Cesari-type Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

An optimal control problem governed by semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.

where y(·) is the solution of (1.1) (called the state corresponding to control u(·)). Our optimal control problem is as follows.
Problem (C). Find aū(·) ∈ U such that J(ū(·)) = inf u(·)∈U J(u(·)). (1.3) Anyū(·) satisfying (1.3) is called an optimal control. It is well-known that optimal control of Problem (C) may fail to exist. When A(x, u) ≡ A(x), a suitable Cesari-type condition and some other mild conditions will guarantee the existence of an optimal control. Cesari-type condition is a natural generalization of optimal control problem with linear state equations and convex cost functionals. Many results are available along these lines. For further detail, see the books by Cesari [6], Li and Yong [11], for examples. For the two phrase case, i.e., U = {0, 1} and A(x, i) ≡ A i (i = 0, 1) with A 0 , A 1 being two constant matrices, Murat and Tartar gave an existence result in the framework of "relaxation" control (see [16]). However, it seems no work devoted to the existence of optimal controls for general cases.
In this paper, we will give a Cesari-type result to ensure the existence of a solution to Problem (C). We always assume Λ and λ be two constants satisfying Λ ≥ λ > 0. Denote by S n + the set of all n × n (symmetric) positive definite matrices and M Λ,λ = Q ∈ S n + λ|ξ| 2 ≤ Qξ · ξ ≤ Λ|ξ| 2 , ∀ ξ ∈ IR n .
For a matrix B, we always denote B ij as its entries.
We recall that a Polish space is a separable completely metrizable topological space. We mention that all (nonempty) closed sets and open sets in lR m are polish spaces.
We make the following assumptions.
(S1) Set Ω is a bounded domain in IR n with a C 2 boundary ∂Ω. (1.4) (S4) Function f (x, y, v) is measurable in x and continuous in (y, v) ∈ IR × U for almost all x ∈ Ω. Moreover, for almost all x ∈ Ω, and for any R > 0, there exists an M R > 0 such that |f (x, y, v)| + |f y (x, y, v)| ≤ M R , ∀ v ∈ U, |y| ≤ R. (1.6) (S5) Function f 0 (x, y, v) is measurable in x, lower semicontinuous in (y, v) ∈ IR × U for almost all x ∈ Ω. Moreover, for almost all x ∈ Ω and for any R > 0, there exists an K R > 0 such that In fact, if necessary, we can replacing ω(·) bỹ ω(r) Denote Z = [0, 1] n and Let e 1 , e 2 , . . . , e n be the canonical basis of IR n . We call a function g(x) is Z-periodic if it admits periodic 1 in the direction e j (j = 1, 2, . . . , n). Denote We define (1.9) Our main result is the following theorem.
Theorem 1.1. Assume (S1) -(S5), and the following condition hold where E(x, y) and GE(x, y) are defined by (1.8) and (1.9), GE(x, y) is the closure of GE(x, y) in S n + × IR × IR, and B δ (y) is the ball centered at y with radius δ. Then Problem (C) admits at least one solution. Remark 1.3. When the leading term is independent of control variable, i.e. A(x, u) ≡ A(x), Theorem 1.1 is equivalent to the classical existence result of optimal control (see Theorem 6.4 in Chapter 3 of [11]). This fact will follows by Proposition 3.4 in Section 3.
When dealing with problems with controls containing in the leading term, we meet a main difficulty that is to find the state equation corresponding to the weak limit of state sequence. This is involved with the H-convergence and G-closure problem. It is known that optimal control usually does not exist for Problem (C) and therefore to seek optimal relaxed control for Problem (C) is more meaningful than to seek a solution for Problem (C). Nevertheless, we think this paper contains some useful ideas for us to get the relaxation of Problem (C), which will be our forthcoming work. In this paper, we will give a local representation of G-closure in Section 2, which is critical in proving the existence theorem. While Section 3 is devoted to a proof of Theorem 1.1 and some propositions.

H-convergence and Local Representation of G-closure
Now, let us recall the notion of H-convergence. This kind of convergence was introduced by Murat and Tartar in [15].

Proposition 2.2. For any sequence
This proposition proves the existence of an H-limit for a subsequence of a bounded sequence, but it delivers no explicit formula for this limit. The next proposition shows that when A ε (·) = A( · ε ) with some periodic matrix valued function A(·), A ε (·) H-converges to an H-limit defined by an explicit formula (up to solving some corresponding cell problems). The proposition can be stated as with A * ∈ M Λ,λ being a constant matrix defined by its entries

