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 \(T\).


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)
Full Text: DOI