Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions

• Published in: Journal of mathematical analysis and applications 2007-05, Vol.329 (2), p.1355-1374
• Main Authors: Krigman, S.S., Wayne, C.E.
• Format: Article
• Language: English
• Subjects: Biorthogonal sequences; Boundary controls; Calculus of variations and optimal control; Damping; Exact sciences and technology; Lack of exact controllability; Mathematical analysis; Mathematics; Maxwell's equations; Partial differential equations; Sciences and techniques of general use
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2006.06.102

This is a second paper in a two part series. In the prequel, [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.06.101], we showed that a system of Maxwell's equations for a homogeneous medium in a cube with nonnegative conductivity possesses the property that any finite combination of eigenfunctions is controllable (spectral controllability) by means of boundary surface currents applied over only one face of the cube. In the present paper it is established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111–124], that spectral controllability is the strongest result possible for this geometry, since the exact controllability fails regardless of the size of the conductivity term. However, we do establish controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at an appropriate exponential rate. This does not contradict the lack of exact controllability since in any Sobolev space there are initial conditions which violate these restrictions.