This section is to show the existence of the optimal control of the approximationproblem corresponding to the phase transition system. To this end, we first show sometechnical lemmas, which are presented below for the sake of completeness.
In order to approximate , we define a nondecreasing function[21] on R by
where denotes the positive part of functions. Then , and
(2.1)
We fix a primitive of such that
(2.2)
Without loss of generality, we may assume ,therefore, the approximation of (1.1) is
(2.3)
Lemma 2.1 Suppose thatsatisfies (2.1)-(2.2),, weaklyinand, are the solutionsof (2.3) corresponding toand,respectively. Then there exists a subsequenceof, still denoted byitself, such that
(2.4)
Proof Replacing and w by and in (2.3),respectively, we obtain
(2.5)
Multiplying (2.5)1 and (2.5)2 by and ,respectively, integrating over Ω and adding the resulting equations, we end up with
(2.6)
Therefore, we conclude with the help of Young’s inequality and the properties ofthat
(2.7)
Here and throughout the proof of Lemma 2.1, we shall denote by C severalpositive constants independent of n. Applying Gronwall’s inequality to(2.7), we derive
(2.8)
Now, testing (2.5)1 by ,we derive
(2.9)
which together with ,(2.2), (2.8), Nirenberg’s inequality and Gronwall’s inequality implies that
(2.10)
Next, multiplying (2.5)1 by ,integrating over and invoking Young’s inequality, wederive
(2.11)
Thanks to (2.8), andGronwall’s inequality, we derive
(2.12)
Inserting (2.10) and (2.12) into (2.5)1, we have
(2.13)
Now, differentiating (2.5)1 with respect to t and multiplying theresult by ,then multiplying (2.5)2 by , adding theresulting equations and integrating over Ω leads to
(2.14)
On the other hand, with the help of (2.8), (2.10), Hölder’s inequality andNirenberg’s inequality, we get
(2.15)
where μ is a small positive constant and ()are independent of n. Inserting (2.15) into (2.14), we derive
(2.16)
Taking the supremum with respect to t in (2.16), choosingsufficiently small and applying Gronwall’s inequality, we end up with
(2.17)
which combined with (2.13) implies that . Therefore, employing thestandard parabolic theory to (2.5)2 leads to
(2.18)
Now we may combine the estimates (2.8), (2.10), (2.12), (2.13) and (2.18) to concludethe results. This completes the proof. □
Lemma 2.2 Suppose thatsatisfies (2.1)-(2.2),letwithweaklyinas,be the solution of(2.3) corresponding to.Then, on some subsequenceof, there exists aquadsuch that
(2.19)
while
(2.20)
(2.21)
(2.22)
(2.23)
asandis a solution of(1.1) satisfying the following estimates:
(2.24)
whereisindependent of ε, n.
Proof Rewrite (2.3) as follows:
(2.25)
Employing almost exactly the same arguments as in the proof of Lemma 2.1, weconclude the results (2.20)-(2.22). Furthermore, by a standard argument in[22], we get a.e. in . This completes theproof. □
Now, we assume that is optimal for problem(P). For each , let be the solution to
(2.26)
It follows from Lemma 2.2 that
(2.27)
Now, the approximating optimal control problems () are asfollows:
where , by
(2.28)
and is the solution of (2.3). Here, denotes the distance of to S,
(2.29)
is the approximations of g[23], where, is amollifier in ,is the projection of H on , which isthe finite dimensional space generated by , isan orthonormal basis in H, is the operator defined by ,.
First of all, we show the existence of optimal solutions for ().
Lemma 2.3 ()has at least one optimal solution.
Proof Let befixed. It is clear that . Let and be aminimizing sequence such that
(2.30)
which together with (H2), (H3) and (2.28) implies that isbounded in . Without loss ofgenerality, we may assume that in. Let and be the solutions of (2.3)corresponding to and, respectively. It follows fromLemma 2.1 that on some subsequence of , still denoted by itself,
(2.31)
With the help of (H2), (2.29) and (2.31), we also obtain
(2.32)
On the other hand, due to (2.31) and (H1), we have that
(2.33)
and therefore
(2.34)
Finally, (2.28) and (2.32)-(2.34) imply that is the optimal pair forproblem ().This concludes the proof of Lemma 2.3. □
Lemma 2.4 Letbeoptimal for problem ()andbe the solution of(2.3) corresponding to.Then, on some subsequence,
(2.35)
(2.36)
Proof Since is asolution to (),we have
(2.37)
which together with (2.27) implies that
(2.38)
which combined with (2.37) implies that
(2.39)
which implies that (2.39), that isbounded in . Without loss generality,we may assume that weakly in, which together withLemma 2.2 implies that there exists a sequence of suchthat
(2.40)
On the other hand, (2.28) and (2.39) imply that
(2.41)
and therefore
(2.42)
Thus, we conclude from (2.28), (2.40) and (2.42) that
(2.43)
Finally, it follows from (2.39), (2.43) and Lemma 2.2 that
(2.44)
This completes the proof. □