×

zbMATH — the first resource for mathematics

A singular perturbation problem in exact controllability of the Maxwell system. (English) Zbl 1030.93025
The author considers the Maxwell system \[ \begin{cases} \varepsilon E_t- \text{rot }H= 0,\\ \mu H_t+ \text{rot }E= 0\quad &\text{in }Q:= \Omega\times (0,T),\\ \nu\wedge E= J\quad &\text{on }\Sigma:= \Gamma\times (0,T),\\ E(0)= E_0,\;H(0)= H_0\quad &\text{in }\Omega,\end{cases}\tag{1} \] and, respectively, the perturbed system \[ \begin{cases} \varepsilon E^\delta_t- \text{rot }H^\delta= 0,\\ \mu H^\delta_t+ \text{rot }E^\delta= 0\quad &\text{in }Q,\\ \nu\wedge E^\delta- \delta\nu\wedge (H^\delta\wedge\nu)=J\quad &\text{on }\Sigma,\;\delta> 0,\\ E^\delta(0)= E_0,\;H^\delta(0)= H_0\quad &\text{in }\Omega,\end{cases}\tag{2} \] where \(\Omega\) is a bounded, open set in \(\mathbb{R}^3\) with smooth boundary \(\Gamma\), \(\nu\) is the exterior pointing unit normal vector to \(\Gamma\), \(\varepsilon= (\varepsilon^{jk}(x))\), \(\mu= (\mu^{jk}(x))\) are positive definite \(3\times 3\) Hermitian matrices with \(C^\infty(\overline\Omega)\) entries, and the function \(J\) is taken from the class \({\mathcal U}= {\mathcal L}^2_T(\Sigma):= \{J\mid J\in L^2(0, T; L^2(\Gamma))\), \(\nu\cdot J(t)= 0\) for a.a. \(x\in\Gamma\) and a.a. \(t\in (0,T)\}\).
Define: \({\mathcal K}={\mathcal L}^2(\Omega)\times{\mathcal L}^2(\Omega)\), \({\mathcal D}_{a,0}(\Omega)= \{\chi\in{\mathcal L}^2(\Omega): \text{div}(a\chi)= 0\}\) for \(a\in L^\infty(\Omega)\), and \({\mathcal K}_0={\mathcal D}_{\varepsilon,0}(\Omega)\times{\mathcal D}_{\mu,0}(\Omega)\), which is a closed subspace of \({\mathcal K}\).
The author considers the problem of exact controllability of the solution of (1) in the space \({\mathcal K}_0\) at time \(T\): given fixed but arbitrary \((E_0, H_0)\), \((E_1,H_1)\in{\mathcal K}_0\), find a control \(J_0\in{\mathcal U}\) such that the solution of (1) satisfies \[ E(T)= E_1,\quad H(T)= H_1.\tag{3} \] Without loss of generality, one may assume that \(E_0= H_0= 0\). It is known that the exact controllability problem has a solution if and only if \({\mathcal K}_0\) is continuously observable, that is, there is a constant \(c^0_T> 0\) such that \[ \|(\phi_0,\psi_0)\|^2_{{\mathcal K}}\leq c^0_T \int_\Sigma |\psi_T|^2 d\Sigma,\quad\forall(\phi_0, \psi_0)\in{\mathcal F}_0,\tag{4} \] where \(\psi_T:= \nu\wedge (\psi\wedge\nu)= \psi-(\psi\cdot\nu)\nu\), \({\mathcal F}_0={\mathcal F}\cap{\mathcal K}_0\), \({\mathcal F}= \{(\phi_0, \psi_0)\in{\mathcal K}: \psi_T|_\Sigma\in{\mathcal L}^2_T(\Sigma)\}\), and where \((\phi,\psi)\) is the solution of the problem \[ \begin{cases} \varepsilon\phi_t- \text{rot }\psi= 0,\\ \mu\psi_t+ \text{rot }\phi= 0\quad &\text{in }Q,\\ \nu\wedge\phi= 0\quad &\text{on }\Sigma,\\\phi(T)= \phi_0,\;\psi(T)= \psi_0\quad &\text{in }\Omega.\end{cases}\tag{5} \] When (4) holds, the control of minimum norm in \({\mathcal L}^2_T(\Sigma)\) with state constraint (3) is given by \[ J^0= -\psi_T|_\Sigma,\tag{6} \] where \((\phi,\psi)\) is the solution of (5) with final data \((\phi_0,\psi_0)\in{\mathcal F}_0\) given by \[ \langle(E_1, H_1), (\phi_0,\psi_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi_T|^2 d\Sigma. \] Thus, the optimality system for the problem of minimum norm is given by (1) and (5), and the minimum norm \(J^0\) is given by (6).
