Controllability and Observability of Matrix Sylvester Adjoint Dynamic Impulsive Systems on Time Scales

This paper investigates the controllability and observability of matrix Sylvester adjoint dynamic impulsive systems within the framework of time scales. By applying the vectorization operator, the system is reformulated into an equivalent Kronecker product-based dynamic impulsive system, enabling more efficient analysis. The study derives necessary and sufficient conditions for controllability and observability through the adjoint matrix approach, highlighting its analytical significance. Additionally, the research incorporates the Gramian matrix to establish results for impulsive dynamic systems on time scales, providing a comprehensive criterion for evaluating system properties. This unified framework bridges discrete and continuous-time dynamics, enabling the modeling and analysis of hybrid systems with abrupt state transitions. The findings advance the mathematical understanding of Sylvester matrix systems and offer practical insights into the design and control of complex impulsive systems across varied applications. This work contributes to the growing field of time-scale calculus and its applications in dynamic system analysis.