A Necessary Optimality Condition on the Control of a Charged Particle

: We consider an optimal control problem with the boundary functional for a Schrödinger equation describing the motion of a charged particle. By using the existence of an optimal solution, we search the necessary optimality conditions for the examined control problem. First, we constitute an adjoint problem by a Lagrange multiplier that is related to constraints of theory on symmetries and conservation laws. The adjoint problem obtained is a boundary value problem with a nonhomo-geneous boundary condition. We prove the existence and uniqueness of the solution of the adjoint problem. Then, we demonstrate the differentiability of the objective functional in the sense of Frechet and get a formula for its gradient. Finally, we give a necessary optimality condition in the form of a variational inequality.


Introduction
The foundations of control theory date back to old times [1][2][3][4], and it has various applications in many fields, such as population dynamics, epidemiology, resource and energy economy, environmental management, optics, communication theory, medical imaging, and astronomy [5][6][7][8][9].In recent years, the necessity of using natural resources, material and technical tools, time, energy, etc., more efficiently has led to an increased relevance of optimal control problems (OCPs).
As we know, an OCP is described by three essential items.The first is the state equation, which describes the behavior of the controlled system.The second is the set of admissible controls, which contains specified functions that take their values in the defined set.The third is a functional of the controls and state variables, which is called the objective function and is determined in accordance with the purpose of the controlled system.When an OCP is examined, first, the existence and uniqueness of the solution of the problem is investigated, then, the necessary and sufficient conditions for the solution are examined, which allows us to provide a method to characterize and find the solutions of the OCPs.
The objective functional and the set of admissible controls are considered as respectively, where W ≡ L 2 (X) × L 2 (X), κ(s, t) ∈ L 2 (S), ω ∈ W, α ≥ 0 is the Tikhonov regularization parameter [10].L 2 (X) is the space of all Lebesgue functions, the squares of which the moduli are integrable over X, W 2 2 (X) is the space of all functions u ∈ L 2 (X) having the generalized derivatives .
Similarly, we denote by W 0,1 2 (Ω) the spaces of all the functions u(x, t) ∈ L 2 (Ω) having first-order generalized partial derivative with respect to variable t in L 2 (Ω) with the norm .
The detailed descriptions of these spaces can be found in [11,12].Thus, we express the OCP investigated in this paper as the problem of finding the minimum of functional (6) on the set Q ad under conditions (1) and (2).
As seen, problems (1) and ( 2) are a Neumann problem, and its solution is defined in the following sense: Definition 1.A solution u of problems (1) and (2) for each q ∈ Q ad is defined to be an element of B 1 ≡ C 0 (Λ T , W 2 2 (X)) ∩C 1 (Λ T , L 2 (X)) satisfying Equation (1) f or a.a x ∈ X and f or any t ∈ Λ T and Equation (2) f or a.a x ∈ X and f or a.a (s, t) ∈ S, where C k (Λ T , Y) for k = 0, 1 is a Banach space of all functions u : Λ T → Y, for which all the derivatives up to order k are continuous Based on the results in [13][14][15], we give the following theorem for problems (1) and ( 2): Theorem 1.If the functions r, p, f , h satisfy conditions (3)-( 5), then problems (1) and ( 2) have a unique solution u in B 1 .Moreover, for any q ∈ Q ad , u satisfies the estimate for any t ∈ Λ T , where the constant c 0 > 0 does not depend on f , h, t.
Also, based on the results in [15], we write the following theorems for the existence of an optimal solution: Theorem 2. If the conditions of Theorem 1 are satisfied, then there exists a unique solution of the OCP on a dense subset W 0 ⊂ W for any ω ∈ W 0 and α > 0 .Theorem 3. If the conditions of Theorem 1 are satisfied, then for any ω ∈ W and α ≥ 0, the OCP has at least one solution.
There is a large amount of research on OCPs for Schrödinger equations without any specific gradient terms: for instance, in [16], the authors demonstrate the existence of an optimal control for the cubic nonlinear Schrödinger equation (NLSE) and give the optimality conditions.In [17], the authors study an OCP with a final functional for a standard linear Schrödinger equation (LSE), give an existence theorem for OCP, and also derive the necessary optimality conditions.In [18], the author gives the results about the internal controllability of the LSE and NLSE.
In [19], the necessary and sufficient conditions for the solution of a bilinear OCP for the LSE are obtained.In [20], the optimality conditions for an LSE with a singular potential are given.In [4,21,22], the authors study OCPs for LSEs.In [22][23][24], the authors prove the existence of solutions of OCPs for systems governed by NLSEs and give the necessary optimality conditions.
As can be seen, all of the aforementioned works are concerned with OCPs for standard Schrödinger equations (linear or nonlinear), that is, the Schrödinger equation does not contain any specific gradient term.But in [13], the authors prove the existence of the optimal solution for an OCP with a Lions-type functional for the LSEwSGT.In [25,26], the existence of the optimal solution and necessary optimality conditions are given for OCPs with a final functional for the NLSEwSGT.Salmanov [27] gives the existence and uniqueness theorems for a solution of an OCP with a Lions-type functional for the NLSEwSGT.
It should be noted here that the OCPs with a boundary functional for the LSEwSGT have been hardly analyzed.In [15,28], the authors demonstrate the existence of optimal solutions for OCPs with a boundary functional for the LSEwSGT.
In the present work, we search the necessary optimality conditions for the OCP with a boundary functional (6) on the admissible controls set for state Equation (1).For this purpose, first, we constitute an adjoint problem.Then, we prove the existence and uniqueness of the solution of the adjoint problem.Later, by showing the differentiability of functional (6) in the sense of Frechet, we obtain a formula for its gradient.Finally, we give a necessary optimality condition in the variational form.

