An Insensitizing Control Problem for the Ginzburg–Landau Equation

Abstract

In this paper, we prove the existence of insensitizing controls for the nonlinear Ginzburg–Landau equation. Here, we have a partially unknown initial data, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. In general, the problem of finding controls with this property is equivalent to prove a partial null controllability result for an optimality system of cascade type. The novelty here is that we consider functionals depending on the gradient of the state, which leads to a null controllability problem for a system with second-order coupling terms. To manage coupling terms of this order, we need a new Carleman estimate for the solutions of the corresponding adjoint system.

Appendix: On the Existence of Solutions

We will show some results about the existence of solutions for system

$$\begin{aligned} \begin{array}{ll} y_{t} - (1+ia) \Delta y +Ry = F^0, &{} \ \text{ in } Q_T\text{, }\\ -z_{t} - (1-ia) \Delta z +Rz= \nabla \cdot ( \nabla y 1_{{\mathcal {O}}}) +F^1&{} \ \text{ in } Q_T\text{, }\\ y = z = 0 &{} \ \text{ on } \Sigma _T, \\ y|_{t =0} = y_0, z|_{t=T}=0 &{} \ \text{ in } \Omega , \end{array} \end{aligned}$$

(75)

where $(F^0,F^1)$ are given. The proofs in here can be adapted to prove existence of solutions for systems (5) and (7).

For $a\in {\mathbb {R}}$, define the linear unbounded operators

$$\begin{aligned} \left\{ \begin{array}{l} D(L_0)=\{u\in H^1_0(\Omega ); \Delta u\in H^1_0(\Omega )\},\\ L_0u=(1+ia)\Delta u - R u \in H^{1}_0(\Omega ), \end{array}\right. \quad \hbox {and}\quad \left\{ \begin{array}{l} D(L_1)=H^1_0(\Omega ),\\ L_1u=(1-ia)\Delta u - R u \in H^{-1}(\Omega )\end{array}\right. \end{aligned}$$

(76)

Both operators are m-dissipative with dense domains; therefore, they generate the $C_0$ semigroups of contractions ${\mathcal {T}}_0$ and ${\mathcal {T}}_1$, respectively.

Here, we will need once again the spaces

$$\begin{aligned}&Y_0= C([0,T]; D(L_0))\cap C^1([0,T]; H^1_0(\Omega ))\quad \hbox {and} \nonumber \\&Y_1=C([0,T]; D(L_1))\cap C^1([0,T]; H^{-1}(\Omega )). \end{aligned}$$

(77)

We start proving the existence of mild solutions for system (75).

Proposition A.1

(Mild Solutions) Assume that $y_0 \in H^1_0(\Omega )$,

$$\begin{aligned} F^0\in L^2(0,T;H^1_0(\Omega ))\quad \hbox {and}\quad F^1\in L^2(0,T;H^{-1}(\Omega )). \end{aligned}$$

Then, problem (75) possesses a unique mild solution $(y,z)\in C([0,T]; H^1_0(\Omega ))\times C([0,T];H^{-1}(\Omega ))$ in the sense that

$$\begin{aligned} y(t)= {\mathcal {T}}_0(t)y_0+\int _0^t{\mathcal {T}}_0(t-s)F^0(s)\,\mathrm{d}s, \end{aligned}$$

(78)

and

$$\begin{aligned} z(t)=\int _t^{T}{\mathcal {T}}_1(s-t)\left( F^1+\nabla \cdot (\nabla y 1_{{\mathcal {O}}})\right) (s)\,\mathrm{d}s. \end{aligned}$$

(79)

Proof

The existence of $y\in C([0,T]; H^1_0(\Omega ))$ satisfying (78) follows directly from the fact that $(L_0, D(L_0))$ is a m-dissipative operator with dense domain (see Lemma 4.1.5 of [18]). Moreover, a simply computation shows that

$$\begin{aligned}&\Vert \nabla \cdot (\nabla y 1_{{\mathcal {O}}})(t)-\nabla \cdot (\nabla y 1_{{\mathcal {O}}})(t_0)\Vert _{H^{-1}(\Omega )}\nonumber \\&\qquad =\sup _{\zeta \in H^1_{0}(\Omega ),\,\, \Vert \zeta \Vert _{H^1_0(\Omega )}=1}\left| \int _{{\mathcal {O}}}(\nabla y(t)-\nabla y(t_0)) \cdot \nabla \zeta \,\mathrm{d}x\right| \nonumber \\&\qquad \le \left( \int _{\Omega }|\nabla y(t)-\nabla y(t_0)|^2\,\mathrm{d}x\right) ^{\frac{1}{2}}, \end{aligned}$$

