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 .