Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T12:09:50.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary control of the Maxwell dynamical system: lack of controllability by topological reasons

Published online by Cambridge University Press:  15 August 2002

Mikhail Belishev  and
Aleksandr Glasman
Mikhail Belishev
Affiliation:
Saint-Petersburg Department of Steklov Mathematical Institute, Fontanka 27, Saint-Petersburg 191011, Russia; belishev@bel.pdmi.ras.ru. Supported by RFBR, grant 98-01-00314.
Aleksandr Glasman
Affiliation:
Saint-Petersburg State University, Saint-Petersburg, Russia. Supported by RFBR, grant 99-01-00107.
Get access

Abstract

The paper deals with a boundary control problem for the Maxwell dynamical system in a bounbed domain Ω ⊂ R3. Let ΩT ⊂ Ω be the subdomain filled by waves at the moment T, T* the moment at which the waves fill the whole of Ω. The following effect occurs: for small enough T the system is approximately controllable in ΩT whereas for larger T < T* a lack of controllability is possible. The subspace of unreachable states is of finite dimension determined by topological characteristics of ΩT.

Keywords

Maxwell's dynamical systemboundary controlunreachable statestopology of a domain.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 207 - 217
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

S. Avdonin, M. Belishev and S. Ivanov, The controllability in the filled domain for the multidimensional wave equation with a singular boundary control. J. Math. Sci. 83 (1997).
Belishev, M.I., Boundary control in reconstruction of manifolds and metrics (the BC-method). Inverse Problems 13 (1997) R1-R45. http://www.iop.org/Journals/ip/. CrossRef
M. Belishev and A. Glasman, Boundary control and inverse problem for the dynamical maxwell system: the recovering of velocity in regular zone. Preprint CMLA ENS Cachan (1998) 9814. http://www.cmla.ens-cachan.fr
M. Belishev and A. Glasman, Vizualization of waves in the Maxwell dynamical system (The BC-method). Preprint POMI (1997) 22. http://www.pdmi.ras.ru/preprint/1997/
M. Belishev, V. Isakov, L. Pestov and V. Sharafutdinov, On reconstruction of gravity field via external electromagnetic measurements. Preprint PDMI (1999) 10. http://www.pdmi.ras.ru/preprint/1999/10-99.ps.gz.
Bykhovskii, E.B. and Smirnov, N.V., On an orthogonal decomposition of the space of square-summable vector- functions and operators of the vector analisys. Proc. Steklov Inst. Math. 59 (1960) 5-36, in Russian.
G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique, Vol. 21 of Travaux et recherches mathématiques. Paris: Dunod. XX (1972).
M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy Problem for Maxwell and elasticity systems. Nonlinear Partial Differential Equations and their applications. College de France Seminar. XIV (1999) to appear.
Lagnese, J., Exact boundary controllability of Maxwell's equations in a general region. SIAM J. Control Optim. 27 (1989) 374-388. CrossRef
I. Lasiecka and R. Triggiani, Recent advances in regularity of second-order hyperbolic mixed problems, and applications, K.R.T. Christopher et al., Eds. Jones, editor. Springer-Verlag, Berlin, Dynam. Report. Expositions Dynam. Systems (N.S.) 3 (1994) 104-162.
R. Leis, Initial boundary value problems in mathematical physics. Teubner, Stuttgart (1972).
V.G. Maz'ya, The Sobolev spaces. Leningrad, Leningrad State University (1985), in Russian.
Nalin, O., Controlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) 811-815.
Russell, D.L., Boundary value control theory of the higher-dimensional wave equation. SIAM J. Control Optim. 9 (1971) 29-42. CrossRef
G. Schwarz, Hodge decomposition. A method for solving boundary value problems. Springer Verlag, Berlin, Lecture Notes in Math. 1607 (1995).
D. Tataru, Unique continuation for solutions to PDE's; between Hoermander's theorem and Holmgren's theorem. Comm. Partial Differential Equations 20 (1995) 855-884.
N. Weck, Exact boundary controllability of a Maxwell problem. SIAM J. Control Optim. (to appear).