Adjoint Problem
In the current section, we constitute an adjoint problem to investigate the differentiability of functional (6).By using a Lagrange multiplier function, we obtain the adjoint problem as follows: where u = u(x, t) ≡ (x, t; q) is a solution of problems ( 1) and ( 2) for any q ∈ Q ad .

The Differentiability of the Objective Functional
In this section, we show that the objective functional J α is differentiable in the meaning of Frechet and get a formula for its gradient with the help of the adjoint problem.

Theorem 7.
Let ω ∈ W be a given function.If the conditions of Theorem 6 are satisfied, then J α is differentiable in the meaning of Frechet on Q ad , and moreover, its gradient is given by the formula where u, Φ are the solutions of problems (1) and ( 2) and ( 8)-( 10) corresponding to q ∈ Q ad , respectively.
Proof.From (6), the enhancement δJ α (q) = J α (q + δq) − J α (q) of J α (q) for any q ∈ Q ad is written as follows where δq = (δq 0 , δq 1 ) is an enhancement given to any q ∈ Q ad such that q + δq ∈ Q ad , and the function δu is a solution of the following problem [15]: where In (86), using the Cauchy-Schwarz inequality, we get In ( 87), if we use the inequalities in [15] and estimates ( 7) and (72), we achieve which shows that R(δq) = o(∥δq∥ B ), where the symbol o(∥δq∥ B ), pronounced "small oh" of ∥δq∥ B , means something for which its ratio with ∥δq∥ B has limit 0, that is, lim where the constants β 1 , β 2 , c 8 > 0 are independent from δq and τ.Thus, by the definition of the differentiability of a functional on closed set Q ad [31], from (85), we can write which implies that J α (q) is a differentiable functional in the meaning of Frechet on Q ad , and its gradient is given by Thus, the proof of Theorem 7 is completed.

The Necessary Optimality Condition
In the last section, we give a necessary optimality condition in the variational form.
Theorem 8. Assume that Theorem 7 holds, and let q * ∈ Q ad be any solution of OCP.Then, for any q ∈ Q ad , the inequality is valid, where u * , Φ * are solutions of problems ( 1) and ( 2) and ( 8)-( 10) corresponding to q * ∈ Q ad , respectively.
Proof.Let q ∈ Q ad be any control and q * ∈ Q ad be solution of OCP, that is, let q * ∈ Q ad be any optimal control.Firstly, let us prove that Q ad is a convex subset of L 2 (X) × L 2 (X).
Dividing both sides of (91) by θ > 0 and then taking the limit as θ → +0, we get J ′ α (q * ), (q − q * ) B ≥ 0 f or any q ∈ Q ad . (92) Considering Formula (75) for q = q * and the integral representation of a linear functional in space B = L ∞ (X) × L ∞ (X), we deduced from (92) that relation (89) is valid, which completes the proof of Theorem 8.

Conclusions
In this work, we present a necessary optimality condition for the problem of controlling a charged particle.We regard an n-dimensional LSEwSGT as the state equation and a boundary functional as the objective functional.We have obtained an adjoint problem with a nonhomogeneous boundary condition.By transforming the adjoint problem into a boundary value problem with a homogeneous boundary condition, we have proved the existence of the solution of the adjoint problem.Also, we have shown that the objective functional is Frechet differentiable.Finally, by proving the convexity of the admissible controls set and by using the results on the existence of the optimal solution, we have produced a necessary optimality condition.
In the literature, OCPs with boundary functionals have been barely studied, and the admissible controls set studied in the present paper contains complex-valued functions whose real and imaginary parts are the measurable bounded functions, which shows that this work is a generalization of previous works.