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Stabilization of heterogeneous Maxwell’s equations by linear or nonlinear boundary feedbacks. (English) Zbl 1030.93026
The attention of the authors is focused on the Maxwell equations with a nonlinear boundary condition \[ \begin{cases} \varepsilon{dE\over\partial t}-\text{curl }H= 0\quad &\text{in }Q:= \Gamma\times\;]0,+\infty[, \\ \mu{\partial H\over\partial t}+ \text{curl }E= 0\quad &\text{in }Q,\\ \text{div}(\varepsilon E)= \text{div}(\mu H)= 0\quad &\text{in }Q,\\ H\times\nu+ g(E\times \nu)x\nu= 0\quad &\text{on }\Sigma:= \Gamma\times\;]0,+\infty[,\\ E(0)= E_0,\;H(0)= H_0\quad &\text{in }\Omega.\end{cases}\tag{1} \] First the authors consider the linear case (i.e. \(g(E)= E\)). In this case the boundary condition is the classical Silver-Müller boundary condition, and the energy \[ \varepsilon(t)= \textstyle{{1\over 2}} \displaystyle{\int_\Omega \{\varepsilon|E(t,x)|^2+ \mu|H(t, x)|^2\} dx} \] is nonincreasing. The domain \(\Omega\) is said to satisfy the \((\varepsilon, \mu)\)-stability estimate if there exist \(T> 0\) and two nonnegative constants \(C_1\), \(C_2\) (which may depend on \(T\)) with \(C_1< T\) such that \[ \int^T_0 \varepsilon(t) dt\leq C_1\varepsilon(0)+ C_2\int^T_0 \int_\Gamma|H(t)\times \nu|^2 d\sigma dt, \] for all solutions \(\left(\begin{smallmatrix} E(t)\\ H(t)\end{smallmatrix}\right)\) of (1) with \(g(E)= E\).
It is stated that \(\Omega\) satisfies the \((\varepsilon,\mu)\)-stability estimate if and only if there exist two positive constants \(M\) and \(\omega\) such that \[ \varepsilon(t)\leq Me^{-\omega t}\varepsilon(0), \] for all solutions \(\left(\begin{smallmatrix} E(t)\\ H(t)\end{smallmatrix}\right)\) of (1) with \(g(E)= E\).
Based on the linear stability estimate a certain controllability result is obtained for the Maxwell system. More exactly, for all \((E_0, H_0)\) in a certain Hilbert space there exist \(T> 0\) and a control \(J\in L^2(\Gamma\times \;]0,T[)^3\) such that the solution \((E,H)\) of \[ \begin{cases} \varepsilon{\partial E\over\partial t}- \text{curl }H= 0\quad &\text{in }Q_T:= \Omega\times\;]0,T[,\\ \mu{\partial H\over\partial t}+ \text{curl }E= 0\quad &\text{in }Q_T,\\ \text{div}(\varepsilon E)= \text{div}(\mu H)= 0\quad &\text{in }Q_T,\\ H\times\nu= J\quad &\text{on }\Sigma_T:= \Gamma\times \;]0,T[,\\ E(0)= E_0,\;H(0)= H_0\quad &\text{in }\Omega\end{cases} \] satisfies \(E(T)= H(T)= 0\).
Finally, for the general system (1), sufficient conditions on \(g\) are given which lead to an explicit decay rate of the energy. For this purpose, the authors use Liu’s principle, based on the \((\varepsilon,\mu)\)-stability estimate, and a certain integral inequality.

MSC:
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
93B05 Controllability
35Q60 PDEs in connection with optics and electromagnetic theory
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