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-algebraicequations (DAEs) defined on a time scale. Controllability is defined as the ability to transfer the system state from any initialcondition to any desired state within a finite time interval using an appropriate control input. Observability is characterized asthe ability to uniquely determine the initial state from output measurements over a finite time interval. The main contributionsare the derivation of necessary and sufficient conditions for both state controllability and state observability. These conditions areformulated in terms of rank-based tests and the properties of Gramian matrices, which are constructed using the Drazin inverseto handle the system’s algebraic constraints and impulsive dynamics. The theoretical results are validated by solving a illustrativecontrollability and observability problem, demonstrating the efficacy of the proposed approach.