Controllability of two-dimensional bilinear systems
DOI:
https://doi.org/10.22199/S07160917.1996.0002.00002Abstract
For bilincar control systems x = Ax + uBx, x ? R2 , A and B 2 x 2 matrices, necessary and sufficient conditions are given for the controllability on R2 -{0}. The method is through Lie theory, and follows the program outlined by this theory which consists in finding first the connected subgroups of the group Gl(2) of all invertible matrices which are transitive on R2 - {0}, and then look at the subsemigroups of these subgroups which are transitive. A detailed and nearly self contained exposition of the determination of the transitive subgroups is presented. It turns out that they are Gl+(2), Sl(2) and the commutative group of nonzero complex numbers. Controllability is analysed by considering these groups separately. In the case of Sl (2) the controllability is decided with the aid of a result of [15] about semigroups in semi-simple Lie groups. A self contained proof specific for Sl(2) is presented. This case by case analysis recovers the necessary and sufficient conditions given by Lepe and Joó and Tuan (see [10]).
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