Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix (Q656487)

Periodic, linear, continuous-time, finite-dimensional control systems with time dependent coefficients are considered. It is assumed that the control is defined in the form of a linear feedback with respect to the state variables, and is periodic with the same period as the system itself. The main purpose is to synthesize this feedback in order to control the spectrum of irregular oscillations. A necessary and sufficient condition for the solution of this problem is formulated and proved. The main requirement is to satisfy certain rank condition. Finally, some remarks and comments on the properties of the spectrum of irregular oscillations are presented. Similar problems have been considered in the paper [\textit{P. Brunovski}, Controllability and linear closed-loop controls in linear periodic systems. J. Differ. Equations, 600, No. 3, 296--313 (1969; Zbl 0176.06301)].