A Necessary Optimality Condition on the Control of a Charged Particle

Abstract

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 nonhomogeneous 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.

1. 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.

In the present paper, we consider the state equation

$$i\frac{\partial u}{\partial t}+\chi \mathrm{\Delta }u+ip(x,t)\nabla u-r\left(x\right)u+q_0\left(x\right)u+i{q}_1\left(x\right)u=h(x,t)$$

(1)

with the initial and boundary conditions

$$u(x,0)=f\left(x\right),x\in X,{\left.\frac{\partial u}{\partial \nu }\right|}_S=0,$$

(2)

where $u=u(x,t)$, $X\subset$ ${\mathbb{R}}^n$ is a convex bounded domain with a smooth boundary $\mathrm{\Gamma },$ $x=(x_1,x_2,...,x_n)\in X,$ $t\in {\mathrm{\Lambda }}_T=\left[0,T\right],$ $(x,t)\in \mathrm{\Omega },$ $\mathrm{\Omega }=X\times \mathrm{\Lambda },$ $\mathrm{\Lambda }=\left(0,T\right),i=\sqrt{-1}$ $T,\chi >0$ are given numbers, $S=\mathrm{\Gamma }\times \mathrm{\Lambda }$ is the lateral surface of $\mathrm{\Omega }$, $\mathrm{\Delta }={\displaystyle \sum _{j=1}^n}\frac{{\partial }^2}{\partial x_j^2}$ is the Laplacian, $\nabla =\left(\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},...,\frac{\partial }{\partial x_n}\right)$ is the gradient, and $\nu$ is the outward unit normal to $\mathrm{\Gamma }.$ Also, $r\left(x\right)$ is a real-valued function satisfying

$$0<{\mu }_0\le r\left(x\right)\le {\mu }_{1}foralmostall(a.a)x\in X,{\mu }_0,{\mu }_1=const.>0.$$

(3)

Further, $p(x,t)=\left(p_1(x,t),p_2(x,t),...,p_n(x,t)\right),$ where the functions $p_j$ for $j=1,2,...,n$ are real-valued measurable functions such that

$$\left.\begin{array}{c}\left|p_j(x,t)\right|\le {\mu }_2,\left|\frac{\partial p_j(x,t)}{\partial x_k}\right|\le {\mu }_3,\left|\frac{\partial p_j(x,t)}{\partial t}\right|\le {\mu }_{4}fora.a\left(x,t\right)\in \mathrm{\Omega },\\ {\left.p\right|}_S=0,{\mu }_2,{\mu }_3,{\mu }_4=const.>0,k=1,2,...,n,\end{array}\right\}$$

(4)

where $f,$ h are complex-valued functions satisfying the conditions

$$f\in W_2^2\left(X\right),{\left.\frac{\partial f}{\partial \nu }\right|}_{\mathrm{\Gamma }}=0,h\in W_2^{0,1}\left(\mathrm{\Omega }\right).$$

(5)

It is obvious that Equation (1), unlike the standard Schrödinger equation, contains the gradient term $ip(x,t)\nabla u$, and this equation is called a linear Schrödinger equation with a specific gradient term (LSEwSGT) [4].

The objective functional and the set of admissible controls are considered as

$$J_{\alpha }\left(q\right)=\left|\right|u-{\kappa \left|\right|}_{L_2\left(S\right)}^2+\alpha \left|\right|q-{\omega \left|\right|}_W^2,$$

(6)

$$Q_{ad}\equiv \left\{\begin{array}{c}q=\left(q_0\left(x\right),q_1\left(x\right)\right):q_0\in L_2\left(X\right),q_1\in L_2\left(X\right),\\ \left|q_0\left(x\right)\right|\le b_0,\left|q_1\left(x\right)\right|\le b_{1}fora.ax\in X,b_0,b_1=const.>0\end{array}\right\},$$

respectively, where $W\equiv L_2\left(X\right)\times L_2\left(X\right),$ $\kappa (s,t)\in L_2\left(S\right),$ $\omega \in W,$ $\alpha \ge 0$ is the Tikhonov regularization parameter [10]. $L_2\left(X\right)$ is the space of all Lebesgue functions, the squares of which the moduli are integrable over $X,$ $W_2^2\left(X\right)$ is the space of all functions $u\in L_2\left(X\right)$ having the generalized derivatives $\frac{{\partial }^{{\alpha }_1+{\alpha }_2+...+{\alpha }_n}}{\partial x_1^{{\alpha }_1}\partial x_2^{{\alpha }_2}...\partial x_n^{{\alpha }_n}}u$ in $L_2\left(X\right)$ for all nonnegative integers ${\alpha }_1,{\alpha }_2,...,{\alpha }_n,$ ${\alpha }_1+{\alpha }_2+...+{\alpha }_n\le 2,$ equipped with the norm

$${∥u∥}_{W_2^2\left(X\right)}={\left(\sum _{{\alpha }_1+{\alpha }_2+...+{\alpha }_n\le 2}\underset{X}{\int }{\left|\frac{{\partial }^{{\alpha }_1+{\alpha }_2+...+{\alpha }_n}}{\partial x_1^{{\alpha }_1}\partial x_2^{{\alpha }_2}...\partial x_n^{{\alpha }_n}}u\right|}^{2}dx\right)}^{\frac{1}{2}}.$$

Similarly, we denote by $W_2^{0,1}\left(\mathrm{\Omega }\right)$ the spaces of all the functions $u(x,t)\in L_2\left(\mathrm{\Omega }\right)$ having first-order generalized partial derivative with respect to variable t in $L_2\left(\mathrm{\Omega }\right)$ with the norm

$${∥u∥}_{W_2^{0,1}\left(\mathrm{\Omega }\right)}={\left(\underset{\mathrm{\Omega }}{\int }\left({\left|u\right|}^2+{\left|u_t\right|}^2\right)dxdt\right)}^{\frac{1}{2}}.$$

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 $q\in Q_{ad}$ is defined to be an element of $B_1\equiv C^0({\mathrm{\Lambda }}_T,W_2^2\left(X\right))$ $\cap C^1({\mathrm{\Lambda }}_T,L_2\left(X\right))$ satisfying Equation (1) $for$ $a.a$ $x\in X$ and $for$ $any$ $t\in {\mathrm{\Lambda }}_T$ and Equation (2) $for$ $a.a$ $x\in X$ and $for$ $a.a$ $\left(s,t\right)\in S$, where $C^k({\mathrm{\Lambda }}_T,Y)$ for $k=0,1$ is a Banach space of all functions $u:{\mathrm{\Lambda }}_T\to Y$, for which all the derivatives up to order k are continuous in ${\mathrm{\Lambda }}_T$ with the norm ${\left|\right|u\left|\right|}_{C^k({\mathrm{\Lambda }}_T,Y)}={\displaystyle \sum _{m=0}^k}{\displaystyle \underset{0\le t\le T}{max}}\left|\right|\frac{d^{m}u\left(t\right)}{d{t}^m}{\left|\right|}_Y$.

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 $any$ $q\in Q_{ad},$ u satisfies the estimate

$${\left|\right|u(.,t)\left|\right|}_{W_2^2\left(X\right)}^2+{∥\frac{\partial u}{\partial t}∥}_{L_2\left(X\right)}^2\le c_0\left({\left|\right|f\left|\right|}_{W_2^2\left(X\right)}^2+{\left|\right|h\left|\right|}_{W_2^{0,1}\left(\mathrm{\Omega }\right)}^2\right)$$

(7)

for $any$ $t\in {\mathrm{\Lambda }}_T,$ 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 $W_0\subset W$ for $any$ $\omega \in W_0$ and $\alpha >0$.

Theorem 3.

If the conditions of Theorem 1 are satisfied, then for $any$ $\omega \in$W and $\alpha \ge 0,$ 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.

2. 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:

$$i\frac{\partial \mathrm{\Phi }}{\partial t}+\chi \mathrm{\Delta }\mathrm{\Phi }+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t)\mathrm{\Phi }\right)-r\left(x\right)\mathrm{\Phi }+q_0\left(x\right)\mathrm{\Phi }-i{q}_1\left(x\right)\mathrm{\Phi }=0$$

(8)

$$\mathrm{\Phi }(x,T)=0,x\in X$$

(9)

$${\left.\frac{\partial \mathrm{\Phi }}{\partial \nu }\right|}_S=\frac{2}{\chi }\left(u(s,t)-\kappa (s,t)\right),(s,t)\in S,$$

(10)

where $u=u(x,t)\equiv (x,t;q)$ is a solution of problems (1) and (2) for any $q\in Q_{ad}.$

Definition 2.

