Search Results (129)

This paper develops an accountability-aware fractional control framework for embodied intelligent systems in shared human environments. The approach combines a Caputo fractional-order stabilizing law, an intent-evidence realization with softmax belief reconstruction, and a conditional proxemic safety layer. Sufficient conditions are established for local Mittag-Leffler stability of the augmented error dynamics and forward invariance of the safe set. Numerical results are presented as a theorem-validation benchmark. For the base case with $\alpha =0.9$, the augmented error norm decays from $1.2359$ to $9.90\times {10}^{-3}$ while the safety margin remains strictly positive, and the robustness condition is satisfied with a margin of $1.8641$. An $\alpha$-sweep and a step-size convergence study further show that the fractional order induces a systematic safety–performance trade-off and that the reported behaviors are numerically stable. Additional simulations with four intent classes, bounded observation noise, and Monte Carlo uncertainty stress tests are included to strengthen the numerical evidence beyond the two-intent theorem-validation case. The manuscript also clarifies the quantitative interpretation of the accountability index, the conditional nature of the safety theorem, and an implementable sampled safety-filter realization for concrete robotic platforms. The results support the proposed framework as a mathematically consistent tool for shaping the balance between regulation and proxemic safety. Full article

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform convergence by Laplace transform and Mittag–Leffler function. Secondly, we establish the existence, uniqueness and Ulam–Hyers stability for the nonlinear scenario via the fixed point theorem and analytical techniques. Eventually, some examples and numerical simulations are provided to examine the effectiveness and availability of the main findings. Full article

This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov exponents, bifurcation diagrams with respect to the variable orders, and the Kaplan–Yorke dimension, are analyzed. A synchronization scheme based on active control is designed for the master–slave configuration, and global Mittag–Leffler stability of the error dynamics is established using a suitable variable-order Lyapunov function. The synchronized states are then applied to an image encryption algorithm. Numerical simulations, security analyses, and NIST randomness tests demonstrate the effectiveness and enhanced performance of the proposed framework compared to existing fixed-order and classical fractional-order methods. Full article

We propose a dimensionally consistent fractional spatio-temporal PDE framework for modelling immune-mediated demyelination in multiple sclerosis (MS). The system couples effector and regulatory T cells, M1/M2 macrophage polarisation, pro- and anti-inflammatory cytokines, oligodendrocyte dynamics, and time-dependent therapeutic controls within a unified distributed-parameter structure. In contrast to ad hoc replacements of integerorder derivatives by Caputo fractional derivatives, the fractional extension proposed here is derived from an underlying continuous-time random walk (CTRW) process with Mittag–Leffler-distributed residence times. This stochastic derivation yields a governing system in which a single commensurate fractional order $\alpha \in (0,1]$, together with a characteristic memory timescale ${\tau }_0$, ensures dimensional consistency and mass balance across all coupled components. The model is formulated as a system of nonlinear reaction–diffusion equations with cross-regulatory and multiplicative interaction terms governing immune amplification, cytokine feedback, and the demyelination–remyelination balance. Analytical interpretation shows how non-Markovian residence times induce Mittag–Leffler-type relaxation and thereby modify effective growth, decay, and stability properties. Numerical simulations compare classical and fractional dynamics, revealing that memory-driven kinetics prolong effector T-cell and M1-macrophage activity, attenuate reparative M2 and oligodendrocyte responses, and extend the effective action of bang–bang therapy inputs representing IFN-$\beta$ and glatiramer acetate beyond their dosing windows. The results indicate that integer-order models may underestimate chronic inflammatory persistence and demyelination severity, while providing a mathematically and physically well-posed platform for memory-aware immune modelling and therapy evaluation in MS. Full article

