Controllability and Observability Analysis of Fractional-Order Dynamical Systems on Time Scales

This paper investigates the controllability and observability of fractional-order dynamical systems on time scales by developing a unified analytical framework based on the Caputo fractional nabla derivative, thereby avoiding separate treatments of continuous and discrete systems. For linear dynamical systems, necessary and sufficient criteria are established through controllability and observability Gramians, together with algebraic rank conditions. Counterexamples demonstrate that the classical full-rank conditions are not sufficient to guarantee controllability and observability on arbitrary time scales. For nonlinear systems, sufficient conditions of controllability and observability are derived using Schauder’s and Banach’s fixed-point theorems. Numerical examples on continuous, uniform discrete, and nonuniform time scales illustrate the applicability and generality of the proposed framework, with potential applications in networked dynamical systems evolving over diverse time scales.