Adaptive Control and Mittag-Leffler Stability of Caputo Fractional Systems with State-Dependent Delays

TL;DR

The paper tackles stability and adaptive control for Caputo fractional-order nonlinear systems with state-dependent delays by developing a singular-kernel Lyapunov–Krasovskii framework that yields Mittag-Leffler stability via tractable LMIs. It then designs an adaptive controller with fractional-order parameter updates and a filter-based delay estimator that avoids reliance on classical derivatives, achieving robust performance under Hölder regularity. Key contributions include explicit delay margins, a concrete Lyapunov-based convergence analysis, and numerical validation on a three-neuron fractional Hopfield network showing substantial reductions in control energy and improved asymptotic accuracy compared with a fractional sliding-mode controller. The work provides a rigorous, implementable toolkit for stability analysis and adaptive control of fractional-delay systems with state-dependent delays, with potential impact in neural networks and other memory-rich dynamical systems.

Abstract

This paper establishes new sufficient conditions for Mittag-Leffler stability of Caputo fractional-order nonlinear systems with state-dependent delays. The central analytical tool is a class of Lyapunov-Krasovskii functionals that incorporate singular kernels of the form $ξ^{α-1}$ for $α\in (0,1)$, coupling fractional memory effects with delay-induced dynamics in a unified framework. We prove that the resulting stability conditions reduce to computationally tractable linear matrix inequalities and derive explicit formulas for the maximum tolerable delay perturbation. Building on this stability foundation, we design an adaptive controller governed by fractional-order parameter update laws with $σ$-modification and a filter-based delay estimation mechanism that circumvents the need for classical state derivatives, which may not exist for fractional-order trajectories. The convergence analysis establishes ultimate boundedness of the closed-loop system with a computable bound that vanishes as the regularization parameters approach zero. Numerical validation on a three-neuron fractional Hopfield network with state-dependent transmission delays demonstrates that the proposed adaptive scheme reduces cumulative control energy by 99.3\% and achieves an asymptotic state error two orders of magnitude smaller than a comparable fractional sliding mode controller.

Figures (7)

• Figure 1: Mittag-Leffler function $E_{\alpha}(-t^{\alpha})$. (a) Linear scale showing monotone decrease. (b) Logarithmic scale revealing algebraic tails (dashed: asymptotic approximation $t^{-\alpha}/\Gamma(1-\alpha)$).
• Figure 2: Uncontrolled system response ($\alpha = 0.95$). (a) State trajectories. (b) State norm (solid) and Mittag-Leffler bound (dashed) on logarithmic scale. (c) Phase portrait in the $x_1$--$x_2$ plane. (d) State-dependent delay evolution.
• Figure 3: Adaptive control performance. (a) Controlled state trajectories. (b) Adaptive control signals. (c) State norm: controlled (solid) vs. uncontrolled (dashed), with settling time annotated. (d) Lyapunov function evolution.
• Figure 4: Adaptive control vs. fractional sliding mode control. (a) State norm evolution. (b) Instantaneous control effort. (c) Cumulative control energy $\int_0^t \|u(s)\|^2 ds$.
• Figure 5: Sensitivity analysis. (a) Effect of fractional order $\alpha \in \{0.70, 0.80, 0.90, 0.95, 0.99\}$ on the state norm. (b) Effect of delay bound $\bar{\tau} \in \{0, 0.1, 0.3, 0.5, 0.7\}$ s. (c) Summary of settling times.

Theorems & Definitions (22)

• Definition 1: Riemann-Liouville Fractional Integral kilbas2006
• Definition 2: Caputo Fractional Derivative podlubny1999
• Definition 3: Mittag-Leffler Function gorenflo2014
• Lemma 1: Caputo Derivative of Quadratic Forms aguila2014
• Lemma 2: Fractional Comparison Principle li2010
• Theorem 3: Existence and Uniqueness
• proof
• Definition 4: Mittag-Leffler Stability li2009
• Proposition 4: Well-Posedness
• proof