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order $\kappa \in (0,1]$. The population is stratified into five epidemiological classes, namely susceptible $\left(\mathcal{S}\right)$, asymptomatic $\left(\mathcal{A}\right)$, symptomatic $\left(\mathcal{I}\right)$, hospitalised $\left(\mathcal{H}\right)$, and recovered $\left(\mathcal{R}\right)$, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number ${\mathcal{R}}_0$ is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium ${\mathcal{E}}_0$ (when ${\mathcal{R}}_0\le 1$) and the endemic equilibrium ${\mathcal{E}}^{*}$ (when ${\mathcal{R}}_0>1$) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of ${\mathcal{R}}_0$ are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network ($\mathcal{PINN}$) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. $\mathcal{PINN}$ embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters $\mathbf{\Theta }$ = $\{\varphi ,\mu ,\sigma ,\psi ,\beta \}$ across three fractional orders $\kappa \in \{1.0,0.95,0.9\}$. The estimated parameters show strong agreement with the true values at the classical limit $\kappa =1.0$ ($\mathrm{MAPE}=2.27\%$), with the natural mortality rate $\mu$ recovered with $\mathrm{APE}\le 0.51\%$ and the transmission rate $\beta$ with $\mathrm{APE}\le 3.63\%$ across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that $\beta$ can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of $1\%$, $3\%$, and $5\%$ demonstrate graceful degradation ($\mathrm{MAPE}$: $0.82\%\to 3.10\%\to 7.31\%$), confirming the reliability of the proposed framework under realistic observational conditions. Full article

This paper presents a unified theoretical and numerical investigation of Hilfer-type fuzzy fractional control systems with infinite continuous delay. By employing contraction mapping principles and compact semigroup theory, we establish rigorous solvability conditions together with Ulam–Hyers–Rassias stability results expressed in terms of Mittag–Leffler functions. To complement the analytical framework, we design and implement numerical schemes based on Euler and IMEX approaches, which confirm the theoretical predictions through simulations. The computational experiments demonstrate the robustness of the proposed framework under delayed feedback and fractional memory effects, highlighting its relevance to practical domains such as biological regulation, porous media transport, and intelligent traffic systems. The contribution of this study lies in the bridge between mathematical rigor and computational implementation, thus advancing the theory of fractional differential inclusions and providing a versatile tool for the stability analysis and control of complex systems with uncertainty and hereditary dynamics. Full article

This article focuses on the notions of third-order fuzzy differential subordination and superordination associated with the generalized Mittag-Leffler operator. Methods emphasizing the key concept of admissible functions are implemented to investigate several third-order fuzzy differential subordination and superordination results. Sandwich-type outcomes are established based on the adopted methodology, linking the dual fuzzy theoretical frameworks. In addition, the applications of fuzzy differential subordination are discussed in the context of decision making problems. The proposed approach provides the mathematical mechanism that ensures the stability and preservation of the decision under changes in criteria and preference evaluations, highlighting the importance of the developed theory. Full article

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights ($w_i\left(t\right)=w\left(t\right)$), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct $w_i\left(t\right))$, a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on $[a,t_1]$) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics. Full article

In recent years, fractional HIV models have received increasing attention. This study derives a fractional HIV model using the continuous-time random walk (CTRW) method, endowing the mathematical model with physical significance. Based on the transmission characteristics of HIV, the proposed model considers extrinsic infectivity, intrinsic infectivity, and drug control, specifically as follows: the extrinsic infectivity is a constant independent of the infection time; the intrinsic infectivity is a power-law function that depends on drug efficacy and infection time; the drug efficacy rate follows a Mittag–Leffler distribution with a long-term effect. Based on these considerations, a fractional HIV model with drug control is established in this paper. In addition, the global asymptotic stability of the equilibrium and the sensitivity analysis of the basic reproduction number $R_0$ are studied, and the theoretical results are verified by numerical simulations. The results show that reducing extrinsic infectivity, controlling intrinsic infectivity, and the drug efficacy rate are crucial in controlling the spread of HIV. Full article

This paper addresses the stabilization problem for a class of fractional-order memristive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays under an intermittent boundary control framework. Two scenarios are considered: systems without parametric uncertainties, for which asymptotic stability is established, and systems with uncertainties, for which robust asymptotic stability is ensured. By constructing appropriate Lyapunov functionals and employing Wirtinger-type inequalities, the fractional Razumikhin approach, and key properties of the Mittag–Leffler function, sufficient stability conditions are derived in terms of linear matrix inequalities with reduced conservatism. Furthermore, the effects of time-varying delays and control activation intervals on the stabilization performance are systematically investigated. The effectiveness and advantages of the proposed control methodology are validated through numerical simulations. Full article

