplemented by
). The *graininess*
is given by
. A point
is said to be *right-dense* if
, *right-scattered* if
, *left-dense* if
, *left-scattered* if
, and *isolated* if
is right-scattered and left-scattered. For every
, by
, we mean the set
. The set
is defined to be
if
does not have a left-scattered maximum; otherwise, it is
without this left-scattered maximum. Let
be a function defined on
, valued in
. We say that
is *delta differentiable* (or simply: *differentiable*) at
provided there exists a vector
, called the derivative of
, such that for all
there is a neighborhood
around
with
for all
. If
is differentiable for every
, then
is said to be *differentiable on*
. If
, then delta derivative is
from continuous calculus; if
, the delta derivative is the forward difference,
, from discrete calculus. A function
defined on
is *rd-continuous* if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of all rd-continuous functions from
to a Banach space
is denoted by
. A matrix function
from
to
is said to be *regressive* if
for all
, and denote
the set of regressive functions from
to
. Moreover, denote
the set of *positively regressive* functions from
to
, that is, the set
.

has a unique solution defined on . Further, if is regressive, this solution exists on .

The solution of the corresponding matrix-valued IVP , always exists for , even is not regressive. In this case, is defined only with (see [12, 13]) and is called the Cauchy operator of the dynamic equation (2.1). If we suppose further that is regressive, the Cauchy operator is defined for all .

where is the derivative (in the second variable of the function ) in normal meaning and is the scalar product.

We refer to [12, 15] for more information on the analysis on time scales.

### 2.2. Linear Equations with Small Nonlinear Perturbation

which has been well studied. If there is at least a such that is singular, we cannot solve explicitly the leading term . In fact, we are concerned with a so-called ill-posed problem where the solutions of Cauchy problem may exist only on a submanifold or even they do not exist. One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of the equation.

where , are orthogonal matrices and is a diagonal matrix with singular values on its main diagonal. Since , on the above decomposition of , we can choose the matrix to be in (see [16]). Hence, by putting and , we obtain and as the requirement.

Under these notations, we have the following Lemma.

Lemma 2.2.

The following assertions are equivalent

(ii)the matrix is nonsingular;

Proof.

(i) (ii) Let and such that . This equation implies . Since and , it follows that . Hence, which implies . This means that . Thus, , that is, the matrix is nonsingular.

(ii) (iii) It is obvious that . We see that and . Thus, and we have .

Let , that is, and . Since , there is a such that and since , . Therefore, . Hence, which follows that . Thus, and then . So, we have that (iii). (iii) (i) is obvious.

Lemma 2.2 is proved.

Lemma 2.3.

Therefore, . The proof of Lemma 2.3 is complete.

Definition 2.4.

The LIDE (2.4) is said to be *index-1* if for all
, the following conditions hold:

Now, we add the following assumptions.

- (1)
The homogeneous LIDE (2.4) is of index-1.

- (2)

Remark 2.6.

By the item (2.13) of Lemma 2.3, the condition (2.18) is independent from the choice of and .

Note that does not depend on the choice of the projector function since the relations and are true for each two projectors and along the space .

We now describe shortly the decomposition technique for (2.3) as follows.

and it is easy to see that is rd-continuous in .

Applying the global existence theorem (see [12]), we see that (2.31), with the initial condition has a unique solution .

Thus, we get the following theorem.

Theorem 2.7.

where is the solution of (2.31) with .

Lemma 2.8.

There hold the following statements:

Proof.

where we have already used a result of Lemma 2.3 that is a projector onto . So .

(ii) Let . Then and . Since , we have . This means that . From the assumption , it follows that . The fact implies that . Thus . The lemma is proved.

- (1)
By virtue of Lemma 2.8, we find out that the solution space is independent from the choice of projector and operator .

- (2)
Since and , the initial condition (2.33) is equivalent to the condition . This implies that the initial condition is not also dependent on choice of projectors.

- (3)
Noting that if is a solution of (2.3) with the initial condition (2.33), then for all . Conversely, let and let , , be the solution of (2.3) satisfying the initial condition . We see that . This means that there exists a solution of (2.3) passing .

### 2.3. Quasilinear Implicit Dynamic Equations

with and assumed to be continuously differentiable in the variable and continuous in .

Suppose that rank for all . We keep all assumptions on the projector and operator stated in Section 2.2.

where is the tangent space of at the point .

Suppose that (2.43) is of index-1. Then, by Lemma 2.2, this condition is equivalent to one of the following conditions:

(3)Let be a matrix such that the matrix is invertible (we can choose , e.g.). From the relation

is invertible.

Lemma 2.10.

Suppose that the bounded linear operator triplet: is given, where are Banach spaces. Then the operator is invertible if and only if is invertible.

Proof .

