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Zbl 0942.93002
Phung, Kim Dang
Control and stabilization of electromagnetic waves. (Contrôle et stabilisation d'ondes électromagnétiques.) (French)
[J] ESAIM, Control Optim. Calc. Var. 5, 87-137 (2000). ISSN 1292-8119; ISSN 1262-3377/e

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.\par 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 $(E_0,H_0)\in \nu^*\cap ((L^2(\Omega))^3\times{\cal M}_E)$, there exists a control $J\in L^2(0, T; L^2_n(\partial\Omega))$, such that the solution of the problem $$\cases \varepsilon\partial_t E-\text{rot }H= 0,\ \mu\partial_t H+\text{rot } E=0\quad &\text{in }\Omega\times [0,T[\\ \text{div }E= 0,\ \text{div }H= 0\quad &\text{in }\Omega\times [0,T[\\ E(\cdot, 0)= E_0,\ H(\cdot, 0)= H_0\quad &\text{in }\Omega\\ H\wedge n= J\cdot 1|_\Gamma\quad &\text{on }\partial\Omega\times ]0,T[\endcases\tag 1$$ satisfies $(E,H)\equiv 0$ for $t\ge T$, 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 ${\cal M}_E$ 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 $$\int^T_0 \int_\Omega|(E, H)|^2\le c\int^T_0 \int_\Gamma|H|^2,$$ and uses the techniques of the work of Bardos et al. on the wave equation.\par 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 $$\cases \varepsilon\partial_t E-\text{rot } H= J\cdot 1|_{\omega\times ]0,T[},\ \mu\partial_t H+\text{rot }E= 0\quad &\text{in }\Omega\times [0,+\infty[\quad (\omega\subset\Omega)\\ \text{div}(\mu H)= 0\quad &\text{in }\Omega\times [0,+\infty[\\ E\wedge n= 0,\ H\cdot n=0\quad &\text{on }\partial\Omega\times [0,+\infty[\\ E(\cdot, 0)=E_0,\ H(\cdot,0)= H_0\quad &\text{in }\Omega\endcases\tag 2$$ satisfies $(E,H)(\cdot,t)\equiv 0$ for $t\ge T$, where $$\nu= \{f\in (L^2(\Omega))^3; \text{div }f= 0\}\times \{g\in (L^2(\Omega))^3;\ \text{div }g= 0,\ g\cdot n|_{\partial\Omega}= 0\},$$ $$S_H= (L^2(\Omega))^3\times{\cal M}_H,$$ ${\cal M}_H$ 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.\par Next, it is shown that related to the boundary stabilization of $$\cases \varepsilon\partial_t E-\text{rot }H= 0,\ \mu\partial_t H+\text{rot } E= 0\quad & \text{in }\Omega\times [0,+\infty[\\ \text{div}(\varepsilon E)= 0,\ \text{div}(\mu H)= 0\quad &\text{in }\Omega\times [0,+\infty[\\ E(\cdot,0)= E_0,\ H(\cdot,0)= H_0\quad &\text{in }\Omega\\ E\wedge n= 0,\ H\cdot n= 0\quad &\text{on }\Gamma_0\times [0,+\infty[\\ (E\wedge n)\wedge n+ z(H\wedge n)= 0\quad &\text{on }\Gamma\times [0,+\infty[\endcases\tag 3$$ there exists $c>0$ and $\beta>0$ such that for every $(E_0,H_0)\in \nu\cap{\cal M}_\Gamma$, $$\varepsilon(t)\le ce^{-\beta t}\varepsilon(0)\quad\text{holds }\forall t\ge 0,$$ and respectively for every $(E_0,H_0)\in{\cal W}\cap{\cal M}_\Gamma$, $$(\varepsilon+ \varepsilon')(t)\le ce^{-\beta t}(\varepsilon+ \varepsilon')(0)\quad\text{holds }\forall t\ge 0,$$ where $$\varepsilon(t):= \textstyle{{1\over 2}}\displaystyle{\int_\Omega} (\varepsilon|H|^2+ \mu|H|^2)$$ is the energy of system (3), $$\multline{\cal W}= \{(f,g)\in (H^1(\Omega))^6;\ \text{div }f= 0,\ f\wedge n|_{\Gamma_0}= 0,\\ \text{div }g= 0,\ g\cdot n|_{\Gamma_0}= 0,\ (f\wedge n)\wedge n+ z(g\wedge n)|_{\Gamma}= 0\},\endmultline$$ $z= (\mu/\varepsilon)^{1/2}$, $\partial\Omega= \Gamma_0\cup\Gamma$, $\Gamma_0\cap\Gamma= \emptyset$, and ${\cal M}_\Gamma$ is the orthogonal of the stationary solutions for the $(L^2(\Omega))^6$ norm.\par 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.
[M.Ivanovici (Craiova)]

MSC 2000:: *93B05 Controllability
93C20 Control systems governed by PDE
78A40 Waves and radiation
35B40 Asymptotic behavior of solutions of PDE
35Q60 PDE of electromagnetic theory and optics

Keywords: exact controllability; Maxwell equation; boundary controllability; internal controllability; HUM method; boundary stabilization

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