1. Introduction

In the last 10 years, set-valued analysis is interesting with the new field of set differential equations (SDEs). There are many the authors are interesting in field of SDEs, for example, Lakshmikantham V., Gnana T., Kaleva O., Mohapatra R.,... Before we proceed to investigate our problems, let's note the following facts:

In[5], Prof.V. Lakshmikantham and the other authors have studied the set differential equations (SDEs).

In[13] and[16], the authors have considered the set control differential equations (SCDEs), that is SDEs with set controls: and have some important results on existence, stability.

In[11] the author has given many kinds of feedback for problem of global controllability.

In this paper, we present the boundedness of set solutions to SCDEs by the Liapunov-like functions and by feedback.

This paper is organized as follows: in section 2, we recall some basic concepts and notations which are useful in next sections. In section 3 we present the boundedness properties of set solutions to SCDEs and in the last section, we give the conclusion and acknowledgements.

2. Preliminaries

In[5], Prof.V. Lakshmikantham and the other authors have studied the set differential equations (SDEs). In this work the authors have considered the Hausdorff metric space as followings:

Let denote the collection of all nonempty convex subsets of. Given - the Hausdorff distance between A and B is defined by

We define the magnitude of a nonempty subset of A

We define the magnitude of a nonempty subset of A:

(1)

where is the zero element of which is regarded as a one point set. - norm in is finite when the supremum in (1) is attained with. The set, with the metric D defined above, is a complete metric space. It has been proven that becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space, if it is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication.

Let if there exists a set such that, then C is called the Hausdorff difference (the geometric difference) of the sets A and B and is denoted by the symbol A-B The mapping

is said to have a Hukuhara derivative at a point if

exist in the topology of and are equal to.

By embedding as a complete cone in a corresponding Banach space and taking into account the result on the differentiation of Bochner integral, we find that if where is integrable in the sence of Bochner, then exists and the equality

(2)

3. Main Results

Let's consider the set control differential equations (SCDEs):

(3)

where , state and control. If integrable, then it is called an admissible control. Let U be a set of all admissible controls. The mapping is said to be a solution of SCDES (3) on iff it satisfies SCDEs on and is the symbolic representation of the following Hukuhara integral expression:

(4)

Definition 3.1. The set solution of SCDEs (3) is said to be:

a/ (B)- bounded on, if there exists the constant such that, by (4) we have, for all .

b/ (EB)- Exponent bounded on, if there exist the constants such that the supper distance:

Assume that satisfies the followings:

(F1). there exists a constant such that

(F2). there exists constant c>0 such that

Theorem 3.1. Let satisfies hypotheses (F1)-(F2) and U satisfies (U1), then SCDEs (3) has unique B- bounded set solutions.

Proof. We have to prove that:

a) By (F1), there exists the set solutions, which is represented as (4).

b) Uniqueness of. Assume that the other set solutions such that, then in force (F2).

c) A boundedness property of set solution that means there exists such that for all .

We estimate by (4) and (F2):

Putting and, we have .

This Gronwall's inequality implies that.

Choosing , we have for all.

Theorem 3.2. Let and and by contraction feedback and, then SCDEs (3) has the unique (B)- bounded solution in.

Proof. (a) Problems of existence and uniqueness are clear.

(b) Problem of (B)- bounded are proved by integral expression (4) followings:

,and

.

Using Gronwall's inequality, we infer where, we obtain .

Next, we present some results about (B), (EB) of solutions in with using the Lyapunov-like functions.

Theorem 3.3. Assume that the positive Lyapunov - like function which satisfies the following conditions:

(i) , where L is bounded Lipschitz constant, for all

;

(ii) , for,where are increasing functions;

(iii)

where for all and, we have the following affirmations:

a/ If then a set solution of SCDEs (3) is (B)-bounded.

b/ If (or if) then a set solution of SCDEs (3) is (EB)-bounded.

Proof. Setting the function, we have

so, implies that. Since where is maximal solution of ODE:

(5)

then for all.

• Let, be given. Choose a such that. We claim that with this then (B)- bounded solution. If it’s not true, there exists solution of SCDEs (3) and , such that and where for all.

Wherever, because

for all, then :

by is a increasing function, therefore this contradiction proves that (B)-bounded solution .

• In the case, if (or) then we have for all.

If then and if (EB) is not true, given, we choose

then

for all, this contradiction proves that the fuzzy set solution is (EB).

Definition 3.2. The set solutions of SCDEs (3) are said to be:

a/ (B1)- equi - bounded, if for any and, there exists a such that.

b/ (B2)- uniform - bounded, if β in (B1) does not depend on.

Theorem 3.4. Assume that the Lyapunov-like function and feedback

satisfy the following conditions:

(i) ,

where L is bounded Lipschitz constant, for all and;

(ii) add condition

where .

If is any solution of SCDEs (3) existing on such that , then we have where is a maximal solution of ordinary differential equation (ODE) (5)

Proof. Let is any solution of SCDEs (3) existing on.

Define so that. Now, for small, by our assumption it follows that

using the Lipschitz condition give (i), thus we have

and is any solution of SCDEs (3), we find that

and

We therefore have the scalar differential inequality which yields, as before, the estimate where is a maximal solution of ODE (5). This proof is complete.

Corollary 3.1. A function is admissible in the theorem 3.4 to yield the estimate

Next, we have some denotes:

,

is increasing in.

Now, we introduce some results on the boundedness of set solutions for SCDEs (3) by feedback.

Theorem 3.5. Assume that

is Lyapunov-like function and is feedback for SCDEs (3) satisfies the following conditions:

(i)

for

(ii)

(iii)

then, the affirmation (B1) holds.

Proof. Proof of this theorem is analogous proof of theorem 3.3.

Theorem 3.6. Assume that

(i) where ρ may be large, satisfies:

(ii) for,

(iii) and where, which are defined only on,then, (B2) holds.

Proof. We have to prove that (B2) holds. Because implies

Thus for all and there exists estimate then by (iii) of theorem 3.5 the affirmation for (B1) holds, that means (B2) holds.

4. Conclusions

By the Lyapunov like-funcions and by some kinds of feeback we just have investigated the problems of boundedness for set solutions to set control differential equations - SCDES, that is an one of the new trends in set-valued analysis. The boundedness properties of set solutions allows testing the extremal solutions, what is useful in practice of applications SDEs and SCDEs.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the referees for their careful reading and many valuable remarks which improved the presentation of the paper.

Thanks are due to the Vietnam National Foundation for Science and Technology Devolopment (NAFOSTED) for financial support.