Similar considerations can be made for the perturbed system (2). Thus, the problem of exact controllability is: with \(E_0= H_0= 0\) and given fixed but arbitrary \((E_1, H_1)\in{\mathcal K}_0\), find a control \(J^\delta\in{\mathcal U}\) such that the solution of (2) satisfies \(E^\delta(T)= E_1\), \(H^\delta(T)= H_1\).
This problem has a solution if and only if there is a constant \(c^\delta_T> 0\) such that \[ \|(\phi_0, \psi_0)\|^2_{{\mathcal K}}\leq c^\delta_T \int_\Sigma |\psi^\delta_T|^2 d\Sigma,\quad\forall (\phi_0, \psi_0)\in{\mathcal K}_0,\tag{7} \] where \((\phi^\delta, \psi^\delta)\) is the solution of \[ \begin{cases} \varepsilon\phi^\delta_t- \text{rot }\psi^\delta= 0,\\ \mu\psi^\delta_t+ \text{rot }\phi^\delta= 0\quad &\text{in }Q,\\ \nu\wedge\phi^\delta+ \delta\psi^\delta_T= 0\quad &\text{on }\Sigma,\\ \phi^\delta(T)= \phi_0,\;\psi^\delta(T)= \psi_0\quad &\text{in }\Omega.\end{cases}\tag{8} \] The aim of this paper is to investigate the connection between the observability estimates (4) and (7), and between the corresponding optimality systems for small values of \(\delta\).
One of the main results states that if (7) holds for some \(\delta_0> 0\), \((E_1,H_1)\in{\mathcal K}_0\) and \((\phi^\delta,\psi^\delta)\) is the solution of (8) with final data \((\phi^\delta_0, \psi^\delta_0)\in{\mathcal K}_0\) given by \[ \langle(E_1, H_1), (\phi^\delta_0, \psi^\delta_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi^\delta_T|^2 d\Sigma, \] then as \(\delta\to 0\), \[ \begin{alignedat}{2} (\phi^\delta(\cdot), \psi^\delta(\cdot)) &\to (\phi(\cdot),\psi(\cdot))\quad &&\text{weakly}^*\text{ in }L^\infty(0,T;{\mathcal K}),\\ (\phi^\delta_0, \psi^\delta_0) &\to (\phi_0, \psi_0)\quad &&\text{weakly in }{\mathcal K},\end{alignedat} \] where \[ \begin{cases} \varepsilon\phi'- \text{rot }\psi= 0,\\ \mu\psi'+ \text{rot }\phi= 0\quad &\text{in }Q,\\ \nu\wedge\phi= 0\quad &\text{on }\Sigma,\\ \phi(T)= \phi_0,\;\psi(T)= \psi_0\quad &\text{in }\Omega.\end{cases} \] Further, \((\phi_0, \psi_0)\in{\mathcal F}_0\), \(\psi^\delta_t|_\Sigma\to \psi_T|_\Sigma\) strongly in \({\mathcal L}^2_T(\Sigma)\) and \[ \langle(E_1, H_1), (\phi_0,\psi_0)\rangle_{{\mathcal K}}= \int_\Sigma|\psi_T|^2 d\Sigma. \] Another result states that if (7) holds for some \(\delta_0> 0\), \(E_0= H_0= 0\), \((E_1, H_1)\in{\mathcal K}_0\), and \((E^\delta, H^\delta)\) is the solution of (2) with \(J= -\psi_T|_\Sigma\), then \((E^\delta, H^\delta)\to (E, H)\) weakly\(^*\) in \(L^\infty(0, T; \chi')\), where \((E,H)\) is the solution of (1) with \(J=- \psi_T|_\Sigma\), and where \(\chi\hookrightarrow{\mathcal K}\hookrightarrow\chi'\) with \(\chi= {\mathcal R}^0\times ({\mathcal R}\cap{\mathcal D}^0_\mu)\), \({\mathcal R}= \{\phi\in{\mathcal L}^2(\Omega): \text{rot }\phi\in{\mathcal L}^2(\Omega)\}\), \({\mathcal R}^0= \{\chi\in{\mathcal R}: \nu\wedge \chi|_\Gamma= 0\}\), \({\mathcal D}_\mu= \{\chi\in{\mathcal L}^2(\Omega):\text{div}(\mu\chi)\in {\mathcal L}^2(\Omega)\}\), \({\mathcal D}^0_\mu= \{\chi\in{\mathcal B}_\mu: \nu(\mu\chi)|_\Gamma= 0\}\).
