Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces

Abstract

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 Rd. 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.

Keywords

• bilinear control systems
• Lie theory
• control sets
• homogeneous spaces
• Lyapunov exponents
• Floquet spectrum