On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation

In this paper, an optimal control problem for Schrodinger equation with complex coefficient which contains gradient is examined. A theorem is given that states the existence and uniqueness of the solution of the initial-boundary value problem for Schrodinger equation. Then for the solution of the optimal control problem, two different cases are investigated. Firstly, it is shown that the optimal control problem has a unique solution for \begin{document}$α >0$\end{document} on a dense subset $G$ on the space $H$ which contains the measurable square integrable functions on \begin{document}$\left(0,l\right)$\end{document} and secondly the optimal control problem has at least one solution for any \begin{document}$α ≥ 0$\end{document} on the space $H$.

1. Introduction. A great number of theoretical concepts governed by Schrödinger equation [5]- [9] have been examined in the area of optimal control problems. These problems have emerged as a result of the examination process of one dimensional loaded mass' disintegration [3]. Different approaches for these problems have been examined [19]- [10], [1]. The statement of the problem studied in this paper seriously differs from other studies. In this problem, we examined Schrödinger equation with complex coefficient which contains gradient and used Lions functional as cost functional. Optimal control problems with Lions functional have been examined for linear and nonlinear Schrödinger equations with complex coefficient which do not contain gradient in the study of [11]- [16]. As optimal control problems involving gradient have not been studied, this paper plays a significant role both in practical and theoretical aspects.
The methodology of the study follows as: In section 2, we state problem and recall some known spaces. Then generalized solution of the problem is presented and for this solution estimates are given. In section 3, existence and uniqueness theorems particular to solution of this problem are proved. The results are given in section 4.
2. Statement of the optimal control problem. Let l > 0, T > 0 be given is a Banach space the elements of which are the functions defined on [0, T ], k ≥ 0 times continuously differentiable and their values belong to the B Banach space. Here, norm is given as below For k ≥ 0 , m ≥ 0, W k p (0, l) and W k,m p (Ω) are Sobolev spaces defined as in the study [12].
3. Existence and uniqueness of solution of the optimal control problem. Now, existence and uniqueness of the solution for (2)-(6) optimal control problem will be researched. Primarily, we will prove uniqueness of the solution for α > 0.
The following theorem can be written from [4].
Theorem 3.1. Let us accept thatX is a uniformly convex space, U is a closed, bounded set ofX space, I(v) is a lower bounded and lower semi continuous functional defined on U and α > 0, β ≥ 1 are given numbers. Then there is such an almost dense subset G inX space that the functional has its minimum value on the set of U for ∀ω ∈ G. If β > 1, the functional J α (v) has its minimum value at a unique point on U [4].
Proof. Firstly we will prove the continuity of functional on the set of V . Let ∆v ∈ H be an increment on ∀v ∈ V element such that v + ∆v ∈ V . Then the functionsψ , k=1,2 functions are the solutions of (3)- (6) for v ∈ V and v + ∆v ∈ V respectively. From the conditions (3)- (6), it is clear that the functions ∆ψ k = ∆ψ k (x, t), k = 1, 2 are the solutions of the following initial boundary value problem: ∆ψ k (x, 0) = 0, k = 1, 2, x ∈ (0, l), Let us obtain an estimate for the solution of this problem. To this end, by multiplying both sides of equation (3) with ∆ψ (x, t) and integrating over Ω T , we get following equality: In the last equality, if we apply partial integration to the second term in the left side by using (5),(6) boundary conditions, the following equality is acquired: When the complex conjugate of the above equality is subtracted from itself, the following equality is obtained: Let us add the term of Ωt da1(x) dx |∆ψ k | 2 dxdτ on and subtract it from the both sides of this equality. Then we get following equality: ∀t ∈ [0, T ], k = 1, 2. Second term in the left side of this equality converts to zero under the conditions ∆ψ 1 (0, t) = ∆ψ 1 (l, t), a 1 (0) = a 1 (l) = 0. Considering this fact and using the conditions ∆ψ k (x, 0) = 0, k = 1, 2, we can write the last equality as follows: ∀t ∈ [0, T ], k = 1, 2.
When the condition (9) is used for a 1 (x) and the Cauchy-Bunjakovskii inequality is employed, following inequality is obtained: Let us take the second term of the right side of inequality (25). Since ∆v ∈ L 2 (0, l),the term can be written as follows: Since ψ 1 ∈ 0 W 2,1 2 (Ω) and ψ 2 ∈ W 2,1 2 (Ω), according to Embedding theorem ψ 1 ∈ L 2 (0, T ; 0 W 1 2 (0, l)) and ψ 2 ∈ L 2 (0, T ; W 1 2 (0, l)) relations are valid. The spaces 0 W 2,1 2 (0, l) and W 1 2 (0, l) are embedded into C[0, l]. So, we can write following inequalities: where the constants c 3 >0, c 4 > 0 are independent from ψ 1 , ψ 2 , respectively. Using these inequalities, the estimates (12), (13) and the inequality (26), we obtain following inequality: If we take into account this inequality in (25), we obtain: ∀t ∈ [0, T ], k = 1, 2. By applying Gronwall lemma, we get following estimate: ∀t ∈ [0, T ] , k = 1, 2. If (15) is used to evaluate increment of the functional J 0 (v) for∀v ∈ V , we obtain increment of the functional as shown below: Here applying the Cauchy-Bunjakovskii inequality and using the estimates (12), (13) in (32), we obtain following inequality: (Ω) . (33) If we use the estimate (31) in the last inequality, we obtain following relation: where c 9 >0 is independent from ∆v. From the last relation, it is observed that the functional is continuous for ∀v ∈ V . Namely |∆J 0 (v)| → 0, for ∆v L2(0,l) → 0. Since (34) inequality is valid for ∀v ∈ V , the functional J 0 (v) is continuous on the set V . On the other hand, J 0 (v) ≥ 0 condition is satisfied for ∀v ∈ V . Namely, the functional J 0 (v) is lower bounded on the set V . Since the set V is closed, bounded and convex on the space L 2 (0, l)and the space H =L 2 (0, l) is a smooth convex space [16], all the conditions of Theorem 2 hold. Thus, in accordance with this theorem, there is a dense set G ⊂ H, such that (2)-(6) optimal control problem has a unique solution for every ω ∈ G , α > 0. Theorem 3 is proved.

Conclusion.
As a result, we obtain the existence and uniqueness of the solution of an optimal control problem for a Schrödinger equation. Because of the difference of the considered linear Schrödinger equation and conditions, this study is different from previous studies in the literature. As the different from the others, we examined a one-dimensional linear Schrödinger equation which contains a gradient term with complex coefficient. Consequently the goal has been successfully achieved