A solution Φ for problems (8)–(10) is defined to be an element of $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ satisfying the integral identity

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\left[\mathrm{\Phi }\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-ip\nabla {\overline{\eta }}_1-r{\overline{\eta }}_1+q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)\right]dxdt\hfill \\ & =& -2\underset{S}{\int }\left(\left(u(s,t)-\kappa (s,t)\right)\right){\overline{\eta }}_1(s,t)dsdt\hfill \end{array}$$

(11)

$for$ $any$ ${\eta }_1\in W_2^{2,1}\left(\mathrm{\Omega }\right)$ such that ${\eta }_1(x,0)=0$ $for$ $a.a$ $x\in X,$ ${\left.\frac{\partial {\eta }_1}{\partial \nu }\right|}_S=0.$

For convenience, let us denote $g(s,t)=\frac{2}{\chi }\left(u(s,t)-\kappa (s,t)\right).$ Thus, we rewrite problems (8)–(10) as

$$i\frac{\partial \mathrm{\Phi }}{\partial t}+\chi \mathrm{\Delta }\mathrm{\Phi }+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j\mathrm{\Phi }\right)-r\mathrm{\Phi }+q_0\mathrm{\Phi }-i{q}_1\mathrm{\Phi }=0$$

(12)

$$\mathrm{\Phi }(x,T)=0,x\in X$$

(13)

$${\left.\frac{\partial \mathrm{\Phi }}{\partial \nu }\right|}_S=g(s,t),(s,t)\in S.$$

(14)

As seen above, problems (12)–(14) are a boundary value problem with a nonhomogeneous boundary condition. Firstly, we turn problems (12)–(14) into a problem with a homogeneous boundary condition. By using the method in [22], we write problems (12)–(14) as

$$i\frac{\partial w}{\partial t}+\chi \mathrm{\Delta }w+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_{j}w\right)-rw+q_{0}w-i{q}_{1}w=F$$

(15)

$$w(x,T)=0,x\in X$$

(16)

$${\left.\frac{\partial w}{\partial \nu }\right|}_S=0,(s,t)\in S,$$

(17)

where $F=F(x,t)=\left(1-i\right)\frac{\partial z}{\partial t}-i{\displaystyle \sum _{j=1}^n}\frac{\partial }{\partial x_j}\left(p_{j}z\right)-q_{0}z+i{q}_{1}z,$ $w=\mathrm{\Phi }(x,t)-z(x,t)$, and z is a solution of the problem

$$\frac{\partial z}{\partial t}+\chi \mathrm{\Delta }z-r\left(x\right)z=0$$

(18)

$$z(x,T)=0,x\in X$$

(19)

$${\left.\frac{\partial z}{\partial \nu }\right|}_S=g(s,t),(s,t)\in S.$$

(20)

Also, $z(x,t)=z_1(x,t)+i{z}_2(x,t)=Rez+iImz,$ $g(s,t)=g_1(s,t)+i{g}_2(s,t)=Reg+iImg$, and the functions $z_1,$ $z_2$ are the solutions of problems (18)–(20) corresponding to the boundary conditions $g_1$ and $g_2,$ respectively.

In addition to the conditions (3)–(5), let $\frac{{\partial }^2{p}_j}{\partial x_k\partial t}$ $\in L_{\infty }\left(\mathrm{\Omega }\right),$ $j,k=1,2,...,n,$ $r\in C\left(\overline{X}\right),$ $\kappa \in W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right),$ where $W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)$ is a Banach space, and the norm in $W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)$ is defined by

$$\begin{array}{ccc}\hfill {∥\kappa ∥}_{W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)}& =& {\left(\underset{0}{\overset{T}{\int }}\underset{\mathrm{\Gamma }}{\int }\underset{\mathrm{\Gamma }}{\int }{\left|\kappa (x,t)-\kappa (y,t)\right|}^2\frac{1}{{\left({∥x-y∥}_{{\mathbb{R}}^n}\right)}^n}dydxdt\right)}^{\frac{1}{2}}\hfill \\ & & +{\left(\underset{\mathrm{\Gamma }}{\int }\underset{0}{\overset{T}{\int }}\underset{0}{\overset{T}{\int }}{\left|\kappa (x,t)-\kappa (x,\theta )\right|}^2{\left|t-\theta \right|}^{\frac{-3}{2}}d\theta dtdx\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Thus, since $B_1\equiv C^0({\mathrm{\Lambda }}_T,W_2^2\left(X\right))$ $\cap C^1({\mathrm{\Lambda }}_T,L_2\left(X\right))\subset W_2^{2,1}\left(\mathrm{\Omega }\right)\subset W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)$, according to the embedding theorem in [29], $u\in W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)$ for the solution u of problems (1) and (2). Hence, since $g(s,t)=\frac{2}{\chi }\left(u(s,t)-\kappa (s,t)\right),$ $g\in W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right).$

Also, by changing the variable $\tau =T-t$, we rewrite problems (18)–(20) as

$$-\frac{\partial \tilde{z}}{\partial \tau }+\chi \mathrm{\Delta }\tilde{z}-r\left(x\right)\tilde{z}=0$$

(21)

$$\tilde{z}(x,0)=0,x\in X$$

(22)

$${\left.\frac{\partial \tilde{z}}{\partial \nu }\right|}_S=\tilde{g}(s,\tau ),(s,\tau )\in S,$$

(23)

where $\tilde{z}(x,\tau )=z(x,T-\tau )=z(x,t),$ $\tilde{g}(s,\tau )=g(s,T-\tau )=g(s,t).$

Based on the results in [11,22,29], with the assumed conditions, we can easily say that problems (21)–(23) have a unique solution $\tilde{z}$ in $W_2^{2,1}\left(\mathrm{\Omega }\right)$, and

$${∥\tilde{z}∥}_{W_2^{2,1}\left(\mathrm{\Omega }\right)}\le c_1{∥\tilde{g}∥}_{W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)},$$

(24)

where the constant $c_1>0$ is independent of $\tilde{g}(s,\tau ).$ Since problems (21)–(23) are equivalent to problems (18)–(20), it is clear that problems (18)–(20) have a unique solution z in $W_2^{2,1}\left(\mathrm{\Omega }\right)$, and

$${∥z∥}_{W_2^{2,1}\left(\mathrm{\Omega }\right)}\le c_1{∥g∥}_{W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)}.$$

(25)

Under the assumed conditions, from the definition of F in problems (15)–(17) and estimate (25), it seems that $F\in L_2\left(\mathrm{\Omega }\right).$

Definition 3.

A solution w of problems (15)–(17) is defined to be an element of $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ satisfying the integral identity

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\left[w(x,t)\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)\right]dxdt\hfill \\ & & +\underset{\mathrm{\Omega }}{\int }w(x,t)\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt=\underset{\mathrm{\Omega }}{\int }F(x,t){\overline{\eta }}_{1}dxdt\hfill \end{array}$$

(26)

$for$ $any$ ${\eta }_1\in W_2^{2,1}\left(\mathrm{\Omega }\right)$ such that ${\eta }_1(x,0)=0$ $for$ $a.a$ $x\in X,$ ${\left.\frac{\partial {\eta }_1}{\partial \nu }\right|}_S=0.$

As in problems (18)–(20), by changing the variable $\tau =T-t$, we transform problems (15)–(17) to the problem

$$-i\frac{\partial \tilde{w}}{\partial \tau }+\chi \mathrm{\Delta }\tilde{w}+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left({\tilde{p}}_j(x,\tau )\tilde{w}\right)-r\tilde{w}+q_0\tilde{w}-i{q}_1\tilde{w}=\tilde{F}(x,\tau ),$$

(27)

$$\tilde{w}(x,0)=0,x\in X,$$

(28)

$${\left.\frac{\partial \tilde{w}}{\partial \nu }\right|}_S=0,(s,\tau )\in S,$$

(29)

where $\left(x,\tau \right)\in \mathrm{\Omega },$ $\tilde{w}(x,\tau )=w(x,T-\tau )=w(x,t),$ ${\tilde{p}}_j(x,\tau )=p_j(x,T-\tau )=p_j(x,t),$ $j=1,2,...,n,$ $\tilde{F}(x,\tau )=F(x,T-\tau )=F(x,t).$ If we denote the complex conjugate of $\tilde{w}(x,\tau )$ by $\varphi =\varphi (x,\tau )$, we can easily say that $\varphi$ is a solution of problem

$$i\frac{\partial \varphi }{\partial \tau }+\chi \mathrm{\Delta }\varphi -i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left({\tilde{p}}_j(x,\tau )\varphi \right)-r\varphi +q_0\varphi +i{q}_1\varphi =G,$$