In this work, we propose a dynamic cobweb-type market equilibrium model where supply responds to price using a Mittag–Leffler kernel generalized weighted Caputo fractional derivative. By permitting independent fractional orders, a time-varying weight function that reweights historical data, and a monotone economic-time transformation that captures heterogeneous adjustment speeds, the formulation expands upon previous fractional cobweb models. We begin by highlighting several special cases encompassed by our proposed model. Next, we establish well-posedness, covering existence, uniqueness, and continuous dependence on initial data and parameters via an equivalent Volterra integral formulation, alongside a positivity theorem that ensures prices remain economically meaningful. Then, we derive stability conditions for the perturbation dynamics and characterize the constant equilibrium price. To perform the simulation, we constructed an explicit Volterra partition scheme specifically designed for the generalized kernel and established its convergence. In addition, we validated this approach using numerical examples illustrating how fractional orders, weights, and time transformations cause transient oscillations and convergence toward equilibrium. Full article

This paper studies the issues of human balancing and stability in the sagittal plane using fractional and integer order time delay feedback control. The neural-mechanical model of human balance is represented as an inverted pendulum controlled by torque. We present a finite-time stability (FTS) analysis for closed-loop neutral time delay systems (NFOTDSs) with fractional order $1<\beta <\alpha \le 2$. By employing a generalized Gronwall inequality, we derive new FTS criteria for these systems in terms of the Mittag-Leffler function. Finally, a suitable numerical example is presented to show the effectiveness of the proposed method. Full article

In this paper, we investigate a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative under two-point boundary conditions. By combining the Green function of the associated linear problem with a generalized Gronwall inequality, we derive pointwise estimates for solutions expressed explicitly in terms of the Mittag–Leffler function. In contrast to existing Lyapunov-type inequalities, which are mainly restricted to linear equations and rely on global supremum norm estimates, our approach preserves the nonlinear structure of the problem and captures the local behavior of solutions. These pointwise estimates lead to a Lyapunov-type inequality for nonlinear fractional equations, extending the classical result of Jleli and Samet beyond the linear framework. Moreover, we show that the obtained Lyapunov condition serves not only as a necessary condition for the existence of nontrivial solutions, but also as a sufficient criterion ensuring Hyers–Ulam stability and uniqueness. An illustrative example is provided to demonstrate the applicability of the theoretical results. Full article

For the first time, this paper introduces Nabla fractional calculus into the distributed Nash equilibrium (NE) seeking problem of aggregative games (AGs) with partial decision information in undirected communication networks, and proposes two novel fractional-order distributed algorithms. In the considered setting, each agent can access to only local information and collaboratively estimates the global aggregate through communication with its neighbors. Both algorithms adopt a backward-difference scheme followed by an implicit fractional-order gradient descent step. One updates local aggregate estimates via fractional-order dynamic tracking and the other uses fractional-order average dynamic consensus protocols. Under standard assumptions, convergence of both algorithms to the NE is rigorously proved using nabla fractional-order Lyapunov stability theory, achieving a Mittag-Leffler convergence rate. The feasibility of the developed schemes is verified via numerical experiments applied to a Nash-Cournot game and the coordination control of flexible robotic arms. Full article

This study explores finite-time synchronization (FTS) in fractional-order quaternion-valued neural networks (FQVNNs) characterized by discontinuous activation functions and uncertainties in parameters. Initially, leveraging the properties of the Mittag-Leffler function along with fractional-order (F-O) delayed differential inequalities, a novel finite-time stability theorem for F-O systems is established, building upon previous research findings. Next, based on norm definitions, two state feedback controllers employing quaternion 1-norm and quaternion 2-norm are devised to ensure FTS for the system under consideration. Following this, by utilizing differential inclusion theory, examining the quaternion sign function, employing advanced inequality methods, applying principles of F-O differential equations, and using the Lyapunov functional approach, new criteria for achieving FTS in FQVNNs are formulated. Additionally, precise estimates for the settling time are presented. In conclusion, two carefully designed numerical examples are included to corroborate the theoretical results derived. Full article