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Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation
Journal of Inequalities and Applications volume 2010, Article number: 768469 (2010)
Abstract
We deal with the approximate controllability for the nonlinear functional differential equation governed by the variational inequality in Hilbert spaces and present a general theorems under which previous results easily follow. The common research direction is to find conditions on the nonlinear term such that controllability is preserved under perturbation.
1. Introduction
Let and
be two complex Hilbert spaces. Assume that
is a dense subspace in
and the injection of
into
is continuous. If
is identified with its dual space, we may write
densely and the corresponding injections are continuous. The norm on
,
and
will be denoted by
,
and
, respectively. The duality pairing between the element
of
and the element
of
is denoted by
, which is the ordinary inner product in
if
. For
we denote
by the value
of
at
. We assume that
has a stronger topology than
and, for the brevity, we may regard that

Let be a continuous linear operator from
into
which is assumed to satisfy GÃ¥rding's inequality, and let
be a lower semicontinuous, proper convex function, and
is a nonlinear mapping. Let
be some Hilbert space and the controller operator
a bounded linear operator from
to
. Then we study the following variational inequality problem with nonlinear term:

Noting that the subdifferential operator is defined by

where denotes the duality pairing between
and
, the problem (NDE) is represented by the following nonlinear functional differential problem:

The existence and regularity for the parabolic variational inequality in the linear case ( ), which was first investigated by Brézis [1, 2], have been developed as seen in Barbu [4, Section
] (also see [4, Section
]). The regularity for the nonlinear variational inequalities of semilinear parabolic type was studied in [5].
The solution (NCE) is denoted by corresponding to the nonlinear term
and the control
. The system (NCE) is said to be approximately controllable in the time interval
, if for every given final state
,
, and
there is a control function
such that
. Investigations of controllability of semilinear systems found in [6, 7] have been studied by many [6–10], which is shown the relation between the reachable set of the semilinear system and that of its corresponding.
In [7, 11], they dealt with the approximate controllability of a semilinear control system as a particular case of sufficient conditions for the approximate solvability of semilinear equations by assuming that
(1) is compact operator, or the embedding
is compact;
(2) is (locally) Lipschitz continuous (or the sublinear growth condition and
(3)the corresponding linear system (NCE) in case where and
is approximately controllable.
Yamamoto and Park [12] studied the controllability for parabolic equations with uniformly bounded nonlinear terms instead of assumptions mentioned above. As for the some considerations on the trajectory set of (NCE) and that of its corresponding linear system (in case ) as matters connected with (3), we refer to Naito [10] and Sukavanam and Tomar [13], and references therein. In [13] and Zhou [14], they studied the control problems of the semilinear equations by assuming (1), (3), a Lipschitz continuity of
and a range condition of the controller
with an inequality constraint.
In this paper, we no longer require the compact property in (1), the uniform boundedness in (2), and the inequality constraint on the range condition of the controller , but instead we need the regularity and a variation of solutions of the given equations. For the basis of our study, we construct the fundamental solution and establish variations of constant formula of solutions for the linear systems.
This paper is composed of four sections. Section 2 gives assumptions and notations. In Section 3, we introduce the single valued smoothing system corresponding to (NCE). Then in Section 4, the relations between the reachable set of systems consisting of linear parts and possibly nonlinear perturbations are addressed. From these results, we can obtain the approximate controllability for (NCE), which is the extended result of [10, 13, 14] to (NCE).
2. Solvability of the Nonlinear Variational Inequality Problems
Let be a bounded sesquilinear form defined in
and satisfying GÃ¥rding's inequality:

where and
is a real number. Let
be the operator associated with the sesquilinear form
:

Then is a bounded linear operator from
to
by the Lax-Milgram theorem. The realization for the operator
in
which is the restriction of
to

is also denoted by . We also assume that there exists a constant
such that

for every , where

is the graph norm of . Thus, in terms of the intermediate theory, we may assume that

where denotes the real interpolation space between
and
.
Lemma 2.1.
Let . Then

Proof.
Put for
. Then,

As in [15, Theorem , Chapter
], the solution
belongs to
hence we obtain that

Conversely, suppose that and
. Put
. Then since
is an isomorphism operator from
to
there exists a constant
such that

From the assumptions and it follows that

Therefore, .
By Lemma 2.1, from Butzer and Berens [16, Theorem ], we can see that

It is known that generates an analytic semigroup
in both
and
. The following Lemma is from [17, Lemma
].
Lemma.
There exists a constant such that the following inequalities hold for all
and every
:

Lemma.
Suppose that and
for
. Then there exists a constant
such that



Proof.
The assertion (2.14) is immediately obtained by virtue of [8, Theorem ] (or [7, Theorem
]). Since

it follows that

From (2.4), (2.14), and (2.15), it holds that

So, if we take a constant such that

the proof is complete.
Let be a nonlinear mapping satisfying the following:
(G1)for any ,
the mapping
is strongly measurable;
(G2)there exist positive constants such that
(i)
(ii) for all
,
, and
.
For , we set

where belongs to
.
Lemma.
Let and
for any
. Then
and

Moreover, if , then

Proof.
From (G1), (G2), and using the Hölder inequality, it is easily seen that

The proof of (2.23) is similar.
By virtue of [5, Theorems and
], we have the following result on the solvability of (NDE) (see [3, 15] in case of corresponding to equations with
).
Proposition.
Let the assumptions (G1) and (G2) be satisfied. Assume that where
stands for the closure in
of the set
. Then, (NDE) has a unique solution

and there exists a constant depending on
such that

3. Smoothing System Corresponding to (NDE)
For every , define

Then the function is Fréchet differentiable on
and its Fréhet differential
is Lipschitz continuous on
with Lipschitz constant
where
as is seen in [4, Corollary
, Chapter II]. It is also well known results that
and
for every
, where
is the minimum element of
.
Now, we introduce the smoothing system corresponding to (NCE) as follows.

