A nonoverlapping domain decomposition for optimal boundary control of the dynamic Maxwell system (Q2717283)

The author studies the Maxwell system NEWLINE\[NEWLINE\begin{cases} \varepsilon E'- \text{rot }H+ \sigma E= F\\ \mu H'+ \text{rot }E= 0\end{cases}\quad\text{in }Q:\equiv \Omega\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\nu\wedge E- \delta\nu\wedge (H\wedge \nu)= J\quad\text{on }\Sigma:\equiv \Gamma\times (0,T),\quad\delta> 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEE(0)= E_0,\quad H(0)= H_0\quad \text{in }\Omega,NEWLINE\]NEWLINE where \(\varepsilon= (\varepsilon^{jk}(x))\), \(\mu= (\mu^{jk}(x))\) and \(\sigma= (\sigma^{jk}(x))\) are \(3\times 3\) Hermitian matrices with \(L^\infty(\Omega)\) entries, \(\varepsilon\) and \(\mu\) are uniformly positive definite on \(\Omega\), and \(J\) is a control input which is taken from the class NEWLINE\[NEWLINE{\mathcal U}={\mathcal L}^2_\tau(\Sigma):= \{J\mid J\in L^2(0,T);{\mathcal L}^2(\Gamma)), \nu\cdot J(t)= 0\text{ for a.e. }x\in\Gamma\text{ and a.e. }t\in (0,T)\}.NEWLINE\]NEWLINE It is shown that if \((E_0, H_0)\in{\mathcal H}\), \(F\in{\mathcal L}^2(0,T; {\mathcal L}^2(\Omega))\) and \(J\in{\mathcal L}^2_\tau(\Sigma)\), then the system (1) has a unique solution \((E,H)\in C([0,T];{\mathcal H})\) which satisfies \(\nu\wedge E|_\Sigma\in{\mathcal L}^2_\tau(\Sigma)\), where \(L^2(\Omega)\) and \({\mathcal L}^2(\Omega)\) denote the usual spaces of Lebesgue square integrable \(C\)-valued functions and \(C^3\)-valued functions, respectively, and \({\mathcal H}={\mathcal L}^2(\Omega)\times {\mathcal L}^2(\Omega)\) is endowed with the norm NEWLINE\[NEWLINE\|(\phi,\psi)\|^2_{\mathcal H}= \langle\varepsilon\phi, \phi\rangle+ \langle \mu\psi,\psi\rangle.NEWLINE\]NEWLINE It is considered the optimal control problem NEWLINE\[NEWLINE\inf_{J\in{\mathcal U}} \int_\Sigma|J|^2 d\Sigma+ k\|(E(T), H(T))- (E_1, H_1)\|^2_{\mathcal H},\quad k> 0,NEWLINE\]NEWLINE subject to (1), where \((E_1,H_1)\in{\mathcal H}\) is given. The problem (1), (2) admits unique optimal control \(J_{\text{opt}}\) which verifies the optimality system consisting of (1), NEWLINE\[NEWLINE\begin{cases} \varepsilon P'- \text{rot }Q- \sigma P=0\\ \mu Q'+ \text{rot }P= 0\end{cases}\quad\text{in }Q,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\nu\wedge P+\delta\nu\wedge (Q\wedge \nu)= 0\quad\text{on }\Gamma,\tag{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINE(P(T), Q(T))= k((E(T), H(T))- (E_1, H_1))\quad\text{in }\OmegaNEWLINE\]NEWLINE and NEWLINE\[NEWLINEJ_{\text{opt}}= Q_\tau:= \nu\wedge (Q\wedge \nu)|_\Sigma= Q|_\Sigma- (Q|_\Sigma\cdot \nu)\nu.\tag{4}NEWLINE\]NEWLINE The main purpose is to describe an iterative domain decomposition for the optimality system (1), (3), (4) in order to approximate the solution of this system. The convergence of this procedure is studied later.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00047].