Existence of solution for an optimal control problem associated to the Ginzburg-Landau system in superconductivity

TL;DR

This paper proves the existence of solutions for an optimal control problem involving the Ginzburg-Landau system in superconductivity, considering boundary controls and external magnetic fields, using advanced analysis tools.

Contribution

It establishes a global existence result for the control problem associated with the Ginzburg-Landau equations in superconductivity, incorporating boundary control and external magnetic fields.

Findings

Existence of solutions for the control problem.

Application of Friedrichs Curl Inequality and Rellich-Kondrashov Theorem.

Model includes boundary control and external magnetic field.

Abstract

This article develops a global existence result for the solution of an optimal control problem associated to the Ginzburg-Landau system. This main result is based on standard tools of analysis and functional analysis, such as the Friedrichs Curl Inequality and the Rellich-Kondrashov Theorem. In the concerning model, we consider the presence of an external magnetic field and the control variable is a complex function acting on the super-conducting sample boundary. Finally the state variables are the Ginzburg-Landau order parameter and the magnetic potential, defined on domains properly specified.

Equations

$$F_s(T)=F_n(T)+\frac{\hslash }{4m}{\int }_{\Omega }|\nabla \psi {|}_2^{2}dx+\frac{\alpha (T)}{4}{\int }_{\Omega }|\psi {|}^{4}dx-\frac{\beta (T)}{2}{\int }_{\Omega }|\psi {|}^{2}dx,$$

$$\left\{\begin{array}[]{ll}-\frac{\hbar}{2m}\nabla^{2}\psi+\alpha|\psi|^{2}\psi-\beta\psi=0,&\text{ in }\Omega\\ \\ \frac{\partial{\psi}}{\partial\textbf{n}}=0,&\text{ on }\partial\Omega.\end{array}\right.$$

$$J(\psi ,\text{A})$$

$$U=W^{1,2}(\Omega ;\mathbb{C})\times W^{1,2}({\mathbb{R}}^3;{\mathbb{R}}^3),$$

$$\left\{\begin{array}[]{ll}\frac{1}{2m}\left(-i\hbar\nabla-\frac{2e}{c}\mathbf{A}\right)^{2}\psi+\alpha|\psi|^{2}\psi-\beta\psi=0,&\text{ in }\Omega\\ \\ \left(i\hbar\nabla\psi+\frac{2e}{c}\textbf{A}\psi\right)\cdot\textbf{n}=0,&\text{ on }\partial\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\text{curl }(\text{curl }\textbf{A})=\text{ curl }\textbf{B}_{0}+\frac{4\pi}{c}\tilde{J},&\text{ in }\Omega\\ \\ \text{curl }(\text{curl }\textbf{A})=\text{ curl }\textbf{B}_{0},&\text{ in }\mathbb{R}^{3}\setminus\overline{\Omega},\end{array}\right.$$

$$\tilde{J}=-\frac{ie\hslash }{2m}\left({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast }\right)-\frac{2e^2}{mc}|\psi {|}^2\text{A}.$$

$${\text{B}}_0\in L^2({\mathbb{R}}^3;{\mathbb{R}}^3)$$

$$\Vert |\varphi {|}^2-|{\varphi }_d{|}^2{\Vert }_{0,2,\Omega }^2$$

$$J(\varphi ,\mathbf{A},u)=\frac{\epsilon }{2}\Vert \nabla \varphi {\Vert }_{0,2,\Omega }^2+K_1\Vert |\varphi {|}^2-|{\varphi }_d{|}^2{\Vert }_{0,2,\Omega }^2+K_2\Vert u{\Vert }_{0,2,\partial \Omega }^2,$$

$$\mathcal{C}=\{(\varphi ,\mathbf{A},u)\in W^{1,2}(\Omega ,\mathbb{C})\times W^{1,2}({\Omega }_1,{\mathbb{R}}^3)\times L^2(\partial \Omega ;\mathbb{C}):\text{ such that }(\text{ref}a.1)\text{ and }(\text{ref}a.2)\text{ hold }\},$$

$$\left\{\begin{array}[]{ll}\frac{1}{2m}\left(-i\hbar\nabla-\frac{2e}{c}\mathbf{A}\right)^{2}\phi+\alpha|\phi|^{2}\phi-\beta\phi=0,&\text{ in }\Omega,\\ \\ \left(i\hbar\nabla\phi+\frac{2e}{c}\mathbf{A}\phi\right)\cdot\mathbf{n}=u,&\text{ on }\partial\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\text{ curl }\text{ curl }\mathbf{A}=\text{ curl }\mathbf{B}_{0}+\frac{4\pi}{c}\tilde{J},&\text{ in }\Omega,\\ \\ \text{ curl }\text{ curl }\mathbf{A}=\text{ curl }\mathbf{B}_{0},&\text{ in }\Omega_{1}\setminus\Omega,\\ \\ \text{ div }\mathbf{A}=0,&\text{ in }\Omega_{1},\\ \\ \mathbf{A}\cdot\mathbf{n}=0,&\text{ on }\partial\Omega_{1}\end{array}\right.$$

