Integrable Systems, Analysis And Probability Seminar
Thursday, March 4, 2004, 4:00pm, 120 Physics
Seymon Tsynkov (NC State)
A Mathematical Model for Active Control of Sound
Abstract:- A problem of eliminating the unwanted part of a time-harmonic wave field
(noise) on a predetermined region of interest is solved by active means,
i.e., by introducing the additional sources of field, called controls,
that generate the appropriate annihilating signal (anti-sound). The general
solution for controls is obtained using Calderon's boundary projections
that render decomposition of a given wave field into its incoming and
outgoing components on the perimeter of the protected domain. A theory
fully analogous to that of Calderon's potentials and projections also
exists for finite differences; and it is used for building the discrete
active controls.
Once the general solution for controls is available, it can be optimized
using various criteria. The problems of minimizing the $L_1$ norm of
the control sources, their $L_2$ norm, and the power consumption by the
control system have been analyzed. Physical interpretations that can be
associated with these three formulations appear to have relatively little
in common, finding the respective minima requires noticeably dissimilar
solution approaches, and the resulting optimal controls also happen to
differ quite drastically from one another.
Joint work with V. Ryaben'kii (Russian Ac. Sci.) and J. Loncaric (LANL).
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