Controllability and Observability Conditions for Impulsive DAEs on Time Scales

ABSTRACT

This paper establishes criteria for controllability and observability in a class of linear time‐invariant impulsive differential‐algebraic equations (DAEs) defined on a time scale. Controllability is defined as the ability to transfer the system state from any initial condition to any desired state within a finite time interval using an appropriate control input. Observability is characterized as the ability to uniquely determine the initial state from output measurements over a finite time interval. The main contributions are the derivation of necessary and sufficient conditions for both state controllability and state observability. These conditions are formulated in terms of rank‐based tests and the properties of Gramian matrices, which are constructed using the Drazin inverse to handle the system's algebraic constraints and impulsive dynamics. The theoretical results are validated by solving a illustrative controllability and observability problem, demonstrating the efficacy of the proposed approach.