(80)

for every $t,t_0\in [0,T]$, which means that $\nabla \cdot (\nabla y 1_{{\mathcal {O}}})\in C([0,T]; H^{-1}(\Omega ))$. Now, using that $(L_1, D(L_1))$ is m-dissipative with dense domain and that $F^1+\nabla \cdot (\nabla y 1_{{\mathcal {O}}})\in L^2(0,T; H^{-1}(\Omega ))$, we have (again by Lemma 4.1.5 of [18]) the existence of $z\in C([0,T]; H^{-1}(\Omega ))$ satisfying (79).$\square$

The next result is dedicated to prove the existence of regular solutions for (75).

Proposition A.2

(Regular Solutions) Assume that $y_0 \in D(L_0)$,

$$\begin{aligned}&F^0\in C([0,T], H^1_0(\Omega ))\cap W^{1,1}(0,T;H^1_0(\Omega ))\quad \hbox {and} \\&\qquad F^1\in C([0,T], H^{-1}(\Omega ))\cap W^{1,1}(0,T;H^{-1}(\Omega )). \end{aligned}$$

Then, problem (75) possesses a unique regular solution in the sense that

$$\begin{aligned} \begin{array}{l} (y,z)\in Y_0\times Y_1,\\ y_{t} - (1+ia) \Delta y +Ry =F^0, \\ -\,z_{t} - (1-ia) \Delta z +Rz= \nabla \cdot ( \nabla y 1_{{\mathcal {O}}}) +F^1,\\ y|_{t =0} = y_0,\, z|_{t=T}=0. \end{array} \end{aligned}$$

(81)

Proof

We start by using Proposition 4.1.6. in [18], and we get that the mild solution (78) satisfies

$$\begin{aligned} \begin{array}{l} y\in Y_0,\\ y_{t} - (1+ia) \Delta y +Ry =F^0, \\ y|_{t =0} = y_0. \end{array} \end{aligned}$$

(82)

Now, it is not difficult to see that

$$\begin{aligned} \int _0^T\Vert \nabla \cdot \nabla (y_t1_{{\mathcal {O}}})(s)\Vert ^2_{H^{-1}(\Omega )}\,\mathrm{d}s= & {} \int _0^T\left| \sup _{\zeta \in H^1_{0}(\Omega ),\,\, \Vert \zeta \Vert _{H^1_0(\Omega )}=1}\int _{{\mathcal {O}}}\nabla y_t\cdot \nabla \zeta \,\mathrm{d}x\right| ^2\mathrm{d}t\\\le & {} \iint _{Q_T}|\nabla y_t|^2\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

and hence $\nabla \cdot (\nabla y 1_{{\mathcal {O}}})\in C([0,T], H^{-1}(\Omega ))\cap W^{1,1}(0,T; H^{-1}(\Omega ))$. Then, we apply again Proposition 4.1.6. of [18], and we get that the mild solution (79) satisfies

$$\begin{aligned} \begin{array}{l} z\in Y_1,\\ -\,z_{t} - (1-ia) \Delta z +Rz= \nabla \cdot ( \nabla y 1_{{\mathcal {O}}}) +F^1, \\ z|_{t = T} = 0. \end{array} \end{aligned}$$

(83)

Combining (82) and (83), we obtain (81). $\square$

Remark A.1

If we assume in Proposition A.1 that $y_0\in L^2(\Omega )$ and that $(F^0,F^1)\in [L^2(0,T; H^{-1}(\Omega ))]^2$, we still can prove the existence of mild solutions for (75). Indeed, let

$$\begin{aligned} \begin{array}{l} D({\tilde{L}}_1)=H^1_0(\Omega ),\\ {\tilde{L}}_1u=(1+ia)\Delta u - R u \in H^{-1}(\Omega ), \end{array} \end{aligned}$$

