Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay ()
1. Introduction
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 [1] and Samoilenko and Perestyuk [2] .
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 [3] studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay and S. Selvi and M. Mallika Arjunan [4] studied the exact controllability for impulsive differential systems with finite delay. For approximate controllability of impulsive semilinear evolution equation, Lizhen Chen and Gang Li [5] 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 [6] - [10] , 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 [11] 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 [12] . This result was used to study the controllability of the nonlinear BBM equations in [13] , which could serve as a basis for studying the BBM equation under the influence of impulses and delays
(1)
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
(2)
(3)
,
,
and
![]()
One natural space to work evolution equations with delay and impulses is the Banach space
![]()
where
and
, endowed with the norm
![]()
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
if for every
and
,
there exists
such that the mild solution
of (1) corresponding to u verifies:
![]()
where
![]()
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
(4)
by applying the classical Unique Continuation Principle for Elliptic Equations (see [14] ) and the following lemma.
Lemma 1.1. (see Lemma 3.14 from [15] , 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
(5)
Theorem 1.1. (Rothe’s Fixed Theorem, [16] - [18] ) 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 [19] ): the spectrum of A consists of eigenvalues
(6)
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
(7)
where
is the inner product in Z and
(8)
So,
is a family of complete orthogonal projections in Z and
(9)
c)
generates the analytic semigroup
given by
(10)
(11)
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
:
(12)
(13)
Since
and
(
is the resolvent set of A), then the operator:
(14)
is invertible with bounded inverse
(15)
Therefore, the systems (11) and its linear part can be written as follows, for ![]()
(16)
(17)
Moreover,
and
can be written in terms of the eigenvalues of A:
(18)
(19)
Therefore, if we put
and
, systems (16) and (17) can be written in the form:
(20)
(21)
and the functions F defined above satisfy:
. (22)
Now, we formulate two simple propositions.
Proposition 2.2. ( [12] ) The operators
and
are given by the following expressions
(23)
(24)
Moreover, the following estimate holds
(25)
where
(26)
Observe that, due to the above notation, systems (20)-(21) can be written as follows
(27)
(28)
where
.
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
(29)
admits only one mild solution given by
(30)
Definition 3.1. For the system (29) we define the following concept: The controllability map (for
)
is given by
(31)
whose adjoint operator
is given by
(32)
The following lemma holds in general for a linear bounded operator
between Hilbert spaces W and Z.
Lemma 3.1. (see [15] [20] [21] and [22] ) The Equation (28) is approximately controllable on
if and only if one of the following statements holds:
a)
.
b)
.
c)
,
in Z.
d)
.
e)
.
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
(33)
is an approximate inverse for the right of the operator G in the sense that
(34)
Proposition 3.4. (see [21] ) If
, then
(35)
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
.
Suppose that
![]()
Then we have that
![]()
where
, which satisfies the conditions:
(36)
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 [14] ), 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
(37)
and
by
(38)
Also, the Controllability Grammian operator is still the same ![]()
(39)
Finally, the operators
defined for
by
(40)
is an approximate inverse for the right of the operator G in the sense that
(41)
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
(42)
admits only one mild solution given by the formula
(43)
![]()
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:
![]()
where
(44)
and
(45)
with
is given by
(46)
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
![]()
where
(47)
![]()
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:
![]()
and
![]()
define above are continuous. The continuity of
follows from the continuity of the nonlinear functions
,
and the following estimate
![]()
On the other hand,
![]()
Therefore,
![]()
where
and
.
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:
![]()
and
![]()
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.
Claim 3.
![]()
where
is the norm in the space
. In fact, consider the following estimates:
![]()
where
![]()
![]()
and
![]()
Therefore,
![]()
where
is given by:
![]()
Hence
![]()
and
(48)
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
(49)
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
![]()
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
converges to y, we get that
![]()
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
and a final state
and
the operator
given by (44)- (46) has a fixed point and the sequence
converges. i.e.,
![]()
![]()
5. Conclusions
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
.
Acknowledgements
We thank the Editor and the referee for their comments. This research was funded by the BCV. This support is greatly appreciated.
Competing Interests
The authors declare that there is not competing of interests.