This report provides a description of unbunched beam stochastic cooling in
the framework of control theory. The main interest in the investigation is
concentrated on the beam stability in an active cooling system. A stochastic
cooling system must be considered as a closed-loop, similar to the feedback
systems used to damp collective instabilities. These systems, which are able to
act upon themselves, are potentially unstable.
The self-consistent solution for the beam motion is derived by means of a
mode analysis of the collective beam motion. This solution yields a criterion
for the stability of each collective mode. The expressions also allow for
overlapping frequency bands in the beam spectrum and thus are valid over the
entire frequency range.
Having established the boundaries of stability in this way, the Fokker-Planck
equation is used to describe the cooling process. This description does not
include collective effects and thus a stable beam must be assumed. Hence the
predictions about the cooling process following from the Fokker-Planck equation
only make physical sense within the boundaries of beam stability. Finally it is
verified that the parameters of the cooling system which give the best cooling
results are compatible with the stability of the beam.Comment: 64 pages, latex, 11 eps-figures appended as uuencoded file, german
hyphenation corrected I