(84)

m-dissipative with dense domain, and $\widetilde{{\mathcal {T}}}_1: H^{-1}(\Omega )\rightarrow H^{-1}(\Omega )$ the semigroup generated by it. Then, the mild solution

$$\begin{aligned} y(t)= \widetilde{{\mathcal {T}}}_1(t)y_0+\int _0^t\widetilde{{\mathcal {T}}_1}(t-s)F^0(s)\,\mathrm{d}s, \end{aligned}$$

is such that $y\in C([0,T]; H^{-1}(\Omega ))$. Using a density argument from regular solutions, we can prove that $y\in C([0,T]; L^2(\Omega ))\cap L^2(0,T; H^1_0(\Omega ))$, and hence

$$\begin{aligned} \nabla \cdot (\nabla y1_{{\mathcal {O}}})\in L^2(0,T; H^{-1}(\Omega )). \end{aligned}$$

Going back to formula (79), we can use again a density argument from regular solutions, and we obtain that $z\in C([0,T]; L^2(\Omega ))\cap L^2(0,T; H^1_0(\Omega ))$.

Remark A.2

The proofs in here can be adapted to prove the existence of regular and mild solutions for system (7).

Regular Solutions If we assume that $\psi _0 \in D(L_0)$, $g^0\in C([0,T], H^{-1}(\Omega ))\cap W^{1,1}(0,T;H^{-1}(\Omega ))$, and that $g^1\in C([0,T], H^1_0(\Omega ))\cap W^{1,1}(0,T;H^1_0(\Omega ))$, we can prove existence of regular solution for (7), in the sense that

$$\begin{aligned} \begin{array}{l} (\varphi ,\psi )\in Y_1\times Y_0,\\ -\,\varphi _{t} - (1-ia) \Delta \varphi +R\varphi = \nabla \cdot (\nabla \psi 1_{{\mathcal {O}}})+g^0, \\ \psi _{t} - (1+ia) \Delta \psi +R\psi = g^1,\\ \psi |_{t =0} = \psi _{0}, \varphi |_{t=T}=0. \end{array} \end{aligned}$$

(85)

Mild Solutions If we assume that $\psi _0 \in L^2(\Omega )$ and $(g^0,g^1)\in [L^2(0,T;H^{-1}(\Omega ))]^2$, there exists a unique mild solution $(\varphi ,\psi )\in [C([0,T]; H^{-1}(\Omega ))]^2$ of (7). In the same way as in Remark A.1, we can prove that $(\varphi ,\psi )\in [C([0,T]; L^2(\Omega ))\cap L^2(0,T; H^1_0(\Omega ))]^2$.

In what follows, we will talk about transposition solutions that are of particular interest for the purposes of this paper.

Let $y_0\in L^2(\Omega )$ and $(F^0,F^1)\in [L^2(0,T; H^{-1}(\Omega ))]^2$. We say that a pair $(y,z)\in L^2(0,T;H^1_0(\Omega ))\times L^2(Q_T)$ is a solution in the transposition sense of (75), if it satisfies

$$\begin{aligned} \begin{array}{l} \displaystyle \int _{0}^T\langle G^0, y\rangle _{H^{-1}H^1_0}\,\mathrm{d}t=Re\int _{\Omega } \varphi (0)\overline{y_0}\,\mathrm{d}x+\int _0^T\langle F^0,\varphi \rangle _{H^{-1}H^1_0}\,\mathrm{d}t,\\ \displaystyle Re\iint _{Q_T} G^1{\bar{z}}\,\mathrm{d}x\mathrm{d}t=-Re\int _0^T\int _{{\mathcal {O}}}\nabla y\cdot \overline{\nabla \psi }\,\mathrm{d}x\mathrm{d}t+\int _{0}^T\langle F^1,\psi \rangle _{H^{-1} H^1_0}\,\mathrm{d}t, \end{array} \end{aligned}$$

(86)

for every $(G^0,G^1)\in L^2(0,T; H^{-1}(\Omega ))\times L^2(Q_T)$, where $(\varphi ,\psi )$ is the solution of

$$\begin{aligned} \begin{array}{lll} -\,\varphi _{t} - (1-ia) \Delta \varphi +R\varphi =G^0 &{} \text{ in } &{} Q_T,\\ \psi _{t} - (1+ia) \Delta \psi +R\psi = G^1&{} \text{ in } &{}Q_T,\\ \varphi =\psi = 0 &{} \text{ on } &{}\Sigma _T,\\ \psi |_{t =0} = 0, \varphi |_{t=T}=0 &{} \text{ in }&{} \Omega . \end{array} \end{aligned}$$

(87)

We have the following result about the existence and uniqueness of transposition solutions.