$$\tilde{J}=-\frac{ie\hslash }{2m}({\varphi }^{\ast }\nabla \varphi -\varphi \nabla {\varphi }^{\ast })-\frac{2e^2}{mc}|\varphi {|}^2\mathbf{A},$$

$$J({\varphi }_0,{\mathbf{A}}_0,u_0)=\underset{(\varphi ,\mathbf{A},u)\in \mathcal{C}}{\min }J(\varphi ,\mathbf{A},u).$$

$$J({\varphi }_n,{\mathbf{A}}_n,u_n)\to \eta =\underset{(\varphi ,\mathbf{A},u)\in \mathcal{C}}{\inf }J(\varphi ,\mathbf{A},u).$$

$$\Vert \nabla {\varphi }_n{\Vert }_{0,2,\Omega }\le K,$$

$$\Vert {\varphi }_n{\Vert }_{0,4,\Omega }\le K,$$

$$\Vert {\varphi }_n{\Vert }_{0,2,\Omega }\le K,$$

$$\Vert u_n{\Vert }_{0,2,\partial \Omega }\le K,\forall n\in \mathbb{N}$$

$${\varphi }_n\rightharpoonup {\varphi }_0,\text{ weakly in }{W}^{1,2}(\Omega ,\mathbb{C}),$$

$${\varphi }_n\to {\varphi }_0,\text{ in norm in }{L}^2(\Omega ,\mathbb{C})\text{ and }{L}^4(\Omega ,\mathbb{C}),$$

$$u_n\rightharpoonup u_0,\text{ weakly in }{L}^2(\partial \Omega ,\mathbb{C}),\text{ as }n\to \infty .$$

$$0={\rho }_{1,n}$$

$$\Vert {\mathbf{A}}_n{\Vert }_{0,4,{\Omega }_1}^2$$

$$\text{ div }{\mathbf{A}}_n=0,\text{ in }{\Omega }_1,\forall n\in \mathbb{N}.$$

$$0={\rho }_{1,n}$$

$$\Vert {\mathbf{A}}_{n_k}{\Vert }_{0,4,{\Omega }_1}\to \infty ,\text{ as }k\to \infty .$$

$${\rho }_{2,n_k}\to \infty ,\text{ as }k\to \infty ,$$

$${\rho }_{2,n}\le 0,\forall n\in \mathbb{N}.$$

Full text

1 Introduction

This work develops an existence result for an optimal control problem closely related to the Ginzburg-Landau system in superconductivity. First, we recall that about the year 1950 Ginzburg and Landau introduced a theory to model the super-conducting behavior of some types of materials below a critical temperature $T_{c}$, which depends on the material in question. They postulated the free density energy may be written close to $T_{c}$ as

[TABLE]

where $\psi$ is a complex parameter, $F_{n}(T)$ and $F_{s}(T)$ are the normal and super-conducting free energy densities, respectively (see [2] for details). Here $\Omega\subset\mathbb{R}^{3}$ denotes the super-conducting sample with a boundary denoted by $\partial\Omega=\Gamma.$ The complex function $\psi\in W^{1,2}(\Omega;\mathbb{C})$ is intended to minimize $F_{s}(T)$ for a fixed temperature $T$.

Denoting $\alpha(T)$ and $\beta(T)$ simply by $\alpha$ and $\beta$, the corresponding Euler-Lagrange equations are given by:

[TABLE]

This last system of equations is well known as the Ginzburg-Landau (G-L) one. In the physics literature is also well known the G-L energy in which a magnetic potential here denoted by A is included. The functional in question is given by:

[TABLE]

Considering its minimization on the space $U$, where

[TABLE]

through the physics notation the corresponding Euler-Lagrange equations are:

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

is a known applied magnetic field.

Existence of a global solution for a similar problem has been proved in [3].

2 An existence result for a related optimal control problem

Let $\Omega\subset\mathbb{R}^{3}$, $\Omega_{1}\subset\mathbb{R}^{3}$ be open, bounded and connected sets with Lipschitzian boundaries, where $\overline{\Omega}\subset\Omega_{1}$ and $\Omega_{1}$ is convex. Let $\phi_{d}:\Omega\rightarrow\mathbb{C}$ be a known function in $L^{4}(\Omega;\mathbb{C})$ and consider the problem of minimizing

[TABLE]

with $(\phi,\mathbf{A},u)$ subject to the satisfaction of the Ginzburg-Landau equations, indicated in $(\ref{a.1})$ and $(\ref{a.2})$ in the next lines.

For such a problem, the control variable is $u\in L^{2}(\partial\Omega;\mathbb{C})$ and the state variables are the Ginzburg-Landau order parameter $\phi\in W^{1,2}(\Omega,\mathbb{C})$ and the magnetic potential $\mathbf{A}\in W^{1,2}(\Omega_{1},\mathbb{R}^{3}).$

Our main existence result is summarized by the following theorem.