(30)

$$\varphi (x,0)=0,x\in X,$$

(31)

$${\left.\frac{\partial \varphi }{\partial \nu }\right|}_S=0,(s,\tau )\in S,$$

(32)

where G is the complex conjugate of $\tilde{F}$. For convenience, if we denote the variable $\tau$ by t and ${\tilde{p}}_j(x,\tau )$ by $p_j(x,t)$, we rewrite problems (30)–(32) as

$$i\frac{\partial \varphi }{\partial t}+\chi \mathrm{\Delta }\varphi -i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t)\varphi \right)-r\varphi +q_0\varphi +i{q}_1\varphi =G(x,t),$$

(33)

$$\varphi (x,0)=0,x\in X,$$

(34)

$${\left.\frac{\partial \varphi }{\partial \nu }\right|}_S=0,(s,t)\in S.$$

(35)

As seen, analyzing the solution of problems (15)–(17) in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ is equivalent to analyzing the solution of problems (33)–(35) in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$. Because problems (15)–(17) are a boundary value problem with final time, and also, by applying the variable transformation $\tau =T-t$ to problems (15)–(17), we obtain an initial boundary value problem, i.e., problems (15)–(17) and (33)–(35) are symmetric.

Definition 4.

A function $\varphi (x,t)$ in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ satisfying the integral identity

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\left[\varphi (x,t)\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1+i\sum _{j=1}^n{p}_j(x,t)\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)\right]dxdt\hfill \\ & & +\underset{\mathrm{\Omega }}{\int }\varphi (x,t)\left(q_0{\overline{\eta }}_1+i{q}_1{\overline{\eta }}_1\right)dxdt=\underset{\mathrm{\Omega }}{\int }G(x,t){\overline{\eta }}_{1}dxdt\hfill \end{array}$$

(36)

$for$ $any$ ${\eta }_1\in W_2^{2,1}\left(\mathrm{\Omega }\right)$ such that ${\eta }_1(x,T)=0$ $for$ $a.a$ $x\in X,$ ${\left.\frac{\partial {\eta }_1}{\partial \nu }\right|}_S=0$ will be called a solution of problems (33)–(35) in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$.

Now, to prove the existence and uniqueness of the solution of problems (33)–(35), we consider the auxiliary problem

$$i\frac{\partial u}{\partial t}+\chi \mathrm{\Delta }u-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t)u\right)-ru+q_{0}u+i{q}_{1}u=h(x,t),$$

(37)

$$u(x,0)=0,x\in X,$$

(38)

$${\left.\frac{\partial u}{\partial \nu }\right|}_S=0,(s,t)\in S,$$

(39)

where $h\in W_2^{0,1}\left(\mathrm{\Omega }\right),$ $q=\left(q_0,q_1\right)\in Q_{ad}.$ Also, let the functions $p_j$ and, additionally, (4),

satisfy the condition

$$\left|\frac{{\partial }^2{p}_j(x,t)}{\partial x_k\partial t}\right|\le {\mu }_{7}fora.a\left(x,t\right)\in \mathrm{\Omega },{\mu }_7=const.>0$$

(40)

for $j,k=1,2,...,n.$

Definition 5.

A solution u of problems (37)–(39) is defined to be an element of $W_2^{2,1}\left(\mathrm{\Omega }\right)$ satisfying Equation (37) $for$ $a.a$ $\left(x,t\right)\in \mathrm{\Omega }$, (38) $for$ $a.a$ $x\in X$ and (39) $for$ $a.a$ $\left(s,t\right)\in S$.

For the solution of problems (37)–(39), we can easily prove the following theorem by Galerkin’s method:

Theorem 4.

Let the functions r and $p_j,$ $j=1,2,...,n$ satisfy conditions (3), (4), and (40), and let $h\in W_2^{0,1}\left(\mathrm{\Omega }\right),$ $q=\left(q_0,q_1\right)\in Q_{ad}$. Then, problems (37)–(39) have a unique solution u in $W_2^{2,1}\left(\mathrm{\Omega }\right)$ in the meaning of Definition 5. Moreover, there is a constant $c_2>0$, which does not depend on h, such that

$${∥u∥}_{W_2^{2,1}\left(\mathrm{\Omega }\right)}^2\le c_2{∥h∥}_{W_2^{0,1}\left(\mathrm{\Omega }\right)}^2.$$

(41)

Theorem 5.

Let the functions r and $p_j$ for $j=1,2,...,n$ satisfy conditions (3), (4), and (40), and let $G\in L_2\left(\mathrm{\Omega }\right),$ $q\in Q_{ad}$. Then, problems (33)–(35) have a unique solution ϕ in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ in the meaning of Definition 4, and ϕ satisfies the estimate

$${∥\varphi (.,t)∥}_{L_2\left(X\right)}^2\le c_3{∥G∥}_{L_2\left(\mathrm{\Omega }\right)}^{2}foranyt\in {\mathrm{\Lambda }}_T,$$

(42)

where the constant $c_3>0$ is independent from G.

Proof. 

We use the method in [30], p. 115, Theorem 2.3, for the proof of Theorem 5. We approximate the function $G\in L_2\left(\mathrm{\Omega }\right)$ by the functions $G^{\left(m\right)}\in W_2^{0,1}\left(\mathrm{\Omega }\right),$ $m=1,2,...$ such that

$${∥G^{\left(m\right)}-G∥}_{L_2\left(\mathrm{\Omega }\right)}\to 0\mathrm{as}m\to \infty .$$

(43)

Thus, for $m=1,2,...$, we obtain the problem

$$i\frac{\partial {\varphi }^{\left(m\right)}}{\partial t}+\chi \mathrm{\Delta }{\varphi }^{\left(m\right)}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t){\varphi }^{\left(m\right)}\right)-r{\varphi }^{\left(m\right)}+q_0{\varphi }^{\left(m\right)}+i{q}_1{\varphi }^{\left(m\right)}=G^{\left(m\right)}(x,t),$$

(44)

$${\varphi }^{\left(m\right)}(x,0)=0,x\in X,$$

(45)

$${\left.\frac{\partial {\varphi }^{\left(m\right)}}{\partial \nu }\right|}_S=0,(s,t)\in S.$$

(46)

Since $G^{\left(m\right)}\in W_2^{0,1}\left(\mathrm{\Omega }\right)$ for each $m=1,2,...$, we deduce from Theorem 4 that problems (44)–(46) have a unique solution ${\varphi }^{\left(m\right)}(x,t)$ in $W_2^{2,1}\left(\mathrm{\Omega }\right)$ corresponding to $G^{\left(m\right)}$ for each $m=1,2,...$.

It is obvious that the functions ${\varphi }^{\left(m\right)}$ for each $m=1,2,...$ satisfy the integral identity

$$\begin{array}{ccc}& & \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial {\varphi }^{\left(m\right)}}{\partial t}+\chi \mathrm{\Delta }{\varphi }^{\left(m\right)}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,\tau ){\varphi }^{\left(m\right)}\right)\right)\overline{\eta }dxd\tau \hfill \\ \hfill +& & \underset{{\mathrm{\Omega }}_t}{\int }\left(-r{\varphi }^{\left(m\right)}+q_0{\varphi }^{\left(m\right)}+i{q}_1{\varphi }^{\left(m\right)}-G^{\left(m\right)}(x,\tau )\right)\overline{\eta }dxd\tau =0\hfill \end{array}$$

(47)

for any $\eta (x,\tau )\in L_2\left(\mathrm{\Omega }\right)$ and $\tau \in \mathrm{\Lambda },$ ${\mathrm{\Omega }}_t=\left(0,t\right)$, and the conditions

$${\varphi }^{\left(m\right)}(x,0)=0fora.ax\in X,$$

(48)

$${\left.\frac{\partial {\varphi }^{\left(m\right)}}{\partial \nu }\right|}_S=0fora.a(s,t)\in S.$$

(49)

