Second-Order Necessary Conditions for Optimal Control of Semilinear Elliptic Equations with Leading Term Containing Controls

An optimal control problem for a semilinear elliptic equation of divergence form is considered. Both the leading term and the semilinear term of the state equation contain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

For problems of elliptic PDEs with control appearing in the leading term, Casas [5] studied the first-order necessary conditions for the case A(x, u) = uI with quadratic cost functional and with the control being Lipschtz continuous. General case were treated by Lou-Yong in [26], and analogous results for parabolic and hyperboliccases were given by Lou in [25] and Li-Lou in [22]. If the leading term of the state equation (1.1) does not contain controls, i.e., A(x, u) ≡ A(x), then one can establish the second-order necessary conditions for partially singular optimal controls following similar arguments of [24]. However, if the leading term of the equation contains the control, we will see that it is much more complicated, even in defining the partial singularity of the optimal control. It turns out that the construction of a proper family of perturbations is much more difficult than the case without having control in the leading term, in order to have the first-order term disappeared in the Taylor type expansion. The difficult will be overcome by introducing the notion of weak singularity which involves a proper vector field. Consequently, the results obtained will have some big difference comparing with those for the problems without having the control in the leading term.
The rest of the paper is organized as follows. In Section 2, we will introduce the notions of singularity and weak singularity of the optimal controls. The main result of the paper will be stated, together with a couple of corollaries. Section 3 will be devoted to a review of the proof for the first-order necessary condition for Problem (C), which will inspire the second-order necessary condition. Section 4 is devoted to a proof of a result crucial for the proof our main result. A proof of the second-order necessary condition will be presented in Section 5.
(S1) Set Ω is a bounded domain in R n (n 2) with a smooth boundary ∂Ω, and metric space (U, ρ) is separable.
The equality in (2.7) follows from the following simple fact (see [26], Lemma 2.3, for a proof).
We now introduce the following definition.
Let us make some observations on the above notions.
If A(x, v) is independent of v ∈ U , then the right hand side of (2.11) is automatically zero, and (2.12) is true. Thus, in such a case, weak singularity is equivalent to singularity, and V 0 (ū(·)) = U 0 (ū(·)) × L.
To state our main result of the current paper, the second-order necessary condition for optimal control of Problem (C), we need the following further assumption.
(S5) Function y → (f (x, y, v), f 0 (x, y, v)) is twice continuously differentiable. Moreover, for any R > 0, there exists a K R > 0 such that We point out that, unlike most of the literature on PDE controls that we cited, no differentiability condition is assumed for the map u → (f (x, y, u), f 0 (x, y, u)). Actually, our U is just a metric space which does not have a linear structure, in general. In particular, no convexity condition is assumed for U . Now, we state our main result of this paper.

The First-Order Necessary Condition Revisited
In this section, we briefly recall the proof of Theorem 2.1, from which we will find a correct direction approaching the second-order necessary condition for the optimal control. To this end, we first recall the following lemma ([1]).
(ii) There exists two constants Λ > λ > 0, such that (iii) The following holds: Let g ∈ H −1 (Ω) and y ε (·) be the solution of in Ω, Then y ε (·) converges weakly to y(·) in H 1 0 (Ω) where y(·) solves in Ω, Observe that Hence, Also, (3.6) can be written as Note that in general G(x, x ε ) does not necessarily converge strongly in L 2 (Ω) (as ε ↓ 0). Therefore, the above lemma is by no means trivial or obvious. On the other hand, the following result is much easier, which will also be used later, for different situations.
To further characterize the optimal control, the second-order necessary condition will be needed.

Proof of Proposition 3.3.
In this section, we will present a proof of Proposition 3.3. Let us begin with the following lemma.
Proof. The lemma is a consequence of Theorem 1.3.14 and Lemma 1.3.32 in [2]. Here we give a direct proof of it. Let Then C ∈ S n + , and Thus, (4.2) Consequently, we get (4.1) since proving our conclusion. ✷ The following lemma will play an interesting role blow.
Lemma 4.2. Let ν = (ν 1 , · · · , ν n ) ⊤ ∈ Z n \{0} (where Z is the set of all integers). Then for any α ∈ (0, 1), is the decimal part of the real number a. Note that If some of integers ν k are zero, we could drop the corresponding terms and reduce the dimension of z.
Thus, we assume all ν k are non-zero. Also, if some ν k < 0, we may replace corresponding z k by (1 − z k ). Therefore, we may let all ν k > 0. Next, we observe the following (noting the [0, 1] n -periodicity of the maps z → ν, z ): Hence, we need to prove the following: Let us use induction. For n = 1, the above is clearly true. Suppose the above holds for n − 1. Then, for the n-dimensional case, we observe the following: For z 1 ∈ [0, 1), Then for z 1 ∈ [0, α), the following holds if and only if either That is, either Note that the above two cases are mutually exclusive (since α − z 1 < 1 − z 1 ). On the other hand, for Thus the following holds: if and only if That is, Hence, by induction hypothesis, This completes the proof. ✷ The following gives a crucial convergence of the weak solution to the state equation under a suitable perturbation of the leading coefficient.
This yields and for anyz ∈ R n withz 1 ∈ [0, α), As a result, Thus,z 1 → ϕ(x,z 1 e 1 ) is 1-periodic. On the other hand, for any z ∈ [0, 1] n , Hence, Therefore, making use of Lemma 4.2, one has Now, we simplify the expression of G(x), suppressing x, Likewise, Hence, This means that for any x ∈ E k , (4.17) The proof is completed.

Second-Order Necessary Conditions.
In this section, we are going to prove Theorem 2.4. For readers' convenience, we will rewrite the relevant equations when needed. We first establish the following lemma.
This completes the proof. ✷ 6 Concluding Remarks.
We have established the second-order necessary conditions for the optimal controls of Problem (C). There are some challenging problems left open. We list some of them here, for which we are still working on with our great efforts.
• Construction of suitable examples for which our second-necessary conditions could lead to some optimal solutions.
• The second-order necessary conditions that we obtained looks complicated. Is it possible to have some better forms?
• Extension to fully non-linear equations.
We hope to be able to report some further results before long. Also, any participation of other interested researchers are welcome.