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Control and stabilization of electromagnetic waves. (Contrôle et stabilisation d’ondes électromagnétiques.) (French) Zbl 0942.93002
The author deals with the exact controllability and stabilization of the Maxwell equation. First of all, he presents the notions and results on the propagation of singularities of the electromagnetic field in a bounded domain. These results, obtained by microlocal analysis, are used to obtain the main results contained in the present paper.
Thus, for the boundary controllability, it is proved, by assuming a geometrical control condition and using the results of the propagation of singularities, that for every \((E_0,H_0)\in \nu^*\cap ((L^2(\Omega))^3\times{\mathcal M}_E)\), there exists a control \(J\in L^2(0, T; L^2_n(\partial\Omega))\), such that the solution of the problem \[ \begin{cases} \varepsilon\partial_t E-\text{rot }H= 0,\;\mu\partial_t H+\text{rot } E=0\quad &\text{in }\Omega\times [0,T[\\ \text{div }E= 0,\;\text{div }H= 0\quad &\text{in }\Omega\times [0,T[\\ E(\cdot, 0)= E_0,\;H(\cdot, 0)= H_0\quad &\text{in }\Omega\\ H\wedge n= J\cdot 1|_\Gamma\quad &\text{on }\partial\Omega\times ]0,T[\end{cases}\tag{1} \] satisfies \((E,H)\equiv 0\) for \(t\geq T\), where \((E,H)\) is the electromagnetic field, the permittivity \(\varepsilon\) and permeability \(\mu\) are strictly positive constants, \[ L^2_n(\partial\Omega)= \{X\in(L^2(\partial\Omega))^3;\;X\cdot n|_{\partial\Omega}= 0\}, \] \[ \nu^*= \{f\in (L^2(\Omega))^3;\;\text{div }f= 0,\;f\cdot n|_{\partial\Omega}= 0\}\times \{g\in (L^2(\partial\Omega))^3;\;\text{div }g= 0\} \] and \({\mathcal M}_E\) is the orthogonal of \[ H_2(\Omega)= \{f\in (L^2(\Omega))^3;\;\text{div }f= 0,\;\text{rot }f= 0,\;f\wedge n|_{\partial\Omega}= 0\}. \] The proof of this result is based on the following observability estimation \[ \int^T_0 \int_\Omega|(E, H)|^2\leq c\int^T_0 \int_\Gamma|H|^2, \] and uses the techniques of the work of Bardos et al. on the wave equation.
Also, for the internal controllability it is proved that for every \((E_0,H_0)\in \nu\cap S_H\), there exists a control \(J\in L^2(0,T; L^2_{\text{div }D}(\Omega))\), such that the solution of \[ \begin{cases} \varepsilon\partial_t E-\text{rot } H= J\cdot 1|_{\omega\times ]0,T[},\;\mu\partial_t H+\text{rot }E= 0\quad &\text{in }\Omega\times [0,+\infty[\quad (\omega\subset\Omega)\\ \text{div}(\mu H)= 0\quad &\text{in }\Omega\times [0,+\infty[\\ E\wedge n= 0,\;H\cdot n=0\quad &\text{on }\partial\Omega\times [0,+\infty[\\ E(\cdot, 0)=E_0,\;H(\cdot,0)= H_0\quad &\text{in }\Omega\end{cases}\tag{2} \] satisfies \((E,H)(\cdot,t)\equiv 0\) for \(t\geq T\), where \[ \nu= \{f\in (L^2(\Omega))^3; \text{div }f= 0\}\times \{g\in (L^2(\Omega))^3;\;\text{div }g= 0,\;g\cdot n|_{\partial\Omega}= 0\}, \] \[ S_H= (L^2(\Omega))^3\times{\mathcal M}_H, \] \({\mathcal M}_H\) is the orthogonal of \[ H_1(\Omega)= \{g\in (L^2(\Omega))^3;\;\text{div }g= 0,\;g\cdot n|_{\partial\Omega}= 0,\;\text{rot }g= 0\} \] and \[ L^2_{\text{div }0}(\Omega)= \{X\in (L^2(\Omega))^3;\;\text{div }X= 0\}. \] The control is constructed as above, by using the HUM method.
Next, it is shown that related to the boundary stabilization of \[ \begin{cases} \varepsilon\partial_t E-\text{rot }H= 0,\;\mu\partial_t H+\text{rot } E= 0\quad & \text{in }\Omega\times [0,+\infty[\\ \text{div}(\varepsilon E)= 0,\;\text{div}(\mu H)= 0\quad &\text{in }\Omega\times [0,+\infty[\\ E(\cdot,0)= E_0,\;H(\cdot,0)= H_0\quad &\text{in }\Omega\\ E\wedge n= 0,\;H\cdot n= 0\quad &\text{on }\Gamma_0\times [0,+\infty[\\ (E\wedge n)\wedge n+ z(H\wedge n)= 0\quad &\text{on }\Gamma\times [0,+\infty[\end{cases}\tag{3} \] there exists \(c>0\) and \(\beta>0\) such that for every \((E_0,H_0)\in \nu\cap{\mathcal M}_\Gamma\), \[ \varepsilon(t)\leq ce^{-\beta t}\varepsilon(0)\quad\text{holds }\forall t\geq 0, \] and respectively for every \((E_0,H_0)\in{\mathcal W}\cap{\mathcal M}_\Gamma\), \[ (\varepsilon+ \varepsilon')(t)\leq ce^{-\beta t}(\varepsilon+ \varepsilon')(0)\quad\text{holds }\forall t\geq 0, \] where \[ \varepsilon(t):= \textstyle{{1\over 2}}\displaystyle{\int_\Omega} (\varepsilon|H|^2+ \mu|H|^2) \] is the energy of system (3), \[ \begin{split}{\mathcal W}= \{(f,g)\in (H^1(\Omega))^6;\;\text{div }f= 0,\;f\wedge n|_{\Gamma_0}= 0,\\ \text{div }g= 0,\;g\cdot n|_{\Gamma_0}= 0,\;(f\wedge n)\wedge n+ z(g\wedge n)|_{\Gamma}= 0\},\end{split} \] \(z= (\mu/\varepsilon)^{1/2}\), \(\partial\Omega= \Gamma_0\cup\Gamma\), \(\Gamma_0\cap\Gamma= \emptyset\), and \({\mathcal M}_\Gamma\) is the orthogonal of the stationary solutions for the \((L^2(\Omega))^6\) norm.
The case of internal stabilization is more special. This problem is treated with more attention because the condition \(\text{div }E= 0\) is not preserved by Maxwell’s system with Ohm’s law.

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
78A40 Waves and radiation in optics and electromagnetic theory
35B40 Asymptotic behavior of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
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