A robust uniform practical stability approach for Caputo fractional hybrid systems

This work introduces a novel approach for analyzing the uniform practical stability (UPS) and strong uniform practical stability (SUPS) of Caputo fractional dynamic equations on time scales, using two measures ( $m,m_0$). The method employs an extended derivative, the Caputo fractional delta Dini derivative (CFr $\mathrm{\Delta }$DiD) of order $\zeta \in (0,1)$, addressing the gap in unified stability frameworks for fractional hybrid systems that span both continuous and discrete time domains. This generalized framework not only unifies various stability concepts but also makes it applicable to hybrid systems with both gradual and abrupt changes. The UPS and SUPS results are demonstrated through illustrative examples.