Abstract
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.
Highlights
The investigation of stability properties plays a prominent role in the qualitative theory of fractional-order systems, as in the case of the classical theory of integer-order dynamical systems [1,2]
Due to the importance in the description of the solution of linear systems of fractional differential equations (FDEs), in Section 3, we provide a detailed description of the Mittag–Leffler function, of its derivatives and of the corresponding asymptotic behavior
Stability analysis of multi-order systems is discussed in Section 5; since general results are far from being formulated in this case, we focus on some special cases and we separately investigate two-dimensional systems, higher dimensional systems with block-triangular structure, and higher dimensional systems with some special fractional orders
Summary
The investigation of stability properties plays a prominent role in the qualitative theory of fractional-order systems, as in the case of the classical theory of integer-order dynamical systems [1,2]. Necessary and sufficient fractionalorder independent conditions have been presented, in terms of the main diagonal elements and the determinant of the system’s matrix, which guarantee the asymptotic stability or instability of the considered two-dimensional system, regardless of the choice of the fractional-orders considered in the system. These latter results prove to be especially useful in practical applications where the exact fractional orders of the Caputo derivatives are not precisely known. Some concluding remarks are provided in the concluding Section 6
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