These results show that the solution of the optimality system for the problem of minimum norm control of (2) from the rest state \((0,0)\) to the state \((E_1, H_1)\) at time \(T\) converges in the sense described to the solution of the optimality system for the problem of minimum norm control of (1).

MSC:
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35Q60 PDEs in connection with optics and electromagnetic theory
49N10 Linear-quadratic optimal control problems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] K. Ammari and M. Tucsnak , Stabilization of second order evolution equations by a class of unbounded feedbacks . ESAIM: COCV (to appear). Numdam | MR 1836048 | Zbl 0992.93039 · Zbl 0992.93039 · doi:10.1051/cocv:2001114 · numdam:COCV_2001__6__361_0 · eudml:90598
[2] G. Duvaut and J.-L. Lions , Inequalities in Mechanics and Physics . Springer-Verlag, Berlin ( 1976 ). MR 521262 | Zbl 0331.35002 · Zbl 0331.35002
[3] M. Eller (private communication).
[4] M. Eller , Exact boundary controllability of electromagnetic fields in a general region . Appl. Math. Optim. (to appear). MR 1861468 | Zbl 0997.35099 · Zbl 0997.35099 · doi:10.1007/s00245-001-0030-x
[5] E. Hendrickson and I. Lasiecka , Numerical approximations and regularization of Riccati equations arising in hyperbolic dynamics with unbounded control operators . Comput. Optim. and Appl. 2 ( 1993 ) 343 - 390 . MR 1259415 | Zbl 0791.49007 · Zbl 0791.49007 · doi:10.1007/BF01299546
[6] E. Hendrickson and I. Lasiecka , Finite dimensional approximations of boundary control problems arising in partially observed hyperbolic systems . Dynam. Cont. Discrete Impuls. Systems 1 ( 1995 ) 101 - 142 . MR 1361235 | Zbl 0876.93046 · Zbl 0876.93046
[7] V. Komornik , Boundary stabilization, observation and control of Maxwell’s equations . Panamer. Math. J. 4 ( 1994 ) 47 - 61 . Zbl 0849.35136 · Zbl 0849.35136
[8] J. Lagnese , Exact boundary controllability of Maxwell’s equations in a general region . SIAM J. Control Optim. 27 ( 1989 ) 374 - 388 . Zbl 0678.49032 · Zbl 0678.49032 · doi:10.1137/0327019
[9] J. Lagnese , The Hilbert Uniqueness Method: A retrospective , edited by K.-H. Hoffmann and W. Krabs. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 149 ( 1991 ). MR 1178298 | Zbl 0850.93104 · Zbl 0850.93104
[10] I. Lasiecka and R. Triggiani , A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations . Proc. Amer. Math. Soc. 104 ( 1988 ) 745 - 755 . MR 964851 | Zbl 0699.47034 · Zbl 0699.47034 · doi:10.2307/2046785
[11] R. Leis , Initial Boundary Value Problems in Mathematical Physics . B. G. Teubner, Stuttgart ( 1986 ). MR 841971 | Zbl 0599.35001 · Zbl 0599.35001
[12] O. Nalin , Contrôlabilité exacte sur une partie du bord des équations de Maxwell . C. R. Acad. Sci. Paris 309 ( 1989 ) 811 - 815 . MR 1055200 | Zbl 0688.49041 · Zbl 0688.49041
[13] K.D. Phung , Contrôle et stabilisation d’ondes électromagnétiques . ESIAM: COCV 5 ( 2000 ) 87 - 137 . Numdam | Zbl 0942.93002 · Zbl 0942.93002 · doi:10.1051/cocv:2000103 · www.edpsciences.org · eudml:90586
[14] D.L. Russell , A unified boundary controllability theory for hyperbolic and parabolic partial differential equations . Stud. Appl. Math. 52 ( 1973 ) 189 - 211 . MR 341256 | Zbl 0274.35041 · Zbl 0274.35041
[15] M. Tucsnak and G. Weiss , How to get a conservative well-posed linear system out of thin air . Preprint. · Zbl 1125.93383 · doi:10.1137/S0363012901399295
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.