A Sufficient Condition for the Practical Stability of Riemann-Liouville Fractional Nonlinear Systems with Time Delays

This study addresses the practical stability analysis of Riemann-Liouville fractional-order nonlinear systems with time delays. We first establish a rigorous formulation of initial conditions that aligns with the properties of Riemann-Liouville fractional derivatives. Subsequently, a generalized definition of practical stability is introduced, specifically tailored to accommodate the hybrid dynamics of fractional calculus and time-delay phenomena. By constructing appropriate Lyapunov-Krasovskii functionals and employing an enhanced Razumikhin-type technique, we derive sufficient conditions ensuring practical stability in the Lp-norm sense. The theoretical findings are validated through illustrative example for fractional order nonlinear systems with time delays.