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).


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
Full Text: DOI Numdam EuDML


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