Singularities of attainable sets (Q1267431)

Linear, stationary, finite-dimensional, continuous-time control systems are considered. Using methods taken from the theory of optimization, topological properties of the attainable sets are studied in detail. Especially local and global behaviours of singularities of the attainable sets, with respect to time, are investigated. Using the concepts of spherical and projective observability, a necessary and sufficient condition for lack of singularities at any time is formulated and proved. The connections with the well-known Kalman observability criterion are pointed out. Limit behaviour of the attainable sets is also discussed. Moreover, several remarks and comments concerning relationships between attainable sets, controllability and different types of observability are given. Similar problems have been considered in an earlier paper [\textit{A. I. Ovseevich}, in: Modelling, estimation and control of systems with uncertainty, Proc. Conf., Sopron/Hung. 1990, Prog. Syst. Control Theory 10, 324-333 (1991; Zbl 0734.93015)].