Parametrization of attainability sets for a class of homogeneous quasicommutative bilinear systems

The attainability set for a homogeneous quasicommutative bilinear system ẋ = (A + uB)x with a scalar limited control is considered. Under certain assumptions, each point of the attainability set is associated with a piecewise-constant control, for which the estimate of the number of switchings is related to the dimension of the phase space of the system. This fact makes it possible to use the switchings of these controls for description of the boundary and the interior of the attainability set of the system under consideration. A class of quasicommutative bilinear systems for which the numbers of switchings needed for the description of the boundary and the interior of the attainability set differ only slightly is established. Examples of homogeneous bilinear systems for which this result is valid are given. © 2009 Allerton Press, Inc.