Since generates a semigroup
on
, the mild solution of (SCE) can be represented by

In virtue of Proposition 2.5, we know that if the assumptions (G1-G2) are satisfied then for every and every
, (SCE) has a unique solution

and there exists a constant depending on
such that

Now, we assume the hypothesis that and
is uniformly bounded, that is,

Lemma.
Let and
be the solutions of (SCE) with same control
. Then there exists a constant
independent of
and
such that

Proof.
For given , let
and
be the solutions of (SCE) corresponding to
and
, respectively. Then from (SCE), we have

and hence, from (2.13) and multiplying by , it follows that

Let us choose a constant such that
. Then by (G1), we have

Integrating (3.7) over and using the monotonicity of
we have

Here, we used

Since for every
, it follows from (A) and using Gronwall's inequality that

Theorem 3.2.
Let the assumptions (G1-G2) and (A) be satisfied. Then in
is a solution of (NCE), where
is the solution of (SCE).
Proof.
In virtue of Lemma 3.1, there exists such that

From (G1-G2), it follows that

Since are uniformly bounded by assumption (A), from (3.13) we have that

therefore,

Note that . Since
and
is demiclosed, we have that

Thus we have proved that satisfies a.e. on
(NCE).
4. Controllability of the Nonlinear Variational Inequality Problems
Let be a state value of the system (SCE) at time
corresponding to the function
, the nonlinear term
, and the control
. We define the reachable sets for the system (SCE) as follows:

Definition.
The system (NCE) is said to be approximately controllable in the time interval if for every desired final state
and
, there exists a control function
such that the solution
of (NCE) satisfies
, that is, if
where
is the closure of
in
, then the system (NCE) is called approximately controllable at time
.
We need the following hypothesis:
for any and
there exists a
such that

where is a constant independent of
.
As seen in [18], we obtain the following results.
Proposition.
Under the assumptions (G1-G2), (A), and (B), the following system

is approximately controllable on , that is,
.
Let . Then it is well-known that

for almost all point of .
Definition.
The point which permits (4.3) to hold is called the Lebesgue point of
.
Let be a solution of (SCE) such that
in
is a solution of (NCE). First we consider the approximate controllability of the system (SCE) in case where the controller
is the identity operator on
under the Lipschitz conditions (G1-G2) on the nonlinear operator
in Proposition 4.2. So,
obviously.
Proposition.
Let be solution of (4.2) corresponding to a control
. Then there exists a
such that

Proof.
Let be a Lebesgue point of
so that

For a given , we define a mapping

by

It follows readily from definition of and Lemma 2.4 that

By a well-known contraction mapping principle, has a unique fixed point
in
if the condition (4.5) is satisfied. Let

Then from (G1-G2), Lemma 2.4, and Proposition 2.5, it follows that

Thus, from which, we have

And we obtain

If is a Lebesgue point of
, then we can solve the equation in
with the initial value
and obtain an analogous estimate to (4.10) and (4.12). If not, we can choose
to be a Lebesgue point of
. Since the condition (4.5) is independent of initial values, the solution can be extended to the interval
, and so we have showed that there exists a
such that
.
Now, we consider the approximate controllability for the following semilinear controlsystem in case where is the identity operator,

Let us define the reachable sets for the system (4.13) as follows:

Theorem.
Under the assumptions (G1-G2), (A), and (B), we have

Therefore, if the system (4.2) with is approximately controllable, then so is the semilinear system (4.13).
Proof.
Let and let
be a solution of (4.2) corresponding to a control
. Consider the following semilinear system:

The solution of (4.2) and (4.16), respectively, can be written as

Then from Proposition 2.5, it is easily seen that , that is,
as
in
. Let
be given. For
, set

Then we have

So, for fixing , we choose some constant
satisfying

and from (2.13), or (2.16) it follows that

Thus, we know that as
in
for
. Noting that

from (2.13), or (2.16), it follows that

Since the condition (4.20) is independent of , by the step by stem method, we get
as
in
, for all
. Therefore, noting that
,
, every solution of the linear system with control
is also a solution of the semilinear system with control
, that is, we have that
in case where
.
From now on, we consider the initial value problem for the semilinear parabolic equation (SCE). Let be some Banach space and let the controller operator
be a bounded linear operator from
to
.
Theorem.
Let us assume that there exists a constant such that

Assume that assumptions (G1-G2), (A), and (B) are satisfied. Then we have

that is, the system (SCE) is approximately controllable on .
Proof.
Let be a solution of the smoothing system (SCE) corresponding to (NCE). Set
where
is a solution of (4.2) corresponding to a control
. Then as seen in Theorem 4.5, we know that
. Consider the following semilinear system:

If we define as in proof of Theorem 3.2, then we get

So, as similar to the proof of Theorem 3.2, we obtain that .
From Theorems 3.2 and 4.6, we obtain the following results.
Theorem.
Under the assumptions (G1-G2), (A), (B), and (B1), the system (NCE) is approximately controllable on .
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Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2009-0071344).
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Jeong, JM., Ju, E. & Lee, K. Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation. J Inequal Appl 2010, 768469 (2010). https://doi.org/10.1155/2010/768469
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DOI: https://doi.org/10.1155/2010/768469
Keywords
- Variational Inequality
- Functional Differential Equation
- Parabolic Type
- Unique Fixed Point
- Approximate Controllability