3)
where {w i } 1≤i≤n is the family of unique solutions in H 1 # (Z)/IR of the cell problems −∇ · (A(z)(e i + ∇w i (z))) = 0, in Z.
For a proof of the above proposition, see Theorem 1.3.18 of [2] or Theorem 1.3.1 of [3].
The next classical result (see Theorem 1.3.23 in [2], for example) shows the fact that a general H-limit A * (·) can be attained as the limit of a sequence of periodic homogenized matrices.
Proposition 2.4. Assume A ε (·) ∈ L ∞ (Ω; M Λ,λ ) H-converge to a limit A * (·). For any x in Ω and any sufficiently small positive h > 0, let A * ε,h (·) be the periodic homogenized matrix defined by its entries is the family of unique solutions in H 1 # (Z)/IR of the cell problems Then, along a subsequence h → 0, We list some useful properties of H-convergence in follows. For proofs of these results, see  [2].
Proposition 2.5 shows that the value of H-limit A * (·) in a region Ω 0 does not depend on he values of sequence A ε (·) outside of this region, which is precisely what we mean by locality.
This proposition shows that H-convergence is weaker than strong convergence. On the other hand, it is well-known that usually the weak limit of a sequence A ε (·) does not equal to its H-limit.
Proposition 2.7. Let (S1) hold. Then there exist constants C > 0 and δ > 0 such that, for any 1 ≤ p ≤ 1 + δ and two sequences of A ε (·) and B ε (·) in L ∞ (Ω; M Λ,λ ), which H-converge to A * (·) and B * (·), respectively, it holds that Now define We see that G(A) is the set of all possible H-limits of {A(·, u(·))} u(·)∈U . A very important problem called G-closure problem is to find out the structure of G(A). Many works devoted to this problem dealt with two-phrase composite cases(see, for examples, [4], [14] and [17]). In [17], a precise formula of G(A) was given for a special two-phrase case of A taking only αI and βI for some β > α > 0. Unfortunately, in most cases including usual two-phrase cases, precise knowledge of the G-closure are still lacking.
A local representation of G(A) is crucial to our main result. We give a simple lemma related to Assumption (S3) first.
We have the following results.
be a family of measurable decompositions of Ω such that: where |E|, diam (E) denotes the Lebesgue measure and the diameter of E, respectively.
Thus, as a function of z, F (x + hz) − F (x) converges in measure to 0 as h → 0. Since we get (2.9) by Lebesgue's dominated convergence theorem.
(ii) By Remark 1.2, we suppose ω(·) is a continuous module without loss of generality. For Consequently, if we set F (x) = 0 for x ∈ Ω and choose we have Φ ∈ C(Ω; IR m ). Consequently, it follows easily from the uniform continuity of Φ and the assumption (c) that Thus, Therefore, We get the proof.
Now, we will give a local representation of G(A).
Theorem 2.9. Assume (S1)-(S3) hold. Then the G-closure set G(A) is characterized by (2.12) Remark 2.1. It is easy to see that in (2.12), G x (A) can be rewritten as Proof of Theorem 2.9. Denote We need to show G(A) = P(A).
We prove G(A) ⊆ P(A) first. Assume A * (·) ∈ G(A). Then there exists a sequence u ε (·) ∈ U , such that as ε → 0 + , By Proposition 2.4, along a subsequence h → 0, where A * h,ε (·) is defined by On the other hand, define A * h,ε (·) by with w i h,ε (·; x) ∈ H 1 # (Z)/IR being the unique Z-periodic solution of Then, combining (2.16) with (2.18), we get x) and using integration by part, we get from the periodicities ofw i h,ε (·; x) and w i h,ε (·; x) that Then the ellipticity of A yields
We will show the result in three steps.
Step I. Assume A(x, u) ≡ A(u).
For any A ∈ G(A), we have u(·) ∈ U Z such that where w i (·) ∈ H 1 # (Z)/IR solves ∇ · A(u(z))(e i + ∇w i (z)) = 0. Let A * (·) ∈ P(A). Then Since G(A) is closed, A k j is always nonempty. Thus, we can select a constant matrix A k j from A k j . Define Since for almost all x ∈ Ω k j , A * (x) ∈ G(A), by (2.26), there is Thus, by (2.25), Consequently, by Proposition 2.6, we have (2.29) The advantage of replacing A k by A k is that we have A k j ∈ G(A) ⊆ G(A) while we do not always have A k j ∈ G(A). Then, by Proposition 2.5 (local property), A k (·) ∈ G(A). Finally, by (2.29), A * (·) ∈ G(A).

That is, P(A) ⊆ G(A).
Step By what we have proved in Step I and the local property of H-convergence, we can see that

P(A) ⊆ G(A) holds in this case.
Step III. General cases. Let A * (·) ∈ P(A). Then A * (x) ∈ G x (A), a.e. x ∈ Ω. We want to prove A * (x) ∈ G(A). Without loss of generality, we can suppose that While A * k (·) is a measurable selection of the projection of A * (x) on G x (A k ), i.e., A * k (·) is measurable and where G x (A k ) is defined by (2.12). By Filippov's lemma (see [10], or Corollary 2.26 of Chapter Now we will show that A * k (·) → A * (·), strongly in L 1 (Ω; S n + ).
(2.31) By (2.30), there exists a u(·) ∈ U Z , such that where for any x ∈ Ω, w i (·; x) ∈ H 1 # (Z)/IR is the solution of Next, we can define A * k (·) by Thus, similar to the proof of (2.23), we have By Lebesgue's dominated convergence theorem, we deduce which proves (2.31).
Furthermore, noting that A * k (·) is piecewise constant and A * k (·) ∈ P(A k ), by Step II, A * k (·) ∈ G(A k ). Then there exists u k,j (·) ∈ U , such that By Proposition 2.7, we obtain Thus it follows from Lemma 2.8 that, Combining the the above with (2.33), we get Consequently, It follows from A k (·) ∈ G(A) that A * (·) ∈ G(A). This ends the proof.

Proof of the Main Theorem
In this section, we will prove our main result. Before that, we need to show three lemmas. The first is about the well-posedness and regularity of state equation (1.1).
This ends the proof.
The third lemma is about relaxed control defined by finite-additive probability measures.
Now we are at the position to prove Theorem 1.1.
II. Now, we turn to prove E(x, y) ⊆ co E(x, y).
Let U k i 1≤i≤k be a family of measurable decompositions of U , such that (a) if i = j, then U k i U k j = ∅; (b) for any k, Noting that m(E k i ) ≥ 0 and k i=1 m(Z k i ) = 1 for k = 1, 2, · · · , we deduce E(x, y) ⊆ co E(x, y).
Consequently, E(x, y) ⊆ co E(x, y). This end the proof.