If we denote ${\mathrm{\Phi }}_{ml}={\varphi }^{\left(m\right)}-{\varphi }^{\left(l\right)}$ for each $m,l=1,2,...$, it is written that ${\mathrm{\Phi }}_{ml}\in W_2^{2,1}\left(\mathrm{\Omega }\right)$. Thus, the functions ${\mathrm{\Phi }}_{ml}$ for each $m,l=1,2,...$ satisfy (47)–(49), that is,

$$\begin{array}{ccc}& & \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial t}+\chi \mathrm{\Delta }{\mathrm{\Phi }}_{ml}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,\tau ){\mathrm{\Phi }}_{ml}\right)-r{\mathrm{\Phi }}_{ml}\right)\overline{\eta }dxd\tau \hfill \\ \hfill +& & \underset{{\mathrm{\Omega }}_t}{\int }\left(q_0{\mathrm{\Phi }}_{ml}+i{q}_1{\mathrm{\Phi }}_{ml})\overline{\eta }dxd\tau =\underset{{\mathrm{\Omega }}_t}{\int }{G}^{\left(m\right)}-G^{\left(l\right)}\right)\overline{\eta }dxd\tau \hfill \end{array}$$

(50)

$${\mathrm{\Phi }}_{ml}(x,0)=0fora.ax\in X,$$

(51)

$${\left.\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial \nu }\right|}_S=0fora.a(s,t)\in S.$$

(52)

If we substitute ${\mathrm{\Phi }}_{ml}$ for the test function $\eta \in L_2\left(\mathrm{\Omega }\right)$ in (50), we get

$$\begin{array}{ccc}& & \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial t}+\chi \mathrm{\Delta }{\mathrm{\Phi }}_{ml}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,\tau ){\mathrm{\Phi }}_{ml}\right)-r{\mathrm{\Phi }}_{ml}\right){\overline{\mathrm{\Phi }}}_{ml}dxd\tau \hfill \\ \hfill +& & \underset{{\mathrm{\Omega }}_t}{\int }(q_0{\mathrm{\Phi }}_{ml}+i{q}_1{\mathrm{\Phi }}_{ml}){\overline{\mathrm{\Phi }}}_{ml}dxd\tau =\underset{{\mathrm{\Omega }}_t}{\int }\left(G^{\left(m\right)}-G^{\left(l\right)}\right){\overline{\mathrm{\Phi }}}_{ml}dxd\tau .\hfill \end{array}$$

(53)

Using the formula of integration by parts and conditions (4) and (51) in (53), we have

$$\begin{array}{ccc}& & \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial t}{\overline{\mathrm{\Phi }}}_{ml}-\chi \sum _{j=1}^n{\left|\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial x_j}\right|}^2+i\sum _{j=1}^n{p}_j(x,\tau ){\mathrm{\Phi }}_{ml}\frac{\partial {\overline{\mathrm{\Phi }}}_{ml}}{\partial x_j}-r{\left|{\mathrm{\Phi }}_{ml}\right|}^2\right)dxd\tau \hfill \\ \hfill +& & \underset{{\mathrm{\Omega }}_t}{\int }\left(q_0{\left|{\mathrm{\Phi }}_{ml}\right|}^2+i{q}_1{\left|{\mathrm{\Phi }}_{ml}\right|}^2\right)dxd\tau =\underset{{\mathrm{\Omega }}_t}{\int }\left(G^{\left(m\right)}-G^{\left(l\right)}\right){\overline{\mathrm{\Phi }}}_{ml}dxd\tau .\hfill \end{array}$$

If we subtract its complex conjugate from the equality above, we get

$$\begin{array}{ccc}& & \underset{{\mathrm{\Omega }}_t}{\int }\left(i\left(\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial t}{\overline{\mathrm{\Phi }}}_{ml}+\frac{\partial {\overline{\mathrm{\Phi }}}_{ml}}{\partial t}{\mathrm{\Phi }}_{ml}\right)+i\sum _{j=1}^n{p}_j(x,\tau )\left({\mathrm{\Phi }}_{ml}\frac{\partial {\overline{\mathrm{\Phi }}}_{ml}}{\partial x_j}+{\overline{\mathrm{\Phi }}}_{ml}\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial x_j}\right)\right)dxd\tau \hfill \\ \hfill +2i& & \underset{{\mathrm{\Omega }}_t}{\int }{q}_1{\left|{\mathrm{\Phi }}_{ml}\right|}^{2}dxd\tau =2i\underset{{\mathrm{\Omega }}_t}{\int }Im\left(\left(G^{\left(m\right)}-G^{\left(l\right)}\right){\overline{\mathrm{\Phi }}}_{ml}\right)dxd\tau .\hfill \end{array}$$

(54)

In (54), if we use the equalities

$$\begin{array}{ccc}\hfill \sum _{j=1}^n{p}_j(x,\tau )\left({\mathrm{\Phi }}_{ml}\frac{\partial {\overline{\mathrm{\Phi }}}_{ml}}{\partial x_j}+{\overline{\mathrm{\Phi }}}_{ml}\frac{\partial {\mathrm{\Phi }}_{ml}}{\partial x_j}\right)& =& \sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,\tau ){\left|{\mathrm{\Phi }}_{ml}\right|}^2\right)-\sum _{j=1}^n\frac{\partial p_j(x,\tau )}{\partial x_j}{\left|{\mathrm{\Phi }}_{ml}\right|}^2,\hfill \\ \hfill \frac{\partial {\mathrm{\Phi }}_{ml}}{\partial t}{\overline{\mathrm{\Phi }}}_{ml}+\frac{\partial {\overline{\mathrm{\Phi }}}_{ml}}{\partial t}{\mathrm{\Phi }}_{ml}& =& \frac{\partial }{\partial t}{\left|{\mathrm{\Phi }}_{ml}\right|}^2\hfill \end{array}$$

(55)

and the conditions ${\left.p_j\right|}_S=0,$ $j=1,2,...,n$ and (51), we obtain

$$\begin{array}{ccc}\hfill \underset{X}{\int }{\left|{\mathrm{\Phi }}_{ml}(x,t)\right|}^{2}dx& =& -2\underset{{\mathrm{\Omega }}_t}{\int }{q}_1{\left|{\mathrm{\Phi }}_{ml}\right|}^{2}dxd\tau +\underset{{\mathrm{\Omega }}_t}{\int }\sum _{j=1}^n\frac{\partial p_j(x,\tau )}{\partial x_j}{\left|{\mathrm{\Phi }}_{ml}\right|}^{2}dxd\tau \hfill \\ & & +2\underset{{\mathrm{\Omega }}_t}{\int }Im\left(\left(G^{\left(m\right)}-G^{\left(l\right)}\right){\overline{\mathrm{\Phi }}}_{ml}\right)dxd\tau \hfill \end{array}$$

(56)

which implies that

$${∥{\mathrm{\Phi }}_{ml}(.,t)∥}_{L_2\left(X\right)}^2\le \left(2b_1+n{\mu }_3+1\right)\underset{{\mathrm{\Omega }}_t}{\int }{\left|{\mathrm{\Phi }}_{ml}\right|}^{2}dxd\tau +\underset{{\mathrm{\Omega }}_t}{\int }{\left|G^{\left(m\right)}-G^{\left(l\right)}\right|}^{2}dxd\tau$$

by the Cauchy–Schwarz inequality and condition (4), which is equivalent to

$${∥{\mathrm{\Phi }}_{ml}(.,t)∥}_{L_2\left(X\right)}^2\le \left(2b_1+n{\mu }_3+1\right)\underset{0}{\overset{t}{\int }}{∥{\mathrm{\Phi }}_{ml}(.,\tau )∥}_{L_2\left(X\right)}^{2}d\tau +{∥G^{\left(m\right)}-G^{\left(l\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^2$$

(57)

for any $t\in {\mathrm{\Lambda }}_T$ for each $m,l=1,2,....$ In (57), if we use Gronwall’s inequality for any $t\in {\mathrm{\Lambda }}_T$ and $m,l=1,2,...$, we get

$${∥{\mathrm{\Phi }}_{ml}(.,t)∥}_{L_2\left(X\right)}^2\le c_4{∥G^{\left(m\right)}-G^{\left(l\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^2,$$

(58)

where the constant $c_4>0$ is independent of $m,l$. Thus, from (58), it is written that

$${∥{\mathrm{\Phi }}_{ml}∥}_{C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))}^2\le c_4{∥G^{\left(m\right)}-G^{\left(l\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^2,m,l=1,2,...$$

(59)

which is equivalent to

$${∥{\varphi }^{\left(m\right)}-{\varphi }^{\left(l\right)}∥}_{C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))}^2\le c_4{∥G^{\left(m\right)}-G^{\left(l\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^2,m,l=1,2,....$$

(60)

