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On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

An optimal control problem governed by a class of semilinear elliptic equations with nonlinear Neumann boundary conditions is studied in this paper. It is pointed out that the cost functional considered may not be convex. Using a relaxation method, some existence results of an optimal control are obtained.


1.
Introduction. Existence of optimal controls is an important topic in the theory of optimal controls, and it has some actual sense in the practice. It is well known to researchers working in the field of optimal control theory that to guarantee the existence of optimal controls we need an assumption of convexity on a control domain and cost functional. Along this line, many results are available. We refer the readers to the books by Cesari [5], Li and Yong [10] and the papers [11], [15] for further details.
In the absence of convexity, the problem will be more difficult. To the best of our knowledge, the first result on the existence of optimal controls in the absence of convexity was established by Neustadt [19] for the finite-dimensional linear systems. Later, more general cases were studied in the literatures (see [2], [3], [4], [21] and the references cited therein, for example). However, for the infinite-dimensionalcase, relevant results are not rich enough. The readers can refer to [7], [8], [9], [20]. When a convex condition is not assumed, relaxed controls have been proved to be an important tool for studying existence of optimal controls(see [12], [13], [14], [16], [17], for example). The concept of relaxed controls has evolved from generalized curves introduced by Young [24] and McShane [18] in the late 1930s. The largest advantage of relaxation is that the space of admissible controls can be extended to a larger space, and both the control system and the cost functional are convexified.
With the aid of relaxed controls, Lou [12], [13] discussed the existence and nonexistence of optimal controls for linear elliptic equations. Later, motivated by the idea presented in [12], [13], the author [16] considered the case of semilinear elliptic controlled system. This present paper can be considered as a sequent work of [16] and there are three main differences between them. (i) In this paper, we will consider a state equation with a nonlinear Neumann boundary condition not a Dirichlet boundary condition.
(ii) The control domain [0,1] in [16] is generalized to [a, b] here. (iii) The present existence results are more general. The technique used in this paper is similar to that in [12] or [16], however, the presence of the nonlinear Neumann boundary condition will bring out some troubles when analyzing the boundary-value of some functions. For this reason, we cannot obtain the expected nonexistence results and only get the existences. To prove the existence theorems, we first consider the optimal relaxed control problem corresponding to the initial (classical) control problem, and then give the existence and the maximum principle for optimal relaxed controls. This process is standard since the optimal relaxed control problem has been convexified. The next step is devoted to analyzing the maximum principle for an optimal relaxed control and a solution of the adjoint system detailedly. Finally, to obtain the existence of optimal controls it is sufficient to prove that the support of an optimal relaxed control is a singleton almost everywhere. Here we use the fact that when an optimal relaxed control is a Dirac measure almost everywhere, then it essentially becomes an optimal control. The rest of this paper is organized as follows. In Section 2, we give the formulation of the optimal control problem and the main results. Section 3 is devoted to presenting the existence and the maximum principle for an optimal relaxed control. The proofs of the main results are presented in Section 4. Finally, Section 5 is a appendix in which two basic results are proved.
2. Formulation of the optimal control problem and the main results. Let Ω be a bounded domain of R n with smooth boundary Γ. Consider the following elliptic controlled system: where ∂ ν is the derivative in the direction of the outward normal ν on Γ, the function u is a control, and y is a state. Let the control domain U = [a, b], and the set of admissible controls U be defined by We impose the following assumptions on f and g: for some constant r 1 > 0, and there exists a constant C 1 > 0 such that for q > 0, and, moreover, for any K > 0, there exists a constant M K > 0 such that

