- Research Article
- Open Access
Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation
Journal of Inequalities and Applications volume 2010, Article number: 768469 (2010)
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.
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 Gårding'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 .
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  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  and Sukavanam and Tomar , and references therein. In  and Zhou , 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 Gårding'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 .
Let . Then
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
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 ].
There exists a constant such that the following inequalities hold for all and every :
Suppose that and for . Then there exists a constant such that
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
(ii) for all , , and .
For , we set
where belongs to .
Let and for any . Then and
Moreover, if , then
From (G1), (G2), and using the Hölder inequality, it is easily seen that
The proof of (2.23) is similar.
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,
Let and be the solutions of (SCE) with same control . Then there exists a constant independent of and such that
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
Let the assumptions (G1-G2) and (A) be satisfied. Then in is a solution of (NCE), where is the solution of (SCE).
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
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:
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 , we obtain the following results.
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 .
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.
Let be solution of (4.2) corresponding to a control . Then there exists a such that
Let be a Lebesgue point of so that
For a given , we define a mapping
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:
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).
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 .
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 .
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.
Under the assumptions (G1-G2), (A), (B), and (B1), the system (NCE) is approximately controllable on .
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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).