Search Results (44)

This paper presents a hybrid two-stage implicit scheme for the numerical solution of fractional initial value problems involving Caputo derivatives. The proposed formulation incorporates the nonlinear source term directly into the time-stepping procedure, leading to improved stability and accuracy compared with classical fractional implicit schemes. The resulting nonlinear systems are solved using a parallel iterative strategy based on the Weierstrass-type method, combined with OpenMP-style parallelization to ensure efficient workload distribution and accelerated convergence. In addition, a data-driven module is introduced to generate high-quality initial guesses, thereby enhancing the robustness and efficiency of the nonlinear solver. The main contributions include the development of a unified fractional-parallel-data-driven framework, improved stability properties with enlarged real-axis stability regions, and reduced computational cost through parallel implementation and informed initialization. A theoretical analysis establishes consistency, boundedness, and convergence under standard Lipschitz assumptions. Numerical experiments on representative fractional models demonstrate that the proposed schemes achieve higher accuracy and improved efficiency compared with classical implicit methods, with significant reductions in error and iteration counts. The ANN-enhanced variant further attains near machine-precision accuracy for a range of fractional orders. Overall, the proposed approach provides a robust and scalable computational framework for the efficient solution of nonlinear fractional dynamical systems. Full article

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with $\alpha$-cut interval arithmetic using constrained fuzzy arithmetic model parametric uncertainty, with numerical values. Oxygen-dependent carrying capacity follows a Hill-type function; hypoxia-induced angiogenesis follows a decreasing Michaelis–Menten function. The model transitions at $t_1=50$ days from memoryless fuzzy classical derivative to fuzzy ABC fractional derivative of order $\psi$. The transition time $t_1=50$ days is biologically justified based on experimental observations of the angiogenic switch in solid tumors, which typically occurs within 4–8 weeks post-inoculation. Positivity, boundedness, Lipschitz continuity, existence, and uniqueness of fuzzy solutions are proved via Banach fixed-point theorem in a weighted norm. A basic reproduction number interval ${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]$ is derived; local and global stability conditions are established for disease-free and endemic equilibria using fuzzy differential inclusions. Global sensitivity analysis using latin hypercube sampling with $N=500$ samples explores the range of possible outcomes across the fuzzy parameter support. In the numerical implementation, we use a fourth-order fuzzy Runge–Kutta method (Phase I), and a fractional Adams–Bashforth–Moulton predictor-corrector method (Phase II), ensuring preservation of fuzzy number characteristics. Full article

This paper investigates the finite time stability (FTS) of multi-delayed systems with Riemann-Liouville fractional order (RLFO). Firstly, a lemma on the FTS criterion is established for RLFO multi-delay systems, which lays the theoretical groundwork for the subsequent analysis of network synchronization and identification. Secondly, for hybrid coupled complex networks (CNs) with RLFO, multiple delays, and a non-Lipschitz vector field, we explore finite-time synchronization and topology identification (TI) without imposing the linear independence condition (LIC). This is achieved by constructing: (1) a regulated control network with topology observers, and (2) an auxiliary network with isolated nodes. Based on the proposed FTS criterion, along with the designed control protocol and adaptive topology observer, sufficient conditions for the finite time synchronization and TI of multi-delayed CNs are derived as linear matrix inequalities (LMIs). Finally, a numerical simulation on the Lorenz system is carried out to validate the derived results and evaluate the efficacy of the proposed method. Full article

This paper develops a mathematically grounded neuro-fuzzy control framework for IoT-enabled irrigation systems in precision agriculture. A discrete-time, physically motivated model of soil moisture is formulated to capture the nonlinear water dynamics driven by evapotranspiration, irrigation, and drainage in the crop root zone. A Mamdani-type fuzzy controller is designed to approximate the optimal irrigation strategy, and an equivalent Takagi–Sugeno (TS) representation is derived, enabling a rigorous stability analysis based on Input-to-State Stability (ISS) theory and Linear Matrix Inequalities (LMIs). Online parameter estimation is performed using a Recursive Least Squares (RLS) algorithm applied to real IoT field data collected from a drip-irrigated orchard. To enhance prediction accuracy and long-term adaptability, the fuzzy controller is augmented with lightweight artificial neural network (ANN) modules for evapotranspiration estimation and slow adaptation of membership-function parameters. This work provides one of the first mathematically certified neuro-fuzzy irrigation controllers integrating ANN-based estimation with Input-to-State Stability (ISS) and LMI-based stability guarantees. Under mild Lipschitz continuity and boundedness assumptions, the resulting neuro-fuzzy closed-loop system is proven to be uniformly ultimately bounded. Experimental validation in an operational IoT setup demonstrates accurate soil-moisture regulation, with a tracking error below 2%, and approximately 28% reduction in water consumption compared to fixed-schedule irrigation. The proposed framework is validated on a real IoT deployment and positioned relative to existing intelligent irrigation approaches. Full article

