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

TL;DR

This paper develops new stability conditions and an adaptive control scheme for Caputo fractional systems with state-dependent delays, using Lyapunov-Krasovskii functionals and delay estimation techniques.

Contribution

It introduces a unified stability framework with tractable conditions and designs an adaptive controller that avoids classical derivatives for fractional systems.

Findings

Stability conditions reduce to linear matrix inequalities.

The adaptive controller significantly reduces control energy.

Achieves high-precision state regulation in a fractional neural network.

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 $\xi^{\alpha-1}$ for $\alpha \in (0,1)$, 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 $\sigma$-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.