See [17, Lemma 1].

is locally uniquely solvable and the solution of (2.43) with the initial condition (2.33) can be expressed by .

is bounded for all and . Then, the right-hand side of (2.56) also satisfies the Lipschitz condition. Thus, from the global existence theorem (see [12]), (2.56) with the initial condition (2.57) has a unique solution defined on .

Therefore, we have the following theorem.

Theorem 2.11.

where is the solution of (2.56) with .

- (1)
- (2)
The assumption that is bounded for a matrix function seems to be too strong. In Section 3, we show a condition for the global solvability via Lyapunov functions.

- (3)
If , there exists satisfying . Hence, . Therefore, by the same argument as in Section 2.2, we can prove that for every , there is a unique solution passing through .

## 3. Stability Theorems of Implicit Dynamic Equations

For the reason of our purpose, in this section we suppose that is an upper unbounded time scale, that is, . For a fixed , denote .

has a unique solution defined on . The condition ensuring the existence of a unique solution can be refered to Section 2. We denote the solution with the initial condition (3.2) by . Remember that we look for the solution of (3.1) in the space . Let for all , which follows that (3.1) has the trivial solution .

We mention again that . Noting that if is the solution of (3.1) and (3.2) then for all .

Definition 3.1.

The trivial solution of (3.1) is said to be

(1) -stable (resp., -stable) if, for each and , there exists a positive such that (resp., ) implies for all ,

(2) -uniformly (resp., -uniformly) stable if it is -stable (resp., -stable) and the number mentioned in the part (1). of this definition is independent of ,

(3) -asymptotically (resp., -asymptotically) stable if it is stable and for each , there exist positive such that the inequality (resp., ) implies . If is independent of , then the corresponding stability is -uniformly asymptotically ( -uniformly asymptotically) stable,

(4) -uniformly globally asymptotically (resp., -uniformly globally asymptotically) stable if for any there exist functions , such that (resp., ) implies for all and if (resp., ) then for all ,

(5)*P*-exponentially stable if there is positive constant
with
such that for every
there exists an
, the solution of (3.1) with the initial condition
satisfies
. If the constant
can be chosen independent of
, then this solution is called
-uniformly exponentially stable.

Remark 3.2.

From and , the notions of -stable and -stable as well as -asymptotically stable and -asymptotically stable are equivalent. Therefore, in the following theorems we will omit the prefixes and when talking about stability and asymptotical stability. However, the concept of -uniform stability implies -uniform stability if the matrices are uniformly bounded and -uniform stability implies -uniform stability if the matrices are uniformly bounded.

and is the domain of definition of .

Proposition 3.3.

Proof.

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

Sufficiency. Suppose there exists a function satisfying (3.4) for each ; we take such that , that is, . If is an arbitrary solution of (3.1) and , then , for all .

Let be the inverse function of . It is clear that also belongs to .

For , we denote . If , then by (remember that does not imply that ). Consider the case where . If , then by the relations (3.6) and (3.8) we have . In particular, which is a contradiction. Thus , this implies , provided .

The proposition is proved.

Similarly, we have the following proposition.

Proposition 3.4.

Remark 3.5.

Note that when the function is independent of and even if the vector field associated with the implicit dynamic equation (3.1) is autonomous, the derivative may depend on .

Theorem 3.6.

Assume that there exist a constant and a function being rd-continuous and a function , defined on satisfying

Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable, that is, every solution with the initial condition (3.2) is defined on .

Proof.

The last inequality says that the solution can be lengthened on , that is, (3.1) is globally solvable.

Theorem 3.7.

Assume that there exist a function being rd-continuous and a function , defined on satisfying the conditions

Assume further that (3.1) is locally solvable. Then the trivial solution of (3.1) is stable.

Proof.

By virtue of Theorem 3.6 and the conditions (2) and (3), it follows that (3.1) is globally solvable. Suppose on the contrary that the trivial solution of (3.1) is not stable. Then, there exists an such that for all there exists a solution of (3.1) satisfying and for some . Put .

By the assumption that and is rd-continuous, we can find such that if then . With given , let be a solution of (3.1) such that and for some .

Therefore, . Further, and by the condition (2) we have . This is a contradiction. The theorem is proved.

Theorem 3.8.

satisfying the conditions

Further, (3.1) is locally solvable. Then the trivial solution of (3.1) is asymptotically stable.

Proof.

Also from Theorem 3.6 and the conditions (2) and (3), it implies that (3.1) is globally solvable.

as , which gets a contradiction.

which follows that by the condition (2).

Theorem 3.9.

Suppose that there exist a function , defined on , and a function such that

(1) uniformly in and for all and ,

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly stable.

Proof.

The proof is similar to the one of Theorem 3.7 with a remark that since uniformly in , we can find such that if then

The proof is complete.

