Fractional-Order System Stability Analysis

Fractional-order systems extend classical dynamics by allowing differentiation and integration to non-integer orders, thereby capturing memory and hereditary properties in physical, biological and engineering processes. The stability analysis of such systems examines whether solutions remain bounded or converge to an equilibrium under perturbations, accounting for the nonlocal character of fractional operators. Core techniques include spectral methods that locate system eigenvalues in a fractional-order stability region, generalised Lyapunov approaches based on fractional-order derivatives, and comparison principles utilising Mittag-Leffler functions as natural generalisations of exponentials. Extensions address linear and nonlinear dynamics, time-delay effects, variable-order behaviour and distributed parameters. Practical applications range from viscoelastic material modelling and electrical circuit design to neural network control and anomalous diffusion. Recent advances have emphasised unifying disparate stability criteria, refining delay-dependent conditions and reducing conservatism through convex optimisation tools such as linear matrix inequalities.

Research from Nature Portfolio

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Research from all publishers

Recent work has explored stability in generalized fractional integro-differential equations with proportional Caputo derivatives and bounded time-varying delays by constructing quadratic Lyapunov functions and adapting Razumikhin techniques to derive conditions for stability, exponential stability and boundedness in a unified manner. Another development formulates finite-time stability criteria for variable-order fractional nonlinear systems under sliding mode control, employing a variable fractional Lyapunov direct method to design chattering-free controllers that guarantee convergence in finite time. A further line of research has introduced novel delay-dependent asymptotic stability conditions for mixed differential and Riemann-Liouville fractional neutral systems, casting stability criteria as linear matrix inequalities through model transformations and auxiliary zero equations, thus improving allowable delay bounds and robustness against nonlinear perturbations.

Fractional-Order System Stability Analysis publication trend

The graph below shows the total number of articles in fractional-order system stability analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Caputo fractional derivative: A non-integer derivative defined via a convolution with a power-law kernel, suitable for problems with standard initial conditions.

Riemann-Liouville fractional derivative: A fractional differentiation operator based on an integral definition, requiring initial conditions expressed in fractional forms.

Lyapunov function: A scalar energy-like function whose fractional derivative along system trajectories is used to assess stability.

Mittag-Leffler function: A two-parameter series generalising the exponential, frequently appearing in analytic solutions of fractional systems.

Razumikhin method: A comparison technique that employs auxiliary functions to establish stability of delayed fractional systems.

Linear matrix inequality (LMI): A convex matrix-based constraint used to derive tractable stability and control conditions.

References

1. Stability of Fractional Order Systems. Mathematical Problems in Engineering (2013).
2. Mittag‐Leffler Stability Theorem for Fractional Nonlinear Systems with Delay. Abstract and Applied Analysis (2010).
3. Stability for generalized Caputo proportional fractional delay integro-differential equations. Boundary Value Problems (2022).
4. Finite time stability and sliding mode control for uncertain variable fractional order nonlinear systems. Advances in Continuous and Discrete Models (2021).
5. A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation. Mathematics (2020).