Theorem 2.1**.**

Consider the functional

[TABLE]

subject to $(\phi,\mathbf{A},u)\in\mathcal{C},$ where

[TABLE]

where

[TABLE]

and

[TABLE]

where,

[TABLE]

and where $\varepsilon>0$ is a small parameter, $K_{1}>0$ and $K_{2}>0$.

Under such hypotheses, there exists $(\phi_{0},\mathbf{A}_{0},u_{0})\in\mathcal{C}$ such that

[TABLE]

Proof.

Let $\{(\phi_{n},\mathbf{A}_{n},u_{n})\}$ be a minimizing sequence (such a sequence exists from the existence result for $u=0$ in [3], and from the fact that $J$ is lower bounded by [math]).

Thus, such a sequence is such that

[TABLE]

From the expression of $J$, there exists $K>0$ such that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

so that, from the Rellich-Kondrashov Theorem, there exists a not relabeled subsequence, $\phi_{0}\in W^{1,2}(\Omega,\mathbb{C})$ and $u_{0}\in L^{2}(\Omega,\mathbb{C})$ such that

[TABLE]

[TABLE]

[TABLE]

On the other hand, we have from (6), from the generalized Hölder inequality and for constants $\gamma=\frac{4\pi}{c}\left|\frac{-ie\hbar}{2m}\right|>0$ and $\gamma_{1}=\frac{4\pi}{c}\frac{2e^{2}}{mc}>0$ that

[TABLE]

From the Friedrichs Inequality (see [8] for details) and the Sobolev Imbedding Theorem for appropriate constants indicated, we obtain

[TABLE]

since from the London Gauge assumption,

[TABLE]

Summarizing, we have obtained, for some appropriate $K_{5}>0$,

[TABLE]

Now, suppose to obtain contradiction there exists a subsequence $\{n_{k}\}\subset\mathbb{N}$ such that

[TABLE]

From (9) we obtain

[TABLE]

which contradicts

[TABLE]

Hence, there exists $K_{6}>0$ such that

[TABLE]

and

[TABLE]

From this and (7) we have,

[TABLE]

Suppose to obtain contradiction there exists a subsequence $\{n_{k}\}\subset\mathbb{N}$ such that

[TABLE]

From (10) we obtain

[TABLE]

which contradicts

[TABLE]

Hence, there exists $K_{7}>0$ such that

[TABLE]

so that from this, the Friedrichs inequality and the London Gauge hypothesis, we obtain $K_{8}>0$ such that

[TABLE]

So from such a result and the Rellich-Kondrashov Theorem there exists a not relabeled subsequence and $\mathbf{A}_{0}\in W^{1,2}(\Omega_{1},\mathbb{R}^{3})$ such that

[TABLE]

[TABLE]

Moreover from the Sobolev Imbedding Theorem, there exist real constants $\hat{K}>0,\;\hat{K}_{1}>0$ such that

[TABLE]

Thus, from this and the first equation in (5), there exist real constants $\hat{K}_{2}>0,\ldots,\hat{K}_{6}>0,$ such that

[TABLE]

From this, up to a subsequence, we get

[TABLE]

Let

[TABLE]

From the last results, we may easily obtain the following limits

[TABLE] 2. 2.

[TABLE] 3. 3.

[TABLE] 4. 4.

[TABLE] 5. 5.

[TABLE] 6. 6.

[TABLE] 7. 7.

[TABLE] 8. 8.

[TABLE] 9. 9.

[TABLE]

For example, for (4), for an appropriate real $\tilde{K}>0$ we have

[TABLE]

The other items may be proven similarly.

Now let $\varphi\in C^{\infty}(\overline{\Omega},\mathbb{C}).$ Observe that

[TABLE]

From this and from

[TABLE]

we have

[TABLE]

so that in such a distributional sense,

[TABLE]

The other boundary condition may be dealt similarly. Thus, from these last results we may infer that in the distributional sense,

[TABLE]

and

[TABLE]

where,

[TABLE]

Hence $(\phi_{0},\mathbf{A}_{0},u_{0})\in\mathcal{C}.$

Finally, from $\phi_{n}\rightarrow\phi_{0}$ in $L^{2}$ and $L^{4}$, $\phi_{n}\rightharpoonup\phi_{0}\text{ weakly in }W^{1,2},$ $u_{n}\rightharpoonup u_{0}\text{ weakly in }L^{2}(\partial\Omega),$ by continuity in $\phi$ and the convexity of $J$ in $\nabla\phi$ and $u$, we have,

[TABLE]

The proof is complete. ∎

3 Conclusion

In this article we have developed a global existence result for a control problem related to the Ginzburg-Landau system in superconductivity. We emphasize the control variable $u$ acts on the super-conducting sample boundary, whereas the state variables, namely, the order parameter $\phi$ and the magnetic potential $\mathbf{A}$ are defined on $\Omega$ and $\Omega_{1}$, respectively. The problem has non-linear constraints but the cost functional is convex. Finally, we highlight the London Gauge assumption and the Friedrichs Inequality have a fundamental role in the establishment of the main results.