Oscillation Criteria for Matrix Fractional Differential Equations via Riccati Transformation and Integral Averaging

In this work, we use the Riemann–Liouville (R-L) fractional derivative of order α∈(0,1) to study the oscillation criteria for damped matrix fractional differential equations and determine sufficient conditions under which all prepared solutions of the system show oscillatory behaviour. The criteria are novel even for the linear undamped case and extend conventional oscillation results for integer-order matrix differential systems to the fractional setting. The goal of the current effort is to better understand the relationships between solutions and their derivatives. Using the matrix-valued Riccati transformation converts the system into a Riccati-type inequality, and the oscillation conditions are then derived by integrating against a weighted kernel via the operator L. Both results generalise the integer-order oscillation criteria to the fractional matrix setting, extending their applicability to fractional-order control systems, viscoelastic structural models, and anomalous diffusion processes. This work develops new conditions and analytical techniques that deepen insight and provide useful results for analysing oscillatory behaviour and asymptotic stability of of the considered systems. To illustrate the significance of the obtained oscillation results, we give two examples.