Mikhail Zelikin
Department of Mathematics and Mechanics,
Moscow State University, Vorobjevy Gory,
119899 Moscow, Russia
Topological structure of chattering optimal syntheses
The term chattering control means the optimal control with
countable number of switches on a finite time interval. The
theory of chattering control is gathering force as a separate branch
of the geometrical optimal control theory. Up to now the main
attention was paid to control systems with single input.
The multiinput control problems with chattering arcs
are as yet little
understood. It was found an explicit
analitic expression for some optimal trajectories of linear-quadratic
problems having control in the unit disk. The part of switching
accumulation points play here points of discontinuity of the
second kind of the control function.
We investigate a phenomenon inherent to multiinput
problems with the control taking values in polytops. We call
this phenomenon as successively inserted chattering
bundles. We mean by this term an optimal synthesis having the
following structure.
The phase space is fibered over
strata of maximal dimension of the stratified singular manifold W
by fibers containing chattering
trajectories (chattering fibers); each stratum of W, being the
base of the preceding bundle, is fibered in his turn by
chattering fibers over the stratum of less dimension which
belongs to its boundary and so on up to the stratum of minimal
dimension which consists of smooth singular trajectories.
Abstract in Postscript
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