Since

$${∥G^{\left(m\right)}-G∥}_{L_2\left(\mathrm{\Omega }\right)}\to 0\mathrm{as}m\to \infty$$

(61)

by limit relation (43), we obtain

$${∥{\varphi }^{\left(m\right)}-{\varphi }^{\left(l\right)}∥}_{C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))}^2\to 0\mathrm{as}m,l\to \infty$$

(62)

from (60). This shows that the sequence $\left\{{\varphi }^{\left(m\right)}(x,t)\right\}$ converges in the norm of $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$. Because the space $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ is complete, the limit function $\varphi (x,t)$ of $\left\{{\varphi }^{\left(m\right)}\right\}$ is in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$, that is, $\varphi \in C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$. Moreover, $\varphi (x,t)$ satisfies the integral identity (36). To show this, let us substitute the test function ${\eta }_1\in W_2^{2,1}\left(\mathrm{\Omega }\right)$ such that ${\eta }_1(x,T)=0$ $for$ $a.a$ $x\in X,$ ${\left.\frac{\partial {\eta }_1}{\partial \nu }\right|}_S=0$ for test function $\eta \in L_2\left(\mathrm{\Omega }\right)$ in (47). Thus, by the formula of integration by parts, for any $t\in {\mathrm{\Lambda }}_T$ and $m,l=1,2,...$, we get

$$\begin{array}{c}\hfill \underset{{\mathrm{\Omega }}_t}{\int }{\varphi }^{\left(m\right)}\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1+i\sum _{j=1}^n{p}_j(x,\tau )\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxd\tau \\ \hfill +\underset{{\mathrm{\Omega }}_t}{\int }{\varphi }^{\left(m\right)}\left(q_0{\overline{\eta }}_1+i{q}_1{\overline{\eta }}_1\right)dxd\tau =\underset{{\mathrm{\Omega }}_t}{\int }{G}^{\left(m\right)}{\overline{\eta }}_1(x,\tau )dxd\tau -i\underset{X}{\int }{\varphi }^{\left(m\right)}{\overline{\eta }}_1(x,t)dx.\end{array}$$

(63)

If we take the limit of equality (63) as $m\to \infty ,$ we obtain

$$\begin{array}{c}\hfill \underset{{\mathrm{\Omega }}_t}{\int }\varphi \left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1+i\sum _{j=1}^n{p}_j(x,\tau )\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxd\tau \\ \hfill +\underset{{\mathrm{\Omega }}_t}{\int }\varphi \left(q_0{\overline{\eta }}_1+i{q}_1{\overline{\eta }}_1\right)dxd\tau =\underset{{\mathrm{\Omega }}_t}{\int }G(x,\tau ){\overline{\eta }}_1(x,\tau )dxd\tau -i\underset{X}{\int }\varphi {\overline{\eta }}_1(x,t)dx\end{array}$$

(64)

for any $t\in {\mathrm{\Lambda }}_T$ and $m=1,2,....$ In (64), taking $t=T$ and using the condition ${\eta }_1(x,T)=0$ for a.a $x\in X$, we prove that (36) holds for $\varphi$, that is, the limit function $\varphi$ is a solution of problems (33)–(35) in the meaning of Definition 4.

If we substitute ${\varphi }^{\left(m\right)}\in W_2^{2,1}\left(\mathrm{\Omega }\right)$ for the test function $\eta \in L_2\left(\mathrm{\Omega }\right)$ in (47), we get

$$\begin{array}{c}\hfill \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial {\varphi }^{\left(m\right)}}{\partial t}{\overline{\varphi }}^{\left(m\right)}-\chi \sum _{j=1}^n{\left|\frac{\partial {\varphi }^{\left(m\right)}}{\partial x_j}\right|}^2+i\sum _{j=1}^n{p}_j(x,\tau ){\varphi }^{\left(m\right)}\frac{\partial {\overline{\varphi }}^{\left(m\right)}}{\partial x_j}\right)dxd\tau \\ \hfill +\underset{{\mathrm{\Omega }}_t}{\int }\left(-r\left(x\right){\left|{\varphi }^{\left(m\right)}\right|}^2+q_0{\left|{\varphi }^{\left(m\right)}\right|}^2+i{q}_1{\left|{\varphi }^{\left(m\right)}\right|}^2-G^{\left(m\right)}(x,\tau ){\overline{\varphi }}^{\left(m\right)}\right)dxd\tau =0\end{array}$$

(65)

by integration by parts. Subtracting its complex conjugate from (65) and using relation (55) for ${\varphi }^{\left(m\right)}$, we obtain

$$\begin{array}{c}\hfill \underset{{\mathrm{\Omega }}_t}{\int }\left(i\frac{\partial }{\partial t}{\left|{\varphi }^{\left(m\right)}\right|}^2+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,\tau ){\left|{\varphi }^{\left(m\right)}\right|}^2\right)-i\sum _{j=1}^n\frac{\partial p_j(x,\tau )}{\partial x_j}{\left|{\varphi }^{\left(m\right)}\right|}^2\right)dxd\tau \\ \hfill +\underset{{\mathrm{\Omega }}_t}{\int }2i{q}_1\left(x\right){\left|{\varphi }^{\left(m\right)}\right|}^{2}dxd\tau =2i\underset{{\mathrm{\Omega }}_t}{\int }Im\left(G^{\left(m\right)}(x,\tau ){\overline{\varphi }}^{\left(m\right)}\right)dxd\tau \end{array}$$

which is equivalent to

$$\begin{array}{c}\hfill \underset{X}{\int }{\left|{\varphi }^{\left(m\right)}(x,t)\right|}^{2}dx=-2\underset{{\mathrm{\Omega }}_t}{\int }{q}_1\left(x\right){\left|{\varphi }^{\left(m\right)}\right|}^{2}dxd\tau +\underset{{\mathrm{\Omega }}_t}{\int }\sum _{j=1}^n\frac{\partial p_j(x,\tau )}{\partial x_j}{\left|{\varphi }^{\left(m\right)}\right|}^{2}dxd\tau \\ \hfill +2\underset{{\mathrm{\Omega }}_t}{\int }Im\left(G^{\left(m\right)}(x,\tau ){\overline{\varphi }}^{\left(m\right)}\right)dxd\tau foranyt\in {\mathrm{\Lambda }}_T,m=1,2,...\end{array}$$

(66)

by conditions (45) and ${\left.p_j\right|}_S=0$ for $j=1,2,...,n.$ In (66), using the Cauchy–Schwarz inequality and the conditions $\left|q_1\left(x\right)\right|\le b_1$ and (4), we get

$${∥{\varphi }^{\left(m\right)}(.,t)∥}_{L_2\left(X\right)}^2\le \left(2b_1+n{\mu }_3+1\right)\underset{0}{\overset{t}{\int }}{∥{\varphi }^{\left(m\right)}(.,\tau )∥}_{L_2\left(X\right)}^{2}d\tau +{∥G^{\left(m\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^2$$

(67)

for any $t\in {\mathrm{\Lambda }}_T,m=1,2,...$, which implies that

$${∥{\varphi }^{\left(m\right)}(.,t)∥}_{L_2\left(X\right)}^2\le c_5{∥G^{\left(m\right)}∥}_{L_2\left(\mathrm{\Omega }\right)}^{2}foranyt\in {\mathrm{\Lambda }}_T,m=1,2,...$$

(68)

by Gronwall’s inequality, where the constant $c_5>0$ is independent of m. Thus, if we take the limit of (68) as $m\to \infty$, we prove that the limit function $\varphi (x,t)$ of $\left\{{\varphi }^{\left(m\right)}(x,t)\right\}$ satisfies (42), which implies that the solution of (33)–(35) is unique. Thus, the proof of Theorem 5 is completed. □

Since problems (33)–(35) and problem (30)–(32) are equivalent, it is easily written that

$${∥\varphi (.,\tau )∥}_{L_2\left(X\right)}^2\le c_3{∥G∥}_{L_2\left(\mathrm{\Omega }\right)}^{2}forany\tau \in {\mathrm{\Lambda }}_T$$

(69)

from Theorem 5, which implies that

$${∥\tilde{w}(.,\tau )∥}_{L_2\left(X\right)}^2\le c_3{∥\tilde{F}∥}_{L_2\left(\mathrm{\Omega }\right)}^{2}forany\tau \in {\mathrm{\Lambda }}_T$$

(70)

due to the fact that $\varphi =\varphi (x,\tau )$ is the complex conjugate of $\tilde{w}(x,\tau )$. Also, since $\tilde{w}(x,\tau )=w(x,t),$ ${\tilde{p}}_j(x,\tau )=p_j(x,t),$ $j=1,2,...,n,$ $\tilde{F}(x,\tau )=F(x,t)$, we can easily say that problems (15)–(17) have a unique solution w in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ satisfying the estimate

$${∥w(.,t)∥}_{L_2\left(X\right)}^2\le c_3{∥F∥}_{L_2\left(\mathrm{\Omega }\right)}^{2}foranyt\in {\mathrm{\Lambda }}_T.$$

(71)

Now, let us give the next theorem for the solution of the adjoint problem by using the solution of problems (15)–(17):

Theorem 6.

