Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces - Oscar Raúl Condori Mamani; Bartolome Valero Larico; María Luisa Torreblanca; Wolfgang Kliemann (Mathematics, 2025)
Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in <semantics data-eusoft-scrollable-element="1">Rd</semantics>. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system.