Approximate controllability of infinite-dimensional bilinear systems (Q1299863)

Bilinear, time-invariant, infinite-dimensional control systems are considered. First, some preliminaries and results on the existence and uniqueness of solutions to bilinear abstract evolution equations are presented. Next, using the semigroup approach and approximation methods, a necessary and sufficient condition for approximate global controllability in a given time interval is formulated and proved by means of a sequence of suitably defined discrete controllability problems in an infinite-dimensional Banach space. Moreover, a sufficient condition for approximate controllability in a given time interval expressed in terms of attainable sets for a discrete-time system is derived. Finally, some special cases are also discussed. Remarks and comments on approximate controllability in infinite-dimensional Banach spaces are also provided. The paper extends, for bilinear control systems defined in infinite-dimensional spaces, the controllability conditions given in the monograph [\textit{J. Klamka}, Controllability of dynamical systems, Kluwer, Dordrecht (1991; Zbl 0732.93008)].