Let the assumptions of Theorem 1 be fulfilled and $\kappa \in W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)$ be a given function. Also, let the functions $r\left(x\right),$ $p_j(x,t),$ $j=1,2,...,n$ satisfy conditions (3), (4), and (40). Then, problems (8)–(10) have a unique solution $\mathrm{\Phi }(x,t)$ in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right)$. Moreover, there is a constant $c_6>0$ that does not depend on t such that

$${∥\mathrm{\Phi }(.,t)∥}_{L_2\left(X\right)}^2\le c_6{∥u-\kappa ∥}_{W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)}^{2}foranyt\in {\mathrm{\Lambda }}_T.$$

(72)

Proof. 

We have proven above that problems (15)–(17) have a unique solution w in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ in the meaning of Definition 3 satisfying estimate (71), and also, problems (18)–(20) have a unique solution z in $W_2^{2,1}\left(\mathrm{\Omega }\right)$ satisfying estimate (25). Hence, since $w=\mathrm{\Phi }-z$, we come to the conclusion that problems (8)–(10) have a unique solution $\mathrm{\Phi }=w+z$ in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ in the meaning of Definition 2. Thus, if we substitute $w=\mathrm{\Phi }-z$ and $F=\left(1-i\right)\frac{\partial z}{\partial t}-i{\displaystyle \sum _{j=1}^n}\frac{\partial }{\partial x_j}\left(p_{j}z\right)-q_{0}z+i{q}_{1}z$ in (26), we write

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxdt\hfill \\ & & +\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt\hfill \\ & & -\underset{\mathrm{\Omega }}{\int }z\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxdt\hfill \\ & & -\underset{\mathrm{\Omega }}{\int }z\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt\hfill \\ & =& \underset{\mathrm{\Omega }}{\int }\left(\left(1-i\right)\frac{\partial z}{\partial t}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_{j}z\right)-q_{0}z+i{q}_{1}z\right){\overline{\eta }}_{1}dxdt\hfill \end{array}$$

(73)

for any ${\eta }_1\in W_2^{2,1}\left(\mathrm{\Omega }\right).$ Also, since $z\in W_2^{2,1}\left(\mathrm{\Omega }\right),$ ${\left.\frac{\partial z}{\partial \nu }\right|}_S=g(s,t)=\frac{2}{\chi }\left(u(s,t)-\kappa (s,t)\right),$ $(s,t)\in S,$ we can easily obtain the relation

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }z(x,t)\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j(x,t)\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r\left(x\right){\overline{\eta }}_1\right)dxdt\hfill \\ & =& \underset{\mathrm{\Omega }}{\int }\left(i\frac{\partial z}{\partial t}+\chi \mathrm{\Delta }z+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t)z\right)-r\left(x\right)z\right){\overline{\eta }}_{1}dxdt\hfill \\ & & -i\underset{X}{\int }z(x,T){\overline{\eta }}_1(x,T)dx+\underset{X}{\int }z(x,0){\overline{\eta }}_1(x,0)dx+\underset{S}{\int }z\frac{\partial {\overline{\eta }}_1}{\partial \nu }dsdt\hfill \\ & & -\underset{S}{\int }{\overline{\eta }}_1\frac{\partial z}{\partial \nu }dsdt-i\underset{S}{\int }\sum _{j=1}^n{p}_j(s,t)z{\overline{\eta }}_{1}cos\left(\nu ,x_j\right)dsdt\hfill \\ & =& \underset{\mathrm{\Omega }}{\int }\left(i\frac{\partial z}{\partial t}+\chi \mathrm{\Delta }z+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_j(x,t)z\right)-r\left(x\right)z\right){\overline{\eta }}_{1}dxdt\hfill \\ & & -\frac{2}{\chi }\underset{S}{\int }\left(u(s,t)-\kappa (s,t)\right){\overline{\eta }}_1(s,t)dsdt\hfill \end{array}$$

(74)

by integrating by parts and using conditions (4) and (19). If we substitute (74) into (73), we get

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxdt\hfill \\ & & +\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt\hfill \\ & & -\underset{\mathrm{\Omega }}{\int }\left(i\frac{\partial z}{\partial t}+\chi \mathrm{\Delta }z+i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_{j}z\right)-rz\right){\overline{\eta }}_{1}dxdt\hfill \\ & & +\frac{2}{\chi }\underset{S}{\int }\left(u(s,t)-\kappa (s,t)\right){\overline{\eta }}_1(s,t)dsdt-\underset{\mathrm{\Omega }}{\int }z\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt\hfill \\ & =& \underset{\mathrm{\Omega }}{\int }\left(\left(1-i\right)\frac{\partial z}{\partial t}-i\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(p_{j}z\right)-q_{0}z+i{q}_{1}z\right){\overline{\eta }}_{1}dxdt\hfill \end{array}$$

which is equivalent to

$$\begin{array}{ccc}& & \underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(-i\frac{\partial {\overline{\eta }}_1}{\partial t}+\chi \mathrm{\Delta }{\overline{\eta }}_1-i\sum _{j=1}^n{p}_j\frac{\partial {\overline{\eta }}_1}{\partial x_j}-r{\overline{\eta }}_1\right)dxdt\hfill \\ & & +\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }\left(q_0{\overline{\eta }}_1-i{q}_1{\overline{\eta }}_1\right)dxdt\hfill \\ & =& -\frac{2}{\chi }\underset{S}{\int }\left(u(s,t)-\kappa (s,t)\right){\overline{\eta }}_1(s,t)dsdt\hfill \end{array}$$

by (18). Then we deduce from above that $\mathrm{\Phi }(x,t)$ satisfies integral identity (11), that is, $\mathrm{\Phi }$ is a unique solution of problems (8)–(10) in $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ in the meaning of Definition 2. Also, since $w=\mathrm{\Phi }-z$, it is written that

$${∥\mathrm{\Phi }(.,t)∥}_{L_2\left(X\right)}\le {∥z(.,t)∥}_{L_2\left(X\right)}+\sqrt{c_3}{∥F∥}_{L_2\left(\mathrm{\Omega }\right)}foranyt\in {\mathrm{\Lambda }}_T$$

from (71). Here, if we consider the inequality

$${∥z(.,t)∥}_{L_2\left(X\right)}\le c_6{∥z∥}_{W_2^{0,1}\left(\mathrm{\Omega }\right)}foranyt\in {\mathrm{\Lambda }}_T$$

and $F=\left(1-i\right)\frac{\partial z}{\partial t}-i{\displaystyle \sum _{j=1}^n}\frac{\partial }{\partial x_j}\left(p_{j}z\right)-q_{0}z+i{q}_{1}z,$ we easily write the inequality

$${∥\mathrm{\Phi }(.,t)∥}_{L_2\left(X\right)}\le c_7{∥z∥}_{W_2^{2,1}\left(\mathrm{\Omega }\right)}foranyt\in {\mathrm{\Lambda }}_T$$

which implies that

$${∥\mathrm{\Phi }(.,t)∥}_{L_2\left(X\right)}\le c_8{∥g∥}_{W_2^{\frac{1}{2},\frac{1}{4}}\left(S\right)}foranyt\in {\mathrm{\Lambda }}_T$$

by (25). Also, since $g(s,t)=\frac{2}{\chi }\left(u(s,t)-\kappa (s,t)\right)$, we can easily say that function $\mathrm{\Phi }$ provides estimate (72), which completes the proof. □

3. The Differentiability of the Objective Functional

In this section, we show that the objective functional $J_{\alpha }$ is differentiable in the meaning of Frechet and get a formula for its gradient with the help of the adjoint problem.

Theorem 7.

