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

(1.1)

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:

(NDE)

Noting that the subdifferential operator is defined by

(1.2)

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

(NCE)

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 Gårding's inequality:

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

for every , where

(2.5)

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

(2.6)

where denotes the real interpolation space between and .

Lemma 2.1.

Let . Then

(2.7)

Proof.

Put for . Then,

(2.8)

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

(2.9)

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

(2.10)

From the assumptions and it follows that

(2.11)

Therefore, .

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

(2.12)

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 :

(2.13)

Lemma.

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

(2.14)
(2.15)
(2.16)

Proof.

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

(2.17)

it follows that

(2.18)

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

(2.19)

So, if we take a constant such that

(2.20)

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

(2.21)

where belongs to .

Lemma.

Let and for any . Then and

(2.22)

Moreover, if , then

(2.23)

Proof.

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

(2.24)

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

(2.25)

and there exists a constant depending on such that

(2.26)

3. Smoothing System Corresponding to (NDE)

For every , define

(3.1)

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.

(SCE)

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

(3.2)

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

(3.3)

and there exists a constant depending on such that

(3.4)

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

(A)

Lemma.

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

(3.5)

Proof.

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

(3.6)

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

(3.7)

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

(3.8)

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

(3.9)

Here, we used

(3.10)

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

(3.11)

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

(3.12)

From (G1-G2), it follows that

(3.13)

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

(3.14)

therefore,

(3.15)

Note that . Since and is demiclosed, we have that

(3.16)

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:

(4.1)

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

(B)

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

(4.2)

is approximately controllable on , that is, .

Let . Then it is well-known that

(4.3)

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

(4.4)

Proof.

Let be a Lebesgue point of so that

(4.5)

For a given , we define a mapping

(4.6)

by

(4.7)

It follows readily from definition of and Lemma 2.4 that

(4.8)

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

(4.9)

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

(4.10)

Thus, from which, we have

(4.11)

And we obtain

(4.12)

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,

(4.13)

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

(4.14)

Theorem.

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

(4.15)

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:

(4.16)

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

(4.17)

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

(4.18)

Then we have

(4.19)

So, for fixing , we choose some constant satisfying

(4.20)

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

(4.21)

Thus, we know that as in for . Noting that

(4.22)

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

(4.23)

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

(B1)

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

(4.24)

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:

(4.25)

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

(4.26)

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 .