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A singular perturbation problem in exact controllability of the Maxwell system. (English) Zbl 1030.93025
The author considers the Maxwell system $\begin{cases} \varepsilon E_t- \text{rot }H= 0,\\ \mu H_t+ \text{rot }E= 0\quad &\text{in }Q:= \Omega\times (0,T),\\ \nu\wedge E= J\quad &\text{on }\Sigma:= \Gamma\times (0,T),\\ E(0)= E_0,\;H(0)= H_0\quad &\text{in }\Omega,\end{cases}\tag{1}$ and, respectively, the perturbed system $\begin{cases} \varepsilon E^\delta_t- \text{rot }H^\delta= 0,\\ \mu H^\delta_t+ \text{rot }E^\delta= 0\quad &\text{in }Q,\\ \nu\wedge E^\delta- \delta\nu\wedge (H^\delta\wedge\nu)=J\quad &\text{on }\Sigma,\;\delta> 0,\\ E^\delta(0)= E_0,\;H^\delta(0)= H_0\quad &\text{in }\Omega,\end{cases}\tag{2}$ where $$\Omega$$ is a bounded, open set in $$\mathbb{R}^3$$ with smooth boundary $$\Gamma$$, $$\nu$$ is the exterior pointing unit normal vector to $$\Gamma$$, $$\varepsilon= (\varepsilon^{jk}(x))$$, $$\mu= (\mu^{jk}(x))$$ are positive definite $$3\times 3$$ Hermitian matrices with $$C^\infty(\overline\Omega)$$ entries, and the function $$J$$ is taken from the class $${\mathcal U}= {\mathcal L}^2_T(\Sigma):= \{J\mid J\in L^2(0, T; L^2(\Gamma))$$, $$\nu\cdot J(t)= 0$$ for a.a. $$x\in\Gamma$$ and a.a. $$t\in (0,T)\}$$.
Define: $${\mathcal K}={\mathcal L}^2(\Omega)\times{\mathcal L}^2(\Omega)$$, $${\mathcal D}_{a,0}(\Omega)= \{\chi\in{\mathcal L}^2(\Omega): \text{div}(a\chi)= 0\}$$ for $$a\in L^\infty(\Omega)$$, and $${\mathcal K}_0={\mathcal D}_{\varepsilon,0}(\Omega)\times{\mathcal D}_{\mu,0}(\Omega)$$, which is a closed subspace of $${\mathcal K}$$.
The author considers the problem of exact controllability of the solution of (1) in the space $${\mathcal K}_0$$ at time $$T$$: given fixed but arbitrary $$(E_0, H_0)$$, $$(E_1,H_1)\in{\mathcal K}_0$$, find a control $$J_0\in{\mathcal U}$$ such that the solution of (1) satisfies $E(T)= E_1,\quad H(T)= H_1.\tag{3}$ Without loss of generality, one may assume that $$E_0= H_0= 0$$. It is known that the exact controllability problem has a solution if and only if $${\mathcal K}_0$$ is continuously observable, that is, there is a constant $$c^0_T> 0$$ such that $\|(\phi_0,\psi_0)\|^2_{{\mathcal K}}\leq c^0_T \int_\Sigma |\psi_T|^2 d\Sigma,\quad\forall(\phi_0, \psi_0)\in{\mathcal F}_0,\tag{4}$ where $$\psi_T:= \nu\wedge (\psi\wedge\nu)= \psi-(\psi\cdot\nu)\nu$$, $${\mathcal F}_0={\mathcal F}\cap{\mathcal K}_0$$, $${\mathcal F}= \{(\phi_0, \psi_0)\in{\mathcal K}: \psi_T|_\Sigma\in{\mathcal L}^2_T(\Sigma)\}$$, and where $$(\phi,\psi)$$ is the solution of the problem $\begin{cases} \varepsilon\phi_t- \text{rot }\psi= 0,\\ \mu\psi_t+ \text{rot }\phi= 0\quad &\text{in }Q,\\ \nu\wedge\phi= 0\quad &\text{on }\Sigma,\\\phi(T)= \phi_0,\;\psi(T)= \psi_0\quad &\text{in }\Omega.\end{cases}\tag{5}$ When (4) holds, the control of minimum norm in $${\mathcal L}^2_T(\Sigma)$$ with state constraint (3) is given by $J^0= -\psi_T|_\Sigma,\tag{6}$ where $$(\phi,\psi)$$ is the solution of (5) with final data $$(\phi_0,\psi_0)\in{\mathcal F}_0$$ given by $\langle(E_1, H_1), (\phi_0,\psi_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi_T|^2 d\Sigma.$ Thus, the optimality system for the problem of minimum norm is given by (1) and (5), and the minimum norm $$J^0$$ is given by (6).