Let $\omega \in W$ be a given function. If the conditions of Theorem 6 are satisfied, then $J_{\alpha }$ is differentiable in the meaning of Frechet on $Q_{ad}$, and moreover, its gradient is given by the formula

$$\begin{array}{ccc}\hfill J_{\alpha }^{\prime }\left(q\right)& =& \left(J_{\alpha 0}^{\prime }\left(q\right),J_{\alpha 1}^{\prime }\left(q\right)\right)\hfill \\ \hfill J_{\alpha 0}^{\prime }\left(q\right)& =& \underset{0}{\overset{T}{\int }}Re\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_0-{\omega }_0\right),\hfill \\ \hfill J_{\alpha 1}^{\prime }\left(q\right)& =& -\underset{0}{\overset{T}{\int }}Im\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_1-{\omega }_1\right),\hfill \end{array}$$

(75)

where $u,\mathrm{\Phi }$ are the solutions of problems (1) and (2) and (8)–(10) corresponding to $q\in Q_{ad}$, respectively.

Proof. 

From (6), the enhancement $\delta J_{\alpha }\left(q\right)=J_{\alpha }(q+\delta q)-J_{\alpha }\left(q\right)$ of $J_{\alpha }\left(q\right)$ for any $q\in Q_{ad}$ is written as follows

$$\begin{array}{ccc}\hfill \delta J_{\alpha }\left(q\right)& =& 2\underset{S}{\int }Re\left[u(s,t)-\kappa (s,t)\delta \overline{u}(s,t)\right]dsdt\hfill \\ & & +2\alpha \underset{X}{\int }\left(q_0-{\omega }_0\right)\delta q_{0}dx+2\alpha \underset{X}{\int }\left(q_1-{\omega }_1\right)\delta q_{1}dx\hfill \\ & & +{∥\delta u∥}_{L_2\left(S\right)}^2+\alpha {∥\delta q∥}_W^2,\hfill \end{array}$$

(76)

where $\delta q=\left(\delta q_0,\delta q_1\right)\in B=L_{\infty }\left(X\right)\times L_{\infty }\left(X\right)$ is an enhancement given to any $q\in Q_{ad}$ such that $q+\delta q\in Q_{ad}$, and the function $\delta u$ is a solution of the following problem [15]:

$$\begin{array}{c}\hfill i\frac{\partial \delta u}{\partial t}+\chi \mathrm{\Delta }\delta u+ip(x,t)\nabla \delta u-r\delta u+(q_0+\delta q_0)\delta u\\ \hfill +i(q_1+\delta q_1)\delta u=-\delta q_{0}u-i\delta q_{1}u,\end{array}$$

(77)

$$\delta u(x,0)=0,x\in X,{\left.\frac{\partial \delta u}{\partial \nu }\right|}_S=0.$$

(78)

Since $\delta u\in B_1\subset W_2^{2,1}\left(\mathrm{\Omega }\right)$ [15], we can easily write the integral identity

$$\begin{array}{c}\hfill \underset{\mathrm{\Omega }}{\int }\left(i\frac{\partial \delta u}{\partial t}+\chi \mathrm{\Delta }\delta u+ip\nabla \delta u-r\delta u+(q_0+\delta q_0)\delta u\right)\overline{\eta }(x,t)dxd\tau \\ \hfill +i\underset{\mathrm{\Omega }}{\int }(q_1+\delta q_1)\delta u\overline{\eta }(x,t)dxd\tau =-\underset{\mathrm{\Omega }}{\int }\delta q_{0}u\overline{\eta }dxd\tau -i\underset{\mathrm{\Omega }}{\int }\delta q_{1}u\overline{\eta }dxd\tau \end{array}$$

(79)

for any $\eta \in L_2\left(\mathrm{\Omega }\right).$ If we substitute $\overline{\mathrm{\Phi }}\in$ $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))\subset L_2\left(\mathrm{\Omega }\right)$ $for$ $\overline{\eta }(x,t)$ in (79), we get

$$\begin{array}{c}\hfill \underset{\mathrm{\Omega }}{\int }\left(i\frac{\partial \delta u}{\partial t}+\chi \mathrm{\Delta }\delta u+ip\nabla \delta u-r\delta u+(q_0+\delta q_0)\delta u\right)\overline{\mathrm{\Phi }}(x,t)dxd\tau \\ \hfill +i\underset{\mathrm{\Omega }}{\int }(q_1+\delta q_1)\delta u\overline{\mathrm{\Phi }}(x,t)dxd\tau =-\underset{\mathrm{\Omega }}{\int }\delta q_{0}u\overline{\mathrm{\Phi }}dxd\tau -i\underset{\mathrm{\Omega }}{\int }\delta q_{1}u\overline{\mathrm{\Phi }}dxd\tau .\end{array}$$

(80)

Also, it is clear that $\mathrm{\Phi }\in$ $C^0({\mathrm{\Lambda }}_T,L_2\left(X\right))$ provides identity (11). In (11), if we substitute $\delta \overline{u}\in B_1$ for the test function ${\overline{\eta }}_1(x,t)$, we obtain

$$\begin{array}{c}\hfill \underset{\mathrm{\Omega }}{\int }\left[\mathrm{\Phi }\left(-i\frac{\partial \delta \overline{u}}{\partial t}+\chi \mathrm{\Delta }\delta \overline{u}-ip\nabla \delta \overline{u}-r\delta \overline{u}+q_0\delta \overline{u}-i{q}_1\delta \overline{u}\right)\right]dxdt\\ \hfill =-2\underset{S}{\int }\left(\left(u(s,t)-\kappa (s,t)\right)\right)\delta \overline{u}(s,t)dsdt\end{array}$$

(81)

and we write the complex conjugate of (81) as

$$\begin{array}{c}\hfill \underset{\mathrm{\Omega }}{\int }\left[\overline{\mathrm{\Phi }}\left(i\frac{\partial \delta u}{\partial t}+\chi \mathrm{\Delta }\delta u+ip\nabla \delta u-r\delta u+q_0\delta u+i{q}_1\delta u\right)\right]dxdt\\ \hfill =-2\underset{S}{\int }\left(\left(\overline{u}(s,t)-\overline{\kappa }(s,t)\right)\right)\delta u(s,t)dsdt.\end{array}$$

(82)

Subtracting (80) from (82), we get

$$\begin{array}{ccc}& & 2\underset{S}{\int }\left(\left(\overline{u}(s,t)-\overline{\kappa }(s,t)\right)\right)\delta u(s,t)dsdt=\underset{\mathrm{\Omega }}{\int }\delta q_{0}u\overline{\mathrm{\Phi }}dxdt\hfill \\ & & +i\underset{\mathrm{\Omega }}{\int }\delta q_{1}u\overline{\mathrm{\Phi }}dxdt+\underset{\mathrm{\Omega }}{\int }\delta q_0\delta u\overline{\mathrm{\Phi }}dxdt+i\underset{\mathrm{\Omega }}{\int }\delta q_1\delta u\overline{\mathrm{\Phi }}dxdt.\hfill \end{array}$$

(83)

If we sum its complex conjugate with (83), we obtain

$$\begin{array}{ccc}& & 2\underset{S}{\int }Re\left(\left(u(s,t)-\kappa (s,t)\right)\right)\delta \overline{u}(s,t)dsdt=\underset{\mathrm{\Omega }}{\int }\delta q_{0}Re\left(u\overline{\mathrm{\Phi }}\right)dxdt\hfill \\ & & -\underset{\mathrm{\Omega }}{\int }\delta q_{1}Im\left(u\overline{\mathrm{\Phi }}\right)dxdt+\underset{\mathrm{\Omega }}{\int }\delta q_{0}Re\left(\delta u\overline{\mathrm{\Phi }}\right)dxdt-\underset{\mathrm{\Omega }}{\int }\delta q_{1}Im\left(\delta u\overline{\mathrm{\Phi }}\right)dxdt.\hfill \end{array}$$

(84)

Considering (84) in (76), we write

$$\begin{array}{ccc}\hfill \delta J_{\alpha }\left(q\right)& =& \underset{\mathrm{\Omega }}{\int }\delta q_{0}Re\left(u\overline{\mathrm{\Phi }}\right)dxdt-\underset{\mathrm{\Omega }}{\int }\delta q_{1}Im\left(\delta u\overline{\mathrm{\Phi }}\right)dxdt\hfill \\ & & +2\alpha \underset{X}{\int }\left(q_0-{\omega }_0\right)\delta q_{0}dx+2\alpha \underset{X}{\int }\left(q_1-{\omega }_1\right)\delta q_{1}dx+R\left(\delta q\right),\hfill \end{array}$$

