Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

Abstract

A linear continuous-time control system with time-varying linear bounded operator coefficients in a Hilbert space is considered. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment of the upper Bohl exponent by state feedback control. The property of exact controllability for the open-loop system is studied, some necessary and sufficient conditions are obtained for exact controllability. For periodic systems, it is proved that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system by means of linear state feedback with a periodic gain operator function. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.