Remark 3.10.

The conclusion of Theorem 3.9 is still true if the condition (1) is replaced by "*there exist two functions*
*,*
*defined on*
*and a function*
*such that*
*for all*
*and*
".

We present a theorem of uniform global asymptotical stability.

Theorem 3.11.

If there exist functions , defined on , and a function satisfying

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly globally asymptotically stable.

Proof.

which contradicts the definition of in (3.21). The proof is complete.

When is not differentiable, one supposes that there exists a -differentiable projector onto and is rd-continuous on ; moreover, for all . Let .

From now on we remain following the above assumptions on the operators whenever is mentioned.

By the same argument as Theorem 3.6, we have the following theorem.

Theorem 3.12.

Assume that there exist a constant and a function being rd-continuous and a function , defined on satisfying

Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable.

Theorem 3.13.

Assume that (3.1) is locally solvable. Then, the trivial solution of (3.1) is stable if there exist a function being rd-continuous and a function , defined on such that

Proof.

Assume that there is a function satisfying the assertions (1), (2), and (3) but the trivial solution of (3.1) is not stable. Then, there exist a positive and a such that ; there exists a solution of (3.1) satisfying and , for some . Let . Since , it is possible to find a satisfying when . Consider the solution satisfying and for a .

We get a contradiction because when .

The proof of the theorem is complete.

Theorem 3.14.

Assume that (3.1) is locally solvable. If there exist two functions , defined on and a function being rd-continuous such that

then the trivial solution of (3.1) is -uniformly stable.

Proof.

The proof is similar to the one of Theorem 3.9.

Theorem 3.15.

If there exist functions , defined on and a function satisfying

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly globally asymptotically stable.

Proof.

Similarly to the proof of Theorem 3.11.

It is difficult to establish the inverse theorem for Theorems from 3.7 to 3.15, that is, if the trivial solution of (3.1) is stable, there exists a function satisfying the assertions in the above theorems. However, if the structure of the time scale is rather simple we have the following theorem.

Theorem 3.16.

Suppose that contains no right-dense points and the trivial solution of (3.1) is -uniformly stable. Then, there exists a function being rd-continuous satisfying the conditions (1), (2), and (3) of Theorem 3.13, where is an open neighborhood of 0 in .

Proof.

where is the unique solution of (3.1) satisfying the initial condition . It is seen that is defined for all satisfying , , and .

Let . By the definition, . From (2.60), for all . In particular, . Thus, . Hence, we have the assertion (2) of the theorem.

The proof is complete.

Now we give an example on using Lyapunov functions to test the stability of equations. The following result finds out that the stability of a linear equation will be ensured if nonlinear perturbations are sufficiently small Lipschitz.

where is sufficiently small. Let be defined by (2.9) with and . By Theorem 2.7, we see that there exists a unique solution satisfying the condition for any .

in there is denoted the set of right-scattered points of the interval .

Denote . It is easy to show that the trivial solution of (3.35) is -uniformly exponentially stable if and only if , where is the domain of uniform exponential stability of . On the exponential stable domain of a time scale, we can refer to [10, 18, 19]. By the definition of exponential stability, it implies that the graininess function of the time scale is upper bounded. Let .

and suppose . Since , this condition implies that (3.35) is -uniformly exponentially stable.

where the matrix is supposed to be symmetric positive definite. It is clear that is symmetric positive definite.

we can examine easily that the matrix also satisfies (3.42), is symmetric and positive definite.

Theorem 3.17.

Suppose that and the homogeneous equation (3.35) is of index-1 and the constant is sufficiently small. Then, the trivial solution of (3.33) is -uniformly globally asymptotically stable.

Proof.

By Theorem 3.15, (3.33) is -uniformly globally asymptotically stable.

Example 3.18.

It is obvious that . Further, . Thus, according to Theorem 2.7 for each , (3.47) with the initial condition has the unique solution.

It is easy to compute, , , , where , and .

Let the Lyapunov function be .

We have for any solution of (3.47) and (noting that ),

(+) if is right-scattered then ,

(+)if is right-dense then , where .

In both two cases, we have , so the trivial solution of (3.47) is -uniformly stable by Theorem 3.14.

Note that if we let then the result is still true. Indeed, by the simple calculations we obtain

Therefore, having the above result is obvious.

## 4. Conclusion

We have studied some criteria ensuring the stability for a class of quasilinear dynamic equations on time scales. So far, the inverse theorem of the theorems of the stability in Section 3 of this paper is still an open problem for an arbitrary time scale meanwhile it is true for discrete and continuous time scales.

## Declarations

### Acknowledgment

This work was done under the support of NAFOSTED no 101.02.63.09.

## Authors’ Affiliations

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