Proposition A.3

(Transposition Solution) For $y_0\in L^2(\Omega )$ and $(F^0,F^1)\in [L^2(0,T; H^{-1}(\Omega ))]^2$, there exists a unique $(y,z)\in [L^2(0,T; H^1_0(\Omega ))]^2$ satisfying (86) for every $(G^0,G^1)\in L^2(0,T; H^{-1}(\Omega ))\times L^2(Q_T)$, where $(\varphi ,\psi )$ is solution of (87).

Proof

Let $\Phi _1: L^2(0,T; H^1_0(\Omega ))\rightarrow {\mathbb {R}}$ the operator

$$\begin{aligned} \Phi _1(h^0)=Re\int _{\Omega } \varphi (0)\overline{y_0}\,\mathrm{d}x+\int _0^T\langle F^0,\varphi \rangle _{H^{-1}H^1_0}\,\mathrm{d}t, \end{aligned}$$

where $\varphi$ satisfies the first equation of (87) for $G^0=-\Delta h^0\in L^2(0,T;H^{-1}(\Omega ))$. From energy estimates, it is easy to see that $\Phi _1$ is continuous. Then, from Lax–Milgram theorem, there exists $y\in L^2(0,T; H^1_0(\Omega ))$ such that

$$\begin{aligned} \int _{0}^T\langle G^0, y\rangle _{H^{-1}H^1_0}\,\mathrm{d}t=Re\iint _{Q_T}\nabla h^0\cdot \nabla {\bar{y}}\,\mathrm{d}x\mathrm{d}t= \Phi _1(h^0), \end{aligned}$$

for every $G^0\in H^{-1}(\Omega )$, where $-\Delta h^0=G^0$. The existence of $z\in L^2(Q_T)$ satisfying the second equation of (86) follows in a completely analogous way, since the linear form

$$\begin{aligned} \Phi _2(G^1)=-\,Re\int _0^T\int _{{\mathcal {O}}}\nabla y\cdot \overline{\nabla \psi }\,\mathrm{d}x\mathrm{d}t+\int _{0}^T\langle F^1,\psi \rangle _{H^{-1} H^1_0}\,\mathrm{d}t, \end{aligned}$$

is continuous in $L^2(Q_T)$.

To prove that $z\in L^2(0,T; H^1_0(\Omega ))$, we can proceed in the following way. First, we take sequences of regular data such that $y_0^n\rightarrow y_0$ in $L^2(\Omega )$ and $(F^0_n, F^1_n)\rightarrow (F^0, F^1)$ in $L^2(0,T; H^{-1}(\Omega ))\times L^2(Q_T)$. We show that the regular solutions $(y_n,z_n)$ for (75) (whose existence is given in Proposition A.2) with initial data $y_0^n$ and $(F^0_n, F^1_n)$ on the right-hand side, are also a solution in the transposition sense; moreover, it is bounded in $[L^2(0,T; H^1_0(\Omega ))]^2$. Hence, in the limit, we obtain that $(y,z)\in [L^2(0,T; H^1_0(\Omega ))]^2$.

Now, let us prove that the solution (y, z) is unique. If $({\hat{y}}, {\hat{z}})$ is another solution, then

$$\begin{aligned} Re\int _{0}^T\langle G^0, y-{\hat{y}}\rangle _{H^{-1}H^1_0}\,\mathrm{d}t=0,\quad \hbox {for all}\quad G^0\in L^2(0,T; H^{-1}(\Omega )), \end{aligned}$$

and

$$\begin{aligned} Re\iint _{Q_T} G^1(z-{\hat{z}})\,\mathrm{d}x\mathrm{d}t=-Re\int _0^T\int _{{\mathcal {O}}}(\nabla y-\nabla {\hat{y}})\cdot \overline{\nabla \psi }\,\mathrm{d}x\mathrm{d}t,\quad \hbox {for all}\quad G^1\in L^2(Q_T). \end{aligned}$$

Hence, $y={\hat{y}}$ and $z={\hat{z}}$.$\square$