In this section we study new and -posteriori error estimates for the mixed finite element approximation to the semilinear parabolic optimal control problems. Let
(3.1)
(3.2)
It can be shown that (see [13] for some detail discussions)
(3.3)
(3.4)
(3.5)
It is clear that S and are well defined and continuous on K and . Also, the functional can be naturally extended on K. Then (2.1) and (2.27) can be represented as
(3.6)
and
(3.7)
In many applications, is uniform convex near the solution u. The convexity of is closely related to the second-order sufficient conditions of the optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many applications, there is , independent of h, such that
(3.8)
Firstly, let us derive a posteriori error estimates for the control u.
Theorem 3.1 Let u and be the solutions of (3.6) and (3.7), respectively. Assume that , (), and there is such that
(3.9)
Then we have
(3.10)
where
Proof It follows from (3.6) and (3.7) that
(3.11)
(3.12)
Then it follows from assumptions (3.8), (3.9), and the Schwarz inequality that
(3.13)
It is not difficult to show
(3.14)
where is defined in (2.46)-(2.51). Thanks to (3.14), it is easy to derive
(3.15)
Then, by estimates (3.13) and (3.15), we can prove the requested result (3.10). □
To estimate the error , we need the following well-known stability results for the following dual equations:
(3.16)
and
(3.17)
where
(3.18′)
Lemma 3.1 Let κ and ϖ be the solutions of (3.16) and (3.17), respectively [24, 25]. Let Ω be a convex domain. Then
and
where .
Next, we estimate the errors and .
Theorem 3.2 Let and be the solutions of (2.38)-(2.44) and (2.46)-(2.51), respectively. Then we have
(3.19)
where
Proof We define as follows:
(3.20)
Then from (3.20) we deduce that
(3.21)
Combining (3.20)-(3.21) and the definitions of and , we can get the following equality:
(3.22)
Let κ be the solution of (3.16) with , we infer that
Furthermore, using (2.38)-(2.40), (2.46)-(2.48) and (2.56)-(2.58), we can obtain that
(3.23)
When , ,
(3.24)
When ,
(3.25)
Hence
(3.26)
Similar to Theorem 3.2 of reference [16], we have derived the following estimate:
(3.27)
This proves (3.19). □
Now, we are in a position to estimate the errors and .
Theorem 3.3 Let and be the solutions of (2.38)-(2.44) and (2.46)-(2.51), respectively. Then we have the following error estimate:
(3.28)
where are defined in Theorem 3.2, and
Proof We first define as follows:
(3.29)
Then from (2.34) and (3.29) we deduce that
(3.30)
Now, we let
Combining (2.41), (3.30) and the definitions of , , and , we get
(3.31)
Let ϖ be the solution of (3.17) with . Then it follows from (2.41)-(2.43), (2.49)-(2.51) and (2.56)-(2.58) that
(3.32)
To prove (3.28), the first step is to estimate . Let , when , by Lemmas 2.1, 2.2 and 3.1, we have
(3.33)
When ,
(3.34)
Now we estimate . Let again. Similarly, when ,
(3.35)
When ,
(3.36)
Next we estimate , . It follows from Lemma 3.1 that
(3.37)
and
(3.38)
Furthermore, we estimate , . It follows from Lemma 3.1 that
(3.39)
and
(3.40)
Hence, from (3.33)-(3.40) we have that when , ,
(3.41)
When ,
(3.42)
Then it follows from (3.41)-(3.42) that
(3.43)
Similar to (3.27), we can prove that
(3.44)
The triangle inequality and (3.43) yield (3.28). □
Let and be the solutions of (2.5)-(2.11) and (2.38)-(2.44), respectively. We decompose the errors as follows: