On non-variational solutions to optimal boundary control problems for parabolic equations

Дослiджується задача оптимального керування для лiнiйного параболiчного рiвняння з необмеженими коефiцiєнтами в головнiй частинi елiптичного оператора. Особливiсть даного рiвняння полягає в тому, що матриця потоку є кососиметричною, а її коефiцiєнти належать до простору L. Показано, що поставлена задача керування має єдиний розв’язок, який не можна досягти через границю оптимальних розв’язкiв для L∞апроксимованих задач. Ключовi слова: параболiчне рiвняння, оптимальне керування, варiацiйний роз’язок, необмеженi коефiцiєнти, кососиметрична матриця. Изучается задача оптимального управления для линейного параболического уравнения с неограниченными коэффициентами в главной части эллиптического оператора. Особенность данного уравнения состоит в том, что матрица потока является кососимметричной, а ее коэффициенты принадлежат пространству L. Показано, что данная задача управления имеет единственное решение, которое нельзя достичь через предел оптимальных решений для L∞-аппроксимированных задач. Ключевые слова: параболическое уравнение, оптимальное управление, вариационное решение, неограниченные коэффициенты, кососимметрическая матрица. We study an optimal boundary control problem (OCP) associated to the linear parabolic equation yt − div (∇y +A(x)∇y) = f . The characteristic feature of this equation is the fact that the matrix A(x) = [aij(x)]i,j=1,...,N is skew-symmetric, aij(x) = −aji(x) and belongs to L-space (rather than L∞). We show that under special choice of matrix A and distribution f , a unique solution to the original OCP inherits a singular character of the original matrix A and it can not be attainable by the solutions of the similar OCPs with L∞-approximations of matrix A.


Introduction
We consider the optimal boundary control problem for a parabolic equation with unbounded coefficients. The characteristic feature of this problem is the fact that the stream matrix A(x) is skew-symmetric and its coefficients belongs to L 2 -space (rather than L ∞ ). As a result, the existence, uniqueness, and variational properties of the weak solution to optimal control problem (OCP) usually are drastically different from the corresponding properties of solutions to the parabolic equations with L ∞ -matrices in coefficients. In most cases, the situation can change dramatically for the matrices A with unremovable singularity. Typically, in such cases, boundary value problem may admit infinitely many weak solutions which can be divided into two classes: approximable and non-approximable solutions [5], [12], and [13].
The aim of this work is to consider OCP with a well prescribed skew-symmetric L 2matrix A and, using the direct method in the Calculus of variations, to show that this problem admits a unique solution possessing a special singular properties. As a result, we prove that this solution cannot be attained through a sequence of optimal solutions to regularized OCP for boundary value problem with skew-symmetric matrices A k ∈ L ∞ (Ω; S 3 ) such that A k → A strongly in L 2 (Ω; S 3 ). Thus, this result shows that a numerical analysis of optimal control problems for parabolic equations with unbounded coefficients is a non-trivial matter and it requires the elaboration of special approaches.

Notation and Preliminaries
Let Ω be the unit ball in R 3 , Ω = {x ∈ R 3 : x R 3 < 1}. Let C ∞ 0 (Ω; Γ D ) be the set of all infinitely differentiable functions ϕ : Ω → R with compact supports in Ω. Let C ∞ 0 (R N ; Γ D ) = ϕ ∈ C ∞ 0 (R N ) : ϕ = 0 on Γ D . We define the Banach space H 1 0 (Ω; Γ D ) as the closure of C ∞ 0 (Ω; Γ D ) with respect to the norm (see [1]) Let H −1 (Ω; Γ D ) be the dual space to H 1 0 (Ω; Γ D ). Let X be a Banach space and let T > 0 be a given value. We denote by L 2 (0, T ; X) the set of measurable functions y ∈ (0, T ) → X such that u(·) X ∈ L 2 (0, T ). Similarly, one can also define the set of distributions D (0, T ; X) on (0, T ) with values in X. L 2 (0, T ; X) is a Banach space with respect to the norm × Ω such that y(t, ·) ∈ L 2 (Ω) for any t ∈ [0, T ] and such that the map t ∈ [0, T ] → y(t, ·) ∈ L 2 (Ω) is continuous. Let us define the Banach space equipped with the norm of the graph. Here, the derivative ∂y/∂t is the distribution in D (0, T ; H −1 (Ω; Γ D )). Then the following properties holds true (see [4,10]).
(3) For any u, v ∈ W Γ D , one has Let y ∈ L 2 (0, T ; H 1 0 (Ω; Γ D )) ∩ C ([0, T ]; L 2 (Ω)). Then the following density result holds: Skew-Symmetric Matrices. Let S 3 be the set of all skew-symmetric matrices A = [a ij ] 3 i,j=1 , i.e., A is a square matrix with a ij = −a ji and, hence, a ii = 0. Therefore, the set S 3 can be identified with the Euclidean space R 3 .
We define the divergence div A of a skew-symmetric matrix A ∈ L 2 Ω; S 3 as a vector-valued distribution d ∈ H −1 (Ω; R 3 ) by the following rule where a i stands for the i-th row of the matrix A. We say that a matrix A ∈ L 2 Ω; S 3 belongs to the space H(Ω, div; S 3 ) if d := div A ∈ L 1 (Ω; R 3 ), that is,

Setting and Approximation of the Optimal Control Problem
We deal with the following optimal control problem (OCP) for a parabolic equation with unbounded coefficients subject to the constraints where Ω be the unit ball in R 3 and its boundary Here, u is a control, y 0 ∈ L 2 (Ω), y d ∈ L 2 (0, T ; H 1 0 (Ω)) and u d ∈ L 2 (0, T ; L 2 (Γ N )) and f ∈ L 2 (0, T ; H −1 (Ω; Γ D )) are given distributions, and A ∈ L 2 Ω; S 3 is a skew-symmetric matrix.
The optimal control problem which we consider is to minimize the discrepancy (tracking error) between a given distribution y d ∈ L 2 (0, T ; H 1 0 (Ω)) and a solution y of the Neumann-Dirichlet boundary value problem for parabolic equation (7)-(9) by choosing an appropriate boundary control u ∈ L 2 (0, T ; L 2 (Γ N )), where δ ij is the Kronecker's delta, cos(ν, x i ) is the i-th directing cosine of ν, and ν is the outward unit normal vector at Γ N to the ball Ω.
More precisely, we are concerned with OCP (6)- (10). The distinguishing feature of this problem is the special choice of matrix A and distribution f . This entails a number of pathologies with respect to the standard properties of optimal control problems for parabolic equation and leads to the non-uniqueness of weak solutions to the corresponding initial boundary value problem and a singular properties of an optimal pair. As a result, numerical approximation of the solution to OCP (6)-(10) is getting non-trivial.
Note that the function y = y(u) is called an approximable solution to the initialboundary value problem in (7)-(9) if it can be attained by weak solutions to the similar boundary value problems with L ∞ -approximated matrix A. However, this type of solutions does not exhaust all weak solutions to the above problem. There is another type of weak solutions, which cannot be approximated by weak solutions of such regularized problems. Usually, such solutions are called non-variational [7,12,13], singular [14], [2], [8], pathological [9], [11] and others To begin with, we introduce the following notion.
It is worth to note that in view of definition of the space W and Theorem 1, the condition (11) has a sense. Moreover, as was shown in [3], if (u, y) is an admissible pair, then y ∈ D(A).
As immediately follows from (12) and the definition of bilinear form [y, ϕ] (see also the extension rule (3)), every admissible pair (u, y) ∈ Ξ is related by the following energy equality The next question which we are going to discuss is about variational solutions to the problem (6)- (10). Since A ∈ L 2 Ω; S 3 , it follows that there exists a sequence of skew- Hence, it is reasonably, from numerical point of view, to consider the following sequence of constrained minimization problems associated with matrices A k .
Indeed, for every k ∈ N the bilinear form [y, ϕ] k : and, hence, the initial-boundary value problem (16) has a unique solution (see [10] for the details) for every u ∈ L 2 (0, T ; L 2 (Γ N )).
As an obvious consequence of this observation and the properties of lower semicontinuity and strict convexity of the cost functional I k , we have: the corresponding minimization problem (14) admits a unique solution [6] Moreover, having fixed a control u ∈ L 2 (0, T ; L 2 (Γ N )), condition (19) implies the fulfilment of the following identities for every where y k = y k (u) ∈ L 2 (0, T ; H 1 0 (Ω)) are the corresponding solutions to the initialboundary value problems (16). Hence, following the standard technique [10], it is easy to show that the sequence {y k } k∈N is bounded in W Γ D for every fixed u ∈ L 2 (0, T ; L 2 (Γ N )) and due to the a priori estimates where the constant C(Ω) is independent of A k , we arrive at the relation So, by the completeness of W Γ D , we can assume that there exists a pair (u * , y * ) ∈ L 2 (0, T ; L 2 (Γ N )) × W Γ D such that up to a subsequence Hence, y 0 k −→ y * strongly in L 2 (0, T ; L 2 (Γ N )) (25) by compactness of the embedding H 1/2 (Γ N ) → L 2 (Γ N ). It remains to prove the properties (17). To do so, we note that due to the strong A∇ϕ, ∇y * − ∇y 0 k R N dxdt −→ 0 as k → ∞ for every ϕ ∈ C ∞ ([0, T ]; C ∞ 0 (Ω)). Hence, A k ∇y 0 k * A∇y * in L 1 (0, T ; L 1 (Ω; R 3 )). It means that we can pass to the limit in integral identity (20) with u = u 0 k . As a result, we have: the pair (u * , y * ) is related by the integral identity (13), therefore, y * is a weak solution to the original boundary value problem (7)-(9) under u = u * in the sense of Definition 1. Thus, (u * , y * ) ∈ Ξ. Moreover, following [3], we have y * ∈ D (A).
In order to prove the property (17) 2 , we pass to the limit in the energy equality (21). Takin into account the lower semicontinuity of the norm · 2 L 2 (0,T ;H 1 (Ω)) with respect to the weak convergence ∇y 0 k ∇y * in L 2 (0, T ; L 2 (Ω; R 3 )), we obtain Thus, the desired inequality (17) 2 obviously follows from (13) and (26). The proof is complete.
As we will see later on, the pair (u * , y * ) is not optimal, in general, and the pair (u d , y d ) is a unique optimal pair to OCP (6)- (10). Whereas we will shown that [y d , y d ] = −α, where α is a given strictly positive value, in the mean time [y * , y * ] ≥ 0 for any attainable pair (u * , y * ). Thus, for given f, y d , y 0 , u d the optimal pair (u 0 , y 0 ) to OCP (6)-(10) cannot be attained through any L ∞ -approximation of the matrix A ∈ L 2 Ω; S 3 .

Example of the Non-Variational Solution
Our aim in this section is to show that optimal control problem (6)-(10) has a unique non-variational solution. Namely, we will show that for a given positive scalar value α ∈ R there exist a skew-symmetric matrix A ∈ L 2 Ω; S 3 and a function y d ∈ L 2 (0, T ; H 1 0 (Ω)) such that y d ∈ D(A) and [y d , where the bilinear form [y, v] is defined by (2). We divide our analysis into several steps. At the first step we define a skewsymmetric matrix A as follows where a(x) = . Since it follows that a ∈ L 2 (Ω). By analogy, it can be shown that b ∈ L 2 (Ω). Moreover, it is easy to see that the skew-symmetric matrix A, define by (28), satisfies the property A ∈ H(Ω, div; S 3 ), i.e. A ∈ L 2 (Ω; S 3 ) and div A ∈ L 1 (Ω; R 3 ). Indeed, in view of the definition of the divergence div A of a skew-symmetric matrix, we have div A = and a i is i-th column of A. As a result, we get div a i L 1 (Ω) =ˆ1 0ˆ2 π 0ˆπ 0 ρ 2 f i (ϕ, ψ) sin ϕ sin ψ ρ 4 ρ 2 sin ψ dψ dϕ dρ < +∞, for the corresponding f i = f i (ϕ, ψ) (i = 1, 2, 3). Therefore, div A ∈ L 1 (Ω; R 3 ).
Step 2 deals with the choice of the function y d ∈ L 2 (0, T ; H 1 0 (Ω)). We define it by the rule for all (t, x) ∈ (0, T ) × Ω. It is easy to see that with respect to the spherical coordinates. Hence, v 0 ∈ C 1 (∂Ω), and, as immediately follows from (29), it provides that By direct computations, we get Hence, there exists a constant C * > 0 such that Thus, As a result, we infer that ∇y d ∈ L 2 (0, T ; L 2 (Ω; R 3 )), i.e. we finally have y d ∈ L 2 (0, T ; H 1 0 (Ω)).
Step 3. We show that the function y d , which was introduced before, belongs to the set D(A). To do so, we have to prove the estimate ˆT , for all ϕ ∈ C ∞ ([0, T ]; C ∞ 0 (Ω)). To this end, we make use of the following transformationŝ which are obviously true for all ψ, ϕ ∈ C ∞ ([0, T ]; C ∞ 0 (Ω)). Since ˆT , it follows that, using the continuation principle, we can extend the previous equality with respect to ψ to the following onê Let us show that (div A, ∇y d ) R 3 ∈ L ∞ ((0, T ) × Ω). In this case, relation (32) implies the estimate ˆT which means that the element y d belongs to the set D(A). Indeed, as follows from (31), we have the equality Thus, the gradient of the function ∇v 0 ( x x R 3 ) is orthogonal to the vector field Q = x/ x 3 R 3 outside the origin. Therefore, where I 2 = 0 by (34). Since x R 3 = sin ϕ sin ψ with respect to the spherical coordinates, and function v 0 is smooth, it follows that there exists a constant C 0 > 0 such that |(∇y d , div A) R 3 | ≤ C 0 almost everywhere in (0, T )×Ω. Thus, and we have obtained the required property.
Then by continuity, we have Since (div A, ∇y d ) R 3 ∈ L ∞ ((0, T ) × Ω), in view of the property (35), we can pass to the limit in the right-hand side of this relation. As a result, we get Let Ω ε = {x ∈ R 3 | ε < x R 3 < 1} and let Γ ε = { x R 3 = ε} be the sphere of radius ε centered at the origin. Then where b 0 = sin ϕ sin ψ and v 2 0 = 52α it remains to combine this result with (36), (37), and relation T 0ˆΩ div A, ∇y 2 d R 3 dxdt = lim ε→0ˆT 0ˆΩε div A, ∇y 2 d R 3 dxdt.