Common Riccati Stability and Time-Delay Systems

This paper investigates the stability properties of matrix families through the lens of Riccati, Lyapunov, and Schur stability. We focus on establishing connections between these stability concepts, particularly in the context of continuous and discrete-time systems, as well as time-delay systems. The results provide criteria for common Riccati stability, exploring its implications on Lyapunov and Schur stability across matrix families. Furthermore, we examine the effects of scaling transformations and similarity transformations on common Riccati stability, demonstrating its robustness under scalar multiplication and similarity changes. The findings contribute to a deeper understanding of matrix stability in control systems, offering insights into the structural preservation of stability properties under various transformations. This work has potential applications in robust control and the stability analysis of complex systems subject to time delays and structural modifications.