Similar considerations can be made for the perturbed system (2). Thus, the problem of exact controllability is: with $$E_0= H_0= 0$$ and given fixed but arbitrary $$(E_1, H_1)\in{\mathcal K}_0$$, find a control $$J^\delta\in{\mathcal U}$$ such that the solution of (2) satisfies $$E^\delta(T)= E_1$$, $$H^\delta(T)= H_1$$.
This problem has a solution if and only if there is a constant $$c^\delta_T> 0$$ such that $\|(\phi_0, \psi_0)\|^2_{{\mathcal K}}\leq c^\delta_T \int_\Sigma |\psi^\delta_T|^2 d\Sigma,\quad\forall (\phi_0, \psi_0)\in{\mathcal K}_0,\tag{7}$ where $$(\phi^\delta, \psi^\delta)$$ is the solution of $\begin{cases} \varepsilon\phi^\delta_t- \text{rot }\psi^\delta= 0,\\ \mu\psi^\delta_t+ \text{rot }\phi^\delta= 0\quad &\text{in }Q,\\ \nu\wedge\phi^\delta+ \delta\psi^\delta_T= 0\quad &\text{on }\Sigma,\\ \phi^\delta(T)= \phi_0,\;\psi^\delta(T)= \psi_0\quad &\text{in }\Omega.\end{cases}\tag{8}$ The aim of this paper is to investigate the connection between the observability estimates (4) and (7), and between the corresponding optimality systems for small values of $$\delta$$.
One of the main results states that if (7) holds for some $$\delta_0> 0$$, $$(E_1,H_1)\in{\mathcal K}_0$$ and $$(\phi^\delta,\psi^\delta)$$ is the solution of (8) with final data $$(\phi^\delta_0, \psi^\delta_0)\in{\mathcal K}_0$$ given by $\langle(E_1, H_1), (\phi^\delta_0, \psi^\delta_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi^\delta_T|^2 d\Sigma,$ then as $$\delta\to 0$$, \begin{alignedat}{2} (\phi^\delta(\cdot), \psi^\delta(\cdot)) &\to (\phi(\cdot),\psi(\cdot))\quad &&\text{weakly}^*\text{ in }L^\infty(0,T;{\mathcal K}),\\ (\phi^\delta_0, \psi^\delta_0) &\to (\phi_0, \psi_0)\quad &&\text{weakly in }{\mathcal K},\end{alignedat} where $\begin{cases} \varepsilon\phi'- \text{rot }\psi= 0,\\ \mu\psi'+ \text{rot }\phi= 0\quad &\text{in }Q,\\ \nu\wedge\phi= 0\quad &\text{on }\Sigma,\\ \phi(T)= \phi_0,\;\psi(T)= \psi_0\quad &\text{in }\Omega.\end{cases}$ Further, $$(\phi_0, \psi_0)\in{\mathcal F}_0$$, $$\psi^\delta_t|_\Sigma\to \psi_T|_\Sigma$$ strongly in $${\mathcal L}^2_T(\Sigma)$$ and $\langle(E_1, H_1), (\phi_0,\psi_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi_T|^2 d\Sigma.$ Another result states that if (7) holds for some $$\delta_0> 0$$, $$E_0= H_0= 0$$, $$(E_1, H_1)\in{\mathcal K}_0$$, and $$(E^\delta, H^\delta)$$ is the solution of (2) with $$J= -\psi_T|_\Sigma$$, then $$(E^\delta, H^\delta)\to (E, H)$$ weakly$$^*$$ in $$L^\infty(0, T; \chi')$$, where $$(E,H)$$ is the solution of (1) with $$J=- \psi_T|_\Sigma$$, and where $$\chi\hookrightarrow{\mathcal K}\hookrightarrow\chi'$$ with $$\chi= {\mathcal R}^0\times ({\mathcal R}\cap{\mathcal D}^0_\mu)$$, $${\mathcal R}= \{\phi\in{\mathcal L}^2(\Omega): \text{rot }\phi\in{\mathcal L}^2(\Omega)\}$$, $${\mathcal R}^0= \{\chi\in{\mathcal R}: \nu\wedge \chi|_\Gamma= 0\}$$, $${\mathcal D}_\mu= \{\chi\in{\mathcal L}^2(\Omega):\text{div}(\mu\chi)\in {\mathcal L}^2(\Omega)\}$$, $${\mathcal D}^0_\mu= \{\chi\in{\mathcal B}_\mu: \nu(\mu\chi)|_\Gamma= 0\}$$.
These results show that the solution of the optimality system for the problem of minimum norm control of (2) from the rest state $$(0,0)$$ to the state $$(E_1, H_1)$$ at time $$T$$ converges in the sense described to the solution of the optimality system for the problem of minimum norm control of (1).

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35Q60 PDEs in connection with optics and electromagnetic theory 49N10 Linear-quadratic optimal control problems
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