ON THE EXISTENCE OF OPTIMAL CONTROL 495
The function g has the same properties as f with (2) replaced by for some constant r 2 ≥ 0. Now we introduce the following cost functional: where (y, u) is a state-control pair, and A, B, C are given constants.
Our optimal control problem can be stated as follows: (P) Find a controlū ∈ U such that Anyū satisfying (6) is said of an optimal control. The purpose of this paper is to prove the existence of an optimal controlū under the different conditions. Before stating the main results, we first introduce the following two systems: and For the simplification of form, our main results are stated separately according to the signs of A, B being same or not. It is mentioned that when discussing the existence only the case of AB ≥ 0, A + B = 0, C < 0 was studied in [16].
Theorem 2.2. Suppose that (S) holds and C < 0. Let ξ, η, l 1 and l 2 be defined as above. Then any one of the following conditions is satisfied, Problem (P) admits at least one optimal control.
Remark 1. If C ≥ 0, then Cesari's condition holds and Problem (P) admits at least one optimal control.
3. Existence and maximum principle for optimal relaxed controls. This section is mainly devoted to considering the optimal relaxed control problem. Firstly, it is necessary to recall the concept of relaxed controls and the relations between classical controls and relaxed controls. Let C(U ) be the space of continuous functions endowed with the maximum norm, and Each element of R will be called a relaxed control. Respectively, an element of U is called a classical control.
It is well known that the set of relaxed controls R is convex and sequentially compact; moreover, U is dense in R with the weak star topology of L ∞ (Ω; M(U )) (see [22], Theorem IV.2.1, p. 272, and Theorem IV.2.6, p. 275). Furthermore, we mention that σ k → σ in R means that We now state the optimal relaxed control problem corresponding to Problem (P) in the following way.
(RP). Find a relaxed controlσ ∈ R such that and y satisfies the following relaxed controlled system: It is mentioned that U can be embedded in R by identifying each u ∈ U with the Dirac measure-valued function δ u ∈ R. Moreover J(δ u ) defined by (15) coincides with J(u) defined by (5).
The following proposition is concerned with the existence and uniqueness of a solution of (16). This result is basic and the proof is similar to that of Theorem 6.11 in [10] (p.78). For the readers' convenience, we will present the proof in the last section of this paper. Proposition 1. Let (S) hold. Then for any 1 ≤ p < ∞ and σ ∈ R, (16) admits a unique solution y ∈ W 1,p (Ω) ∩ L ∞ (Ω). Moreover, there exists a constant K p independent of σ, such that The existence theorem and maximum principle for optimal relaxed controls can be established in essentially the same way as those for classical control problems. Proof. By (17), we have thatJ Since R is convex and sequentially compact, we can suppose that σ k converges to someσ in R. By (17) and the Sobolev embedding theorem, we can suppose that for someȳ ∈ W 1,p (Ω) with p > n, where y k is the solution of (16) corresponding to σ k . Finally, by (S) and (18), we can get thatȳ is the solution of (16) corresponding toσ, andσ is an optimal relaxed control.
Lemma 3.2. Suppose that (S) holds. Letσ be an optimal relaxed control of Problem (RP), andȳ be the optimal state corresponding toσ, i.e., Then there exists aψ ∈ W 1,p (Ω) (p ≥ 1), such that in Ω, and Sketch of the proof. Letσ be an optimal relaxed control, then for any σ ∈ R, and α ∈ (0, 1), we have σ α =σ + α(σ −σ) ∈ R since R is convex. Let y α be the state corresponding to σ α , then we can prove that On the other hand, by optimality, Passing to the limit, we get Letψ be the solution of the adjoint equation of (20), then we have that Thus, (21) follows from the above inequality immediately.
Before finishing this section, we introduce another useful lemma, which can be found in [12] (see Lemma 2.3).
for any constant M .
4. The proofs of the main results. Before proving the main results, it is necessary to present the following comparison principle which will be proved in the next section.

SHU LUAN
Therefore, by the monotonicity of f , (32) and (9), we have that Combing (29), (31) and (33), for almost all x ∈ E, we have that This is a contradiction. Hence, the measure of E is zero, and we conclude the proof of (i).
Proof of Theorem 2.2. Theorem 2.2 can be proved using the similar arguments in the proof of Theorem 2.1 (i), and therefore we omit it.
Appendix. In this section, we will prove Proposition 1 and Proposition 2.
The proof of Proposition 1.
Step 1. Let 1 ≤ p < ∞ and m > 0. We define In the same way, we define another function g m . Consider the following truncated problem: Now, for any z ∈ L p (Ω), by (3) and the definitions of f m and g m , we see that Moreover, where L p > 0 is a constant, which, in particular, is independent of z. Thus, we see that the map z → z m defined through (35) is continuous and compact from some fixed ball in L p (Ω) into itself. Hence by the Schauder fixed-point theorem, there exists a fixed point y m of this map. Clearly, y m ∈ W 1,p (Ω) is a solution of (35).
Step 2. For any m > 0, we will prove y m L ∞ (Ω) ≤ K , where the constant K > 0 is independent of m. It is mentioned that this uniform estimate can be deduced from [23] (Theorem 4.1) and which is mainly based on the Moser iteration technique. For the readers' convenience, next we only deal with the special cases of q = 1 in (3) and n ≥ 2, however, it is sufficient for the readers to understand the main idea.
In a similar way one shows that y − m = max{−y m , 0} belongs to L ∞ (Ω). This proves that y m = y + m − y − m ∈ L ∞ (Ω) and y m L ∞ (Ω) ≤ K for some constant K > 0 independent of m.
Step 3. Then, if we take m > K, y m = y is a solution of (35) and satisfies (17). Finally, using the monotonicity assumptions on f, g in (S), we obtain the uniqueness of y, and the proof of Proposition 1 is completed.