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
Full Text: DOI EuDML


[1] H. Barucq, Étude asymptotique du système de Maxwell avec conditions aux limites absorbantes. Thèse de l’université de Bordeaux I ( 1993).
[2] N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki. Asterisque 245 ( 1997) 167-195. Zbl0954.35102 MR1627111 · Zbl 0954.35102
[3] H. Barucq et B. Hanouzet, Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller II. C. R. Acad. Sci. Paris Sér. I Math. 316 ( 1993) 1019-1024. Zbl0776.35073 MR1222965 · Zbl 0776.35073
[4] C. Bardos, L. Halpern, G. Lebeau, J. Rauch et E. Zuazua, Stabilisation de l’équation des ondes au moyen d’un feedback portant sur la condition aux limites de Dirichlet. Asymptot. Anal. 4 ( 1991) 285-291. Zbl0764.35055 MR1127003 · Zbl 0764.35055
[5] C. Bardos, G. Lebeau et J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 ( 1992) 1024-1065. Zbl0786.93009 MR1178650 · Zbl 0786.93009
[6] M. Cessenat, Mathematical method in electromagnetism - linear theorie and applications. Ser. Adv. Math. Appl. Sci. 41 ( 1996). Zbl0917.65099 MR1409140 · Zbl 0917.65099
[7] R. Dautray et J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Masson ( 1988). Zbl0642.35001 · Zbl 0642.35001
[8] V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations. Panamer. Math. J. 4 ( 1994) 47-61. Zbl0849.35136 MR1310321 · Zbl 0849.35136
[9] J. Lagnese, Exact boundary controllability of Maxwell’s equations in a general region. SIAM J. Control Optim. 27 ( 1989) 374-388. Zbl0678.49032 MR984833 · Zbl 0678.49032
[10] G. Lebeau, Contrôle et stabilisation hyperboliques. Séminaire E.D.P. École Polytechnique ( 1990). Zbl0706.35004 MR1073191 · Zbl 0706.35004
[11] J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués. RMA, Masson, Paris ( 1988). Zbl0653.93002 MR953547 · Zbl 0653.93002
[12] J.-L. Lions et E. Magenes, Problèmes aux limites non homogènes. Dunod ( 1968). · Zbl 0165.10801
[13] G. Lebeau et L. Robbiano, Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 ( 1997) 465-491. Zbl0884.58093 MR1432305 · Zbl 0884.58093
[14] R. Melrose et J. Sjöstrand, Singularities of boundary value problems I. Comm. Pure Appl. Math. 31 ( 1978) 593-617. Zbl0368.35020 MR492794 · Zbl 0368.35020
[15] R. Melrose et J. Sjöstrand, Singularities of boundary value problems II. Comm. Pure Appl. Math. 35 ( 1982) 129-168. Zbl0546.35083 MR644020 · Zbl 0546.35083
[16] O. Nalin, Contrôlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris Sér. I Math. 309 ( 1989) 811-815. Zbl0688.49041 MR1055200 · Zbl 0688.49041
[17] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York ( 1983). Zbl0516.47023 MR710486 · Zbl 0516.47023
[18] K.-D. Phung, Stabilisation frontière du système de Maxwell avec la condition aux limites absorbante de Silver-Müller. C. R. Acad. Sci. Paris Sér. I Math. 3233 ( 1995) 187-192. Zbl0844.35125 MR1320353 · Zbl 0844.35125
[19] K.-D. Phung, Contrôlabilité exacte et stabilisation interne des équations de Maxwell. C. R. Acad. Sci. Paris Sér. I Math. 3233 ( 1996) 169-174. Zbl0858.93011 MR1402537 · Zbl 0858.93011
[20] J.V. Ralston, Solutions of Wave equation with localized energy. Comm. Pure. Appl. Math. 22 ( 1969) 807-823. Zbl0209.40402 MR254433 · Zbl 0209.40402
[21] J. Rauch et M. Taylor, Penetration into shadow region and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Mathematics J. 22 ( 1972) 277-284. Zbl0227.35064 MR303098 · Zbl 0227.35064
[22] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, N.J. ( 1981). Zbl0453.47026 MR618463 · Zbl 0453.47026
[23] M. Taylor, Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 ( 1975) 457-478. Zbl0332.35058 MR509098 · Zbl 0332.35058
[24] M. Taylor, Grazing rays and reflection of singularities of solutions to wave equations II. Comm. Pure Appl. Math. 29 ( 1976) 463-481. Zbl0335.35059 MR427839 · Zbl 0335.35059
[25] N. Week, Exact boundary controllability for a Maxwell problem, submitted to SIAM J. Control. Optim. Zbl0963.93040 · Zbl 0963.93040
[26] N. Weck et K.J. Witsch, Low frequency asymptotics for dissipative Maxwell’s equations in bounded domains. Math. Methods Appl. Sci. 13 ( 1990) 81-93. Zbl0704.35007 MR1060225 · Zbl 0704.35007
[27] K. Yamamoto, Singularities of solutions to the boundary value problem for elastic and Maxwell’s equations. Japan J. Math. (N.S.) 14 ( 1988) 119-163. Zbl0669.73017 MR945621 · Zbl 0669.73017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.