On the stability of mu-varying dynamic equations on stochastically generated time scales

In their 2001 paper, Potzsche, Siegmund and Wirth gave necessary and sufficient conditions for an LTI system on a time scale to have exponentially stable solutions based on pole placement. We find simple conditions for the stability of mu-varying scalar dynamic equations on time scales which are stochastically generated. As a special case, we examine the region in the complex plane which will guarantee the exponential stability of solutions of LTI systems. Via a decay analysis, we show how the tendency of the solution to grow or decay at each time step is determined by the pole placement within the region of exponential stability¹.