Periodic Solutions of Linear Systems of Dynamic Equations on Isolated Time Scales

The time scales calculus provides a framework that unites discrete and continuous dynamical systems to what are called dynamic equations on time scales. A time scale is a closed subset of the real line, and so it is not generally closed under addition. Thus, periodic functions on time scales have historically been difficult to define. In 2022, a definition of periodic real‐valued functions for isolated time scales was provided which yields the periodicity concept by requiring area to be preserved by forward translation of integrals. Our work generalizes this periodicity concept to matrix‐valued functions. In particular, we derive conditions for the existence of periodic solutions of homogeneous dynamic equations with and without periodic coefficient functions. We then apply the well‐known concept of controllability and derive a simpler condition to check for controllability of our considered systems with periodic coefficient functions.