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Exact boundary controllability of a Maxwell problem. (English) Zbl 0963.93040

The author studies the exact boundary controllability for the following simplified form of a Maxwell problem in the time interval \(I:= [0,T]\): \[ \begin{cases} \partial_t E=\nabla_x\wedge H,\;\partial_t H=-\nabla_x\wedge E\quad &\text{in }I\times \Omega,\\ (E(0,\cdot), H(0,\cdot))= (E^0, H^0),\\ \nu\wedge E= J\quad &\text{on }I\times \Gamma,\end{cases}\tag{1} \] where \(\Omega\subset \mathbb{R}^3\) is a bounded open set lying on one side of its \(C^\infty\)-boundary \(\partial\Omega:= \Gamma\), \((E^0, H^0)\) is the initial state and \(J\) is the boundary control.
The author investigates the problem of steering some initial state \((E^0, H^0)\) of (1) to rest by controlling the lateral boundary condition. Also, one completely identifies those initial states which can be controlled and those which cannot.
Let \({\mathcal L}^2:= \{v: \Omega\to\mathbb{R}^3\mid v_1,v_2,v_3\in L^2\}\) and \({\mathcal H}^s\) denotes, for each \(s\in\mathbb{R}\), the \(L^2\)-Sobolev spaces in the vector-valued case. The author introduces the following spaces: \[ \begin{aligned} {\mathcal R} &:= \{v\in {\mathcal L}^2\mid\nabla\wedge v:= \text{curl }v\in{\mathcal L}^2\};\;{\mathcal D}:= \{v\in{\mathcal L}^2\mid\langle\nabla, v\rangle:= \text{div }v\in{\mathcal L}^2\},\\ {\mathcal R}_0 &:= \{v\in{\mathcal R}\mid\nabla\wedge v= 0\};\;{\mathcal D}_0:= \{v\in{\mathcal D}\mid\langle\nabla, v\rangle= 0\},\\ \mathring{{\mathcal R}} &:= \{v\in{\mathcal R}\mid\langle\nabla\wedge v,\varphi\rangle= \langle v,\nabla\wedge \varphi\rangle\;\forall\varphi\in {\mathcal R}\},\\ \mathring{{\mathcal D}} &:= \{v\in{\mathcal D}\mid \langle\text{div } v,\varphi\rangle= -\langle v,\nabla\varphi\rangle\;\forall\varphi\in H^1\},\\ {\mathcal H}_N &:={\mathcal R}_0\cap{\mathcal D}_0\cap \mathring{{\mathcal D}}\end{aligned} \] and the following spaces of states: \[ \begin{aligned} S_2 &:= (\nabla\wedge{\mathcal R})\times ((\nabla\wedge \mathring{{\mathcal R}})\oplus{\mathcal H}_N\oplus \chi_{1/2}(\Omega));\;S_1:= (\nabla\wedge{\mathcal R})\times ((\nabla \wedge \mathring{{\mathcal R}})\oplus{\mathcal H}_N);\\ S_0 &:= (\nabla\wedge {\mathcal R})\times (\nabla\wedge \mathring{{\mathcal R}}); S^1_k:= ({\mathcal H}^1\times{\mathcal H}^1)\cap S_k, k= 0,1,2; S^D_k:= D(A)\cap S^1_k, k= 0,1,2,\end{aligned} \] where \(A: D(A)\subset \mathring{{\mathcal R}}\times{\mathcal R}\subset{\mathcal L}^2\times{\mathcal L}^2\to{\mathcal L}^2\times{\mathcal L}^2\), \((E, H)\to (i\nabla\wedge H,-i\nabla\wedge E)\) is selfadjoint with respect to the natural scalar product \[ \langle\langle(E^1, H^1),(E^2, H^2)\rangle\rangle:= \langle E^1, E^2\rangle+\langle H^1, H^2\rangle. \] The main theorem states that if \(T> \text{diam}(\Omega)\), then each state in \(S^1_2\) can be steered to rest in time \(T\). In particular, each divergence-free state can be steered to rest in time \(T\) if \(\partial\Omega\) is connected. This result is proved in four steps: The initial state \((\varepsilon^0,H^0)\in S^1_2\) may not be an element of \(D(A)\). Firstly, the author shows that it takes arbitrarily short time to steer it into a state which is in \(S^D_2\). In the second and third steps there are removed extra components of \(H^0\) by steering into \(S^D_1\) and then into \(S^D_0\), again in arbitrarily short time. Finally, granted time \(T> \text{diam}(\Omega)\), the resulting state \((E^0, H^0)\in S^D_0\) may be steered to rest by using the technique of D. L. Russell for the wave equation.
For the class of regions \(\Omega\) which contain a segment of length \(\text{diam }\Omega\), the main theorem is almost optimal in the sense that for \(T< \text{diam}(\Omega)\), even in \(S^D_0\) there are states which cannot be steered to rest in time \(T\).

MSC:

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35B37 PDE in connection with control problems (MSC2000)
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
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