(85)

where

$$\begin{array}{ccc}\hfill R\left(\delta q\right)& =& \underset{\mathrm{\Omega }}{\int }\delta q_{0}Re\left(\delta u\overline{\mathrm{\Phi }}\right)dxdt-\underset{\mathrm{\Omega }}{\int }\delta q_{1}Im\left(\delta u\overline{\mathrm{\Phi }}\right)dxdt\hfill \\ & & +{∥\delta u∥}_{L_2\left(S\right)}^2+\alpha {∥\delta q∥}_W^2.\hfill \end{array}$$

(86)

In (86), using the Cauchy–Schwarz inequality, we get

$$\begin{array}{ccc}& & \left|R\left(\delta q\right)\right|\le {∥\mathrm{\Phi }∥}_{L_2\left(\mathrm{\Omega }\right)}{∥\delta u∥}_{L_2\left(\mathrm{\Omega }\right)}{∥\delta q_0∥}_{L_{\infty }\left(X\right)}\hfill \\ & & +{∥\mathrm{\Phi }∥}_{L_2\left(\mathrm{\Omega }\right)}{∥\delta u∥}_{L_2\left(\mathrm{\Omega }\right)}{∥\delta q_1∥}_{L_{\infty }\left(X\right)}+{∥\delta u∥}_{L_2\left(S\right)}^2+\alpha {∥\delta q∥}_W^2.\hfill \end{array}$$

(87)

In (87), if we use the inequalities

$$\begin{array}{ccc}\hfill {∥\delta u∥}_{W_2^{2,1}\left(\mathrm{\Omega }\right)}^2& \le & {\beta }_1{∥\delta q∥}_B^2,\hfill \\ \hfill {∥\delta u∥}_{L_2\left(S\right)}^2& \le & {\beta }_2{∥\delta q∥}_B^2\hfill \end{array}$$

in [15] and estimates (7) and (72), we achieve

$$\left|R\left(\delta q\right)\right|\le c_8{∥\delta q∥}_B^2$$

(88)

which shows that $R\left(\delta q\right)=o\left({∥\delta q∥}_B\right)$, where the symbol $o\left({∥\delta q∥}_B\right)$, pronounced “small oh” of ${∥\delta q∥}_B$, means something for which its ratio with ${∥\delta q∥}_B$ has limit 0, that is, $\underset{{∥\delta q∥}_B\to 0}{lim}\frac{R}{{∥\delta q∥}_B}=0$, where the constants ${\beta }_1,{\beta }_2,c_8>0$ are independent from $\delta q$ and $\tau$. Thus, by the definition of the differentiability of a functional on closed set $Q_{ad}$ [31], from (85), we can write

$$\begin{array}{ccc}& & \delta J_{\alpha }\left(q\right)=\underset{X}{\int }\left[\underset{0}{\overset{T}{\int }}Re\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_0-{\omega }_0\right)\right]\delta q_{0}dx\hfill \\ & & +\underset{X}{\int }\left[-\underset{0}{\overset{T}{\int }}Im\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_1-{\omega }_1\right)\right]\delta q_{1}dx+o\left({∥\delta q∥}_B\right)\hfill \end{array}$$

$${∥L\delta u(.,t)∥}_{L_2\left(D\right)}^2\le {\beta }_{19}\left({∥\delta v_0∥}_{L_{\infty }\left(D\right)}^2+{∥\delta v_1∥}_{L_{\infty }\left(D\right)}^2\right),$$

which implies that $J_{\alpha }\left(q\right)$ is a differentiable functional in the meaning of Frechet on $Q_{ad}$, and its gradient is given by

$$\begin{array}{ccc}\hfill J_{\alpha }^{\prime }\left(q\right)& =& \left(J_{\alpha 0}^{\prime }\left(q\right),J_{\alpha 1}^{\prime }\left(q\right)\right)\hfill \\ \hfill J_{\alpha 0}^{\prime }\left(q\right)& =& \underset{0}{\overset{T}{\int }}Re\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_0-{\omega }_0\right),\hfill \\ \hfill J_{\alpha 1}^{\prime }\left(q\right)& =& -\underset{0}{\overset{T}{\int }}Im\left(u\overline{\mathrm{\Phi }}\right)dt+2\alpha \left(q_1-{\omega }_1\right).\hfill \end{array}$$

Thus, the proof of Theorem 7 is completed. □

4. 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 $q^{*}\in Q_{ad}$ be any solution of OCP. Then, for any $q\in Q_{ad}$, the inequality

$$\begin{array}{ccc}& & \underset{X}{\int }\left[\underset{0}{\overset{T}{\int }}Re\left(u^{*}{\overline{\mathrm{\Phi }}}^{*}\right)dt+2\alpha \left(q_0^{*}-{\omega }_0\right)\right]\left(q_0-q_0^{*}\right)dx\hfill \\ & & +\underset{X}{\int }\left[-\underset{0}{\overset{T}{\int }}Im\left(u^{*}{\overline{\mathrm{\Phi }}}^{*}\right)dt+2\alpha \left(q_1^{*}-{\omega }_1\right)\right]\left(q_1-q_1^{*}\right)dx\ge 0\hfill \end{array}$$

(89)

is valid, where $u^{*},$ ${\mathrm{\Phi }}^{*}$ are solutions of problems (1) and (2) and (8)–(10) corresponding to $q^{*}\in Q_{ad}$, respectively.

Proof. 

Let $q\in Q_{ad}$ be any control and $q^{*}\in Q_{ad}$ be solution of OCP, that is, let $q^{*}\in Q_{ad}$ be any optimal control. Firstly, let us prove that $Q_{ad}$ is a convex subset of $L_2\left(X\right)\times L_2\left(X\right)$. For this purpose, we show that $\theta q^0+(1-\theta )q^1\in Q_{ad}$, with $\theta \in \left[0,1\right]$ $\left(\theta \in \mathbb{R}\right)$ for any two points $q^0=\left(q_0^0,q_1^0\right),$ $q^1=\left(q_0^1,q_1^1\right)\in Q_{ad}$. Since the space $L_2\left(X\right)$ is the convex, we can write that $\theta q_l^0+\left(1-\theta \right)q_l^1\in L_2\left(X\right)$ for any $\theta \in \left[0,1\right]$ and $l=0,1$, and

$$\begin{array}{ccc}\hfill \left|\theta q_l^0\left(x\right)+\left(1-\theta \right)q_l^1\left(x\right)\right|& \le & \left|\theta q_l^0\left(x\right)\right|+\left|\left(1-\theta \right)q_l^1\left(x\right)\right|\hfill \\ & =& \theta \left|q_l^0\left(x\right)\right|+\left(1-\theta \right)\left|q_l^1\left(x\right)\right|\hfill \\ & \le & \theta b_l+\left(1-\theta \right)b_l\hfill \\ & =& b_l\mathrm{for}l=0,1\hfill \end{array}$$

which shows that set $Q_{ad}$ is a convex subset. Hence, we write

$$q^{*}+\theta \left(q-q^{*}\right)\in Q_{ad},\theta \in \left[0,1\right]$$

for any elements $q^{*},q$ of $Q_{ad}$.

For any $q\in Q_{ad}$, it is clear that

$$J_{\alpha }\left(q^{*}+\theta (q-q^{*})\right)-J_{\alpha }\left(q^{*}\right)\ge 0.$$

(90)

Since $J_{\alpha }$ is a differentiable functional in the meaning of Frechet on $Q_{ad}$, from (90), we write

$$0\le J_{\alpha }\left(q^{*}+\theta (q-q^{*})\right)-J_{\alpha }\left(q^{*}\right)={⟨J_{\alpha }^{\prime }\left(q^{*}\right),\theta (q-q^{*})⟩}_B+o\left(\theta \right)foranyq\in Q_{ad}$$

which is equivalent to

$$\theta {⟨J_{\alpha }^{\prime }\left(q^{*}\right),(q-q^{*})⟩}_{B=L_{\infty }\left(X\right)\times L_{\infty }\left(X\right)}+o\left(\theta \right)\ge 0foranyq\in Q_{ad}.$$

(91)

Dividing both sides of (91) by $\theta >0$ and then taking the limit as $\theta \to +0$, we get

$${⟨J_{\alpha }^{\prime }\left(q^{*}\right),(q-q^{*})⟩}_B\ge 0foranyq\in Q_{ad}.$$

(92)

Considering Formula (75) for $q=q^{*}$ and the integral representation of a linear functional in space $B=L_{\infty }\left(X\right)\times L_{\infty }\left(X\right)$, we deduced from (92) that relation (89) is valid, which completes the proof of Theorem 8. □

5. 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.