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For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. In this paper we apply the Rothe’s Fixed Point Theorem to prove the interior approximate controllability of the following Benjamin Bona-Mohany(BBM) type equation with impulses and delay where and are constants, Ω is a domain in , ω is an open non-empty subset of Ω , denotes the characteristic function of the set ω , the distributed control , are continuous functions and the nonlinear functions are smooth enough functions satisfying some additional conditions.

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. One may easily visualize situations in these examples where abrupt changes such as harvesting, disasters and instantaneous stocking may occur. These problems can be modeled by impulsive differential equations with delays, and one can find information about impulsive differential equations in Lakshmikantham [

The controllability of impulsive evolution equations has been studied recently by several authors, but most of them study the exact controllability only. For example, D. N. Chalishajar [

The approximate controllability of the linear part of the Benjamin-Bona-Mahony (BBM) equation was proved in [

where

and

One natural space to work evolution equations with delay and impulses is the Banach space

where

with

We shall denote by C the space of continuous functions:

endowed with the norm

Definition 1.1. (Approximate Controllability) The system (1) is said to be approximately controllable on

where

As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting

We also study the approximate controllability of the corresponding linear system

by applying the classical Unique Continuation Principle for Elliptic Equations (see [

Lemma 1.1. (see Lemma 3.14 from [

if and only if

The approximate controllability of the system (1) follows from the approximate controllability of (4), the compactness of the semigroup generated by the associated linear operator, the conditions (2) and (3) satisfied by the nonlinear term

Proposition 1.1. Let

Theorem 1.1. (Rothe’s Fixed Theorem, [

a)

b)

Then there is a point

In this section we choose a Hilbert space where system (1) can be written as an abstract differential equation with impulses and delay; to this end, we consider the following notations:

Let

The operator A has the following very well known properties (see N. I. Akhiezer and I. M. Glazman [

each one with finite multiplicity

a) There exists a complete orthonormal set

b) For all

where

So,

c)

Consequently, the system (1) can be written as abstract differential equations with impulses and delay in Z:

where

On the other hand, from conditions (2) and (3) we get the following estimates.

Proposition 2.1. Under the conditions (2)-(3) the functions

Since

is invertible with bounded inverse

Therefore, the systems (11) and its linear part can be written as follows, for

Moreover,

Therefore, if we put

and the functions F defined above satisfy:

Now, we formulate two simple propositions.

Proposition 2.2. ( [

Moreover, the following estimate holds

where

Observe that, due to the above notation, systems (20)-(21) can be written as follows

where

In this section we prove the interior controllability of the linear system (28). To this end, notice that for an arbitrary

admits only one mild solution given by

Definition 3.1. For the system (29) we define the following concept: The controllability map (for

whose adjoint operator

The following lemma holds in general for a linear bounded operator

Lemma 3.1. (see [

a)

b)

c)

d)

e)

f) For all

So,

Remark 3.1. The Lemma 3.1 implies that the family of linear operators

is an approximate inverse for the right of the operator G in the sense that

Proposition 3.4. (see [

Theorem 3.1. The system (28) is approximately controllable on

and the error of this approximation

Proof. It is enough to show that the restriction

whose adjoint operator

Since B is given by the formula

and

Suppose that

Then we have that

where

Hence, following the proof of Lemma 1.1, we obtain that

Now, putting

Then, from the classical Unique Continuation Principle for Elliptic Equations (see [

On the other hand,

Therefore,

Lemma 3.2. Let S be any dense subspace of

where

Proof (Þ) Suppose

Therefore,

(Ü) This side is trivial. W

Remark 3.2 According to the previous Lemma, if the system is approximately controllable, it is approximately controllable with control functions in the following dense spaces of

Moreover, the operators G,

and

Also, the Controllability Grammian operator is still the same

Finally, the operators

is an approximate inverse for the right of the operator G in the sense that

In this section we prove the main result of this paper, the interior controllability of the semilinear BBM Equation with impulses and delay given by (1), which is equivalent to prove the approximate controllability of the system (27). To this end, observe that for all

admits only one mild solution given by the formula

Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the Benjamin-Bona-Mahony (1) with impulses and delay.

Define the operator

where

and

with

Theorem 4.1. The nonlinear system (1) is approximately controllable on

and the error of this approximation

where

Proof. We shall prove this Theorem by claims. Before, we note that

Claim 1. The operator

and

define above are continuous. The continuity of

On the other hand,

Therefore,

where

The continuity of the operator

Claim 2. The operator

Therefore,

Now, consider the following estimate:

Without lose of generality we assume that

and

Since

Consequently, if we take a sequence

Consider the sequence

Continuing this process for the intervals

Claim 3.

where

where

and

Therefore,

where

Hence

and

Claim 4. The operator

Hence, if we denote by

Claim 5. The sequence

On the other hand, from (48) we know for all

Particularly, we have the following situation:

Now, applying Cantor’s diagonalization process, we obtain that

and from (49) we have that

which is evidently a contradiction. Then, the claim is true and there exists

Therefore, without loss of generality, we can assume that the sequence

Then,

Hence,

To conclude the proof of this Theorem, it enough to prove that

From Lemma 3.2.d) we get that

Now, from Proposition 3.1, we get that

Therefore, since

Consequently,

Then,

Therefore,

and the proof of the theorem is completed. W

As a consequence of the foregoing theorem we can prove the following characterization:

Theorem 4.2. The Impulsive Semilinear System (1) is approximately controllable if for all states

Our technique can be applied to those control systems whose linear parts generate a compact semigroup and are under the influence of impulses and delays, as well as the following examples which represent research problems.

Problem 1. It appears that our technique can also be applied to prove the interior controllability of the strongly damped wave equation with impulses and delay

in the space

Problem 2. Our technique may also be applied to a system given by partial differential equations modeling the structural damped vibrations of a string or a beam with impulses and delay

Here

We thank the Editor and the referee for their comments. This research was funded by the BCV. This support is greatly appreciated.

The authors declare that there is not competing of interests.

Leiva, H. and Sanchez, J.L. (2016) Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay. Applied Mathematics, 7, 1748- 1764. http://dx.doi.org/10.4236/am.2016.715147