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A nonoverlapping domain decomposition for optimal boundary control of the dynamic Maxwell system. (English) Zbl 0979.93058
Chen, Goong (ed.) et al., Control of nonlinear distributed parameter systems. Partly proceedings of the conference advances in control of nonlinear distributed parameter systems, Texas A & M Univ., College Station, TX, USA. Dedicated to Prof. David L. Russell on the occasion of his 60th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 218, 157-176 (2001).
The author studies the Maxwell system $\begin{cases} \varepsilon E'- \text{rot }H+ \sigma E= F\\ \mu H'+ \text{rot }E= 0\end{cases}\quad\text{in }Q:\equiv \Omega\times (0,T),$ $\nu\wedge E- \delta\nu\wedge (H\wedge \nu)= J\quad\text{on }\Sigma:\equiv \Gamma\times (0,T),\quad\delta> 0,\tag{1}$ $E(0)= E_0,\quad H(0)= H_0\quad \text{in }\Omega,$ 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 ${\mathcal U}={\mathcal L}^2_\tau(\Sigma):= \{J\mid J\in L^2(0,T);{\mathcal L}^2(\Gamma)), \nu\cdot J(t)= 0\text{ for a.e. }x\in\Gamma\text{ and a.e. }t\in (0,T)\}.$ 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 $\|(\phi,\psi)\|^2_{\mathcal H}= \langle\varepsilon\phi, \phi\rangle+ \langle \mu\psi,\psi\rangle.$ It is considered the optimal control problem $\inf_{J\in{\mathcal U}} \int_\Sigma|J|^2 d\Sigma+ k\|(E(T), H(T))- (E_1, H_1)\|^2_{\mathcal H},\quad k> 0,$ 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), $\begin{cases} \varepsilon P'- \text{rot }Q- \sigma P=0\\ \mu Q'+ \text{rot }P= 0\end{cases}\quad\text{in }Q,$ $\nu\wedge P+\delta\nu\wedge (Q\wedge \nu)= 0\quad\text{on }\Gamma,\tag{3}$ $(P(T), Q(T))= k((E(T), H(T))- (E_1, H_1))\quad\text{in }\Omega$ and $J_{\text{opt}}= Q_\tau:= \nu\wedge (Q\wedge \nu)|_\Sigma= Q|_\Sigma- (Q|_\Sigma\cdot \nu)\nu.\tag{4}$ 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.
For the entire collection see [Zbl 0959.00047].

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 49M27 Decomposition methods