On frequency distribution of impulsive feedback control times

The present work considers an impulsive feedback control problem governed by a system of ODEs defined on a class of isolated time scales. The control is activated once the unique solution's trajectory leaves a bounded connected domain. A periodicity of the controlled system's solution in the context of shift-invariant time scales are discussed. Making use of the Δ-measure theory, a cumulative distribution of the control times is introduced. Then with the help of dynamic inequalities on time scales such as Opial-type inequality, upper estimations for the occurrence frequency of the control times are obtained. A uniform distribution of the control times mod 1 is studied and a necessary condition for the uniformity of distribution of control times in terms of critical dynamic of the controlled system is obtained.