Stability regions for fractional differential systems with a time delay

• Differential systems with fractional derivatives and time delays are studied.
• Optimal stability conditions are derived.
• The Laplace transform method is utilised.
• Stability switches with respect to increasing delay are described.
• Stability behaviour of the fractional delay Duffing’s equation is analysed.

The paper investigates stability and asymptotic properties of autonomous fractional differential systems with a time delay. As the main result, necessary and sufficient stability conditions are formulated via eigenvalues of the system matrix and their location in a specific area of the complex plane. These conditions represent a direct extension of Matignon’s stability criterion for fractional differential systems with respect to the inclusion of a delay. For planar systems, our stability conditions can be expressed quite explicitly in terms of entry parameters. Applicability of these results is illustrated via stability investigations of the fractional delay Duffing’s equation.