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  and Samoilenko and Perestyuk  .
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  studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay and S. Selvi and M. Mallika Arjunan  studied the exact controllability for impulsive differential systems with finite delay. For approximate controllability of impulsive semilinear evolution equation, Lizhen Chen and Gang Li  studied the approximate controllability of impulsive differential equations with nonlocal conditions, using measure of noncompactness and Monch Fixed Point Theorem, and assuming that the nonlinear term does not depend on the control variable. Recently, in  -  , the approximate controllability of semilinear evolution equations with impulses has been studied by applying Rothe’s Fixed Point Theorem, showing that the influence of impulses do not destroy the controllability of some known systems like the heat equation, the wave equation, the strongly damped wave equation. More recently, in  the approximate controllability of the heat equation with impulses and delay has been studied.
The approximate controllability of the linear part of the Benjamin-Bona-Mahony (BBM) equation was proved in  . This result was used to study the controllability of the nonlinear BBM equations in  , which could serve as a basis for studying the BBM equation under the influence of impulses and delays
where and are constants, is a domain in, is an open non- empty subset of, denotes the characteristic function of the set, the distributed control, are continuous functions. Here is the delay and the nonlinear functions are smooth enough and satisfy
One natural space to work evolution equations with delay and impulses is the Banach space
where and, endowed with the norm
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 if for every and, there exists such that the mild solution of (1) corresponding to u verifies:
As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting and.
We also study the approximate controllability of the corresponding linear system
by applying the classical Unique Continuation Principle for Elliptic Equations (see  ) and the following lemma.
Lemma 1.1. (see Lemma 3.14 from  , p. 62) Let and be sequences of real numbers such that:. Then
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 and the following results:
Proposition 1.1. Let be a measure space with and. Then and
Theorem 1.1. (Rothe’s Fixed Theorem,  -  ) Let E be a Banach space and be a closed convex subset such that the zero of E is contained in the interior of B. Consider be a continuous mapping with
a) is compact.
b) (, where denotes the boundary of B.
Then there is a point such that
Let and consider the linear unbounded operator defined by, where
The operator A has the following very well known properties (see N. I. Akhiezer and I. M. Glazman  ): the spectrum of A consists of eigenvalues
each one with finite multiplicity equal to the dimension of the corresponding eigenspace. Therefore:
a) There exists a complete orthonormal set of eigenvectors of A.
b) For all we have
where is the inner product in Z and
So, is a family of complete orthogonal projections in Z and
c) generates the analytic semigroup given by
where, , , is a bounded linear operator, is defined by and the functions
, are defined by
On the other hand, from conditions (2) and (3) we get the following estimates.
Proposition 2.1. Under the conditions (2)-(3) the functions, , defined above satisfy and:
Since and (is the resolvent set of A), then the operator:
is invertible with bounded inverse
Therefore, the systems (11) and its linear part can be written as follows, for
Moreover, and can be written in terms of the eigenvalues of A:
Therefore, if we put and, systems (16) and (17) can be written in the form:
and the functions F defined above satisfy:
Now, we formulate two simple propositions.
Proposition 2.2. (  ) The operators and are given by the following expressions
Moreover, the following estimate holds
Observe that, due to the above notation, systems (20)-(21) can be written as follows
3. Preliminaries on Controllability of the Linear Equation
In this section we prove the interior controllability of the linear system (28). To this end, notice that for an arbitrary and the initial value problem
admits only one mild solution given by
Definition 3.1. For the system (29) we define the following concept: The controllability map (for) is given by
whose adjoint operator is given by
The following lemma holds in general for a linear bounded operator between Hilbert spaces W and Z.
Lemma 3.1. (see    and  ) The Equation (28) is approximately controllable on if and only if one of the following statements holds:
c), in Z.
f) For all we have, where
So, and the error of this approximation is given by
Remark 3.1. The Lemma 3.1 implies that the family of linear operators , defined for by
is an approximate inverse for the right of the operator G in the sense that
Proposition 3.4. (see  ) If, then
Theorem 3.1. The system (28) is approximately controllable on. Moreover, a sequence of controls steering the system (28) from initial state to an neighborhood of the final state at time is given by the formula
and the error of this approximation is given by the expression
Proof. It is enough to show that the restriction of G to the space has range dense, i.e., or. Consequently, takes the following form
whose adjoint operator is given by
Since B is given by the formula
and by (24), we get that and.
Then we have that
where, which satisfies the conditions:
Hence, following the proof of Lemma 1.1, we obtain that
Now, putting, we obtain that
Then, from the classical Unique Continuation Principle for Elliptic Equations (see  ), it follows that. So,
On the other hand, is a complete orthonormal set in, which implies that.
Therefore, , which implies that. So,. Hence, the system (29) is approximately controllable on, and the remainder of the proof follows from Lemma 3.1. W
Lemma 3.2. Let S be any dense subspace of. Then, system (29) is approximately controllable with control if, and only if, it is approximately controllable with control. i.e.,
where is the restriction of G to S.
Proof (Þ) Suppose and. Then, for a given and there exits and a sequence such that
Therefore, and for n big enough. Hence, .
(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 are well define in the space of continuous functions: by
Also, the Controllability Grammian operator is still the same
Finally, the operators defined for by
is an approximate inverse for the right of the operator G in the sense that
4. Main Result
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 and the initial value problem
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 by the following formula:
with is given by
Theorem 4.1. The nonlinear system (1) is approximately controllable on. Moreover, a sequence of controls steering the system (1) from initial state to an -neighborhood of the final state at time is given by
and the error of this approximation is given by
Proof. We shall prove this Theorem by claims. Before, we note that and.
Claim 1. The operator is continuous. In fact, it is enough to prove that the operators:
define above are continuous. The continuity of follows from the continuity of the nonlinear functions, and the following estimate
On the other hand,
The continuity of the operator follows from the continuity of the operators and define above.
Claim 2. The operator is compact. In fact, let D be a bounded subset of . It follows that, we have
Therefore, is uniformly bounded.
Now, consider the following estimate:
Without lose of generality we assume that. On the other hand we have:
Since is a compact operator for, then we know that the function is uniformly continuous. So,
Consequently, if we take a sequence on, this sequence is uniformly bounded and equicontinuous on the interval and, by Arzela theorem, there is a subsequence of, which is uniformly convergent on.
Consider the sequence on the interval. On this interval the sequence is uniformly bounded and equicontinuous, and for the same reason, it has a subsequence uniformly convergent on.
Continuing this process for the intervals, , ∙∙∙, , we see that the sequence converges uniformly on the interval. This means that is compact, which implies that the operator is compact.
where is the norm in the space. In fact, consider the following estimates:
where is given by:
Claim 4. The operator has a fixed point. In fact, for a fixed, there exists big enough such that
Hence, if we denote by the ball of center zero and radius, we get that. Since is compact and maps the sphere into the interior of the ball, we can apply Rothe’s fixed point Theorem 1.1 to ensure the existence of a fixed point such that
Claim 5. The sequence is bounded. In fact, for the purpose of contradiction, let us assume that is unbounded. Then, there exits a subsequence such that
On the other hand, from (48) we know for all that
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 such that
Therefore, without loss of generality, we can assume that the sequence converges to. So, if
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 converges to y, we get that
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 and a final state and the operator given by (44)- (46) has a fixed point and the sequence converges. i.e.,
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, where is a bounded domain in, is an open nonempty subset of, denotes the characteristic function of the set, the distributed control, are continuous functions, and, are positive numbers.
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 is a bounded domain in, is an open nonempty subset of, denotes the characteristic function of the set, the distributed control , are continuous functions and .
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.