This paper investigates a novel class of fractional Langevin equations, which introduces a time-varying p(t)-Laplacian operator and parameterized anti-periodic boundary conditions. This approach overcomes the limitations of traditional models characterized by constant diffusion exponents and fixed boundary locations. Under non-compactness conditions, the existence of solutions is established by applying Schaefer’s fixed-point theorem, which significantly relaxes the conventional constraints on the nonlinear term. Moreover, by imposing a Lipschitz condition on the nonlinear term, a Ulam–Hyers-type stability criterion for the coupled system is derived. This work not only extends the relevant stability theory but also provides a rigorous theoretical foundation for error control in practical applications. The effectiveness of the theoretical results is validated through numerical examples. Full article

This study looks at the stability and stabilization issues concerning the nonlinear time-delay systems specified by conformable derivatives. These requirements can be used for many useful applications. Through the construction of appropriate Lyapunov–Krasovskii functionals, we develop novel linear matrix inequality (LMI) conditions for the exponential stability of autonomous systems and practical exponential stability for systems subject to bounded perturbations. Furthermore, we propose state-feedback stabilization strategies that transform the controller design problem into a convex optimization framework solvable via efficient LMI techniques. The theoretical developments are comprehensively validated through numerical examples that demonstrate the effectiveness of the proposed stability and stabilization criteria. The results establish a rigorous framework for analyzing and controlling conformable fractional-order systems with time delays, bridging theoretical advances with practical implementation considerations. Full article

Maximum power point tracking (MPPT) under partial shading is a nonconvex, rapidly varying control problem that challenges multi-agent policies deployed on photovoltaic modules. We present Fuzzy–MAT3D, a fuzzy-augmented multi-agent TD3 (Twin-Delayed Deep Deterministic Policy Gradient) controller trained under centralized training/decentralized execution (CTDE). On the theory side, we prove that differentiable fuzzy partitions of unity endow the actor–critic maps with global Lipschitz regularity, reduce temporal-difference target variance, enlarge the input-to-state stability (ISS) margin, and yield a global $L_{\infty }$$\gamma$-contraction of fixed-policy evaluation (hence, non-expansive with $\kappa =\gamma <1$). We further state a two-time-scale convergence theorem for CTDE-TD3 with fuzzy features; a PL/last-layer-linear corollary implies point convergence and uniqueness of critics. We bound the projected Bellman residual with the correct contraction factor (for $L^{\infty }$ and $L^2\left(\rho \right)$ under measure invariance) and quantified the negative bias induced by $min\{Q_1,Q_2\}$; an N-agent extension is provided. Empirically, a balanced common-random-numbers design across seven scenarios and 20 seeds, analyzed by ANOVA and CRN-paired tests, shows that Fuzzy–MAT3D attains the highest mean MPPT efficiency (92.0% ± 4.0%), outperforming MAT3D and Multi-Agent Deep Deterministic Policy Gradient controller (MADDPG). Overall, fuzzy regularization yields higher efficiency, suppresses steady-state oscillations, and stabilizes learning dynamics, supporting the use of structured, physics-compatible features in multi-agent MPPT controllers. At the level of PV plants, such gains under partial shading translate into higher effective capacity factors and smoother renewable generation without additional hardware. Full article

We develop a comprehensive dynamic Walrasian framework entirely in $L^{\infty }$ so that prices and allocations are essentially bounded, and market clearing holds pointwise almost everywhere. Utilities are allowed to be locally Lipschitz and quasi-concave; we employ Clarke subgradients to derive generalized quasi-variational inequalities (GQVIs). We endogenize inventories through a capital-accumulation constraint, leading to a differential QVI (dQVI). Existence is proved under either strong monotonicity or pseudo-monotonicity and coercivity. We establish Walras’ law, and the complementarity, stability, and sensitivity of the equilibrium correspondence in $L^2$-metrics, incorporate time-discounting and uncertainty into $\Omega \times [0,T]$, and present convergent numerical schemes (Rockafellar–Wets penalties and extragradient). Our results close the “in mean vs pointwise” gap noted in dynamic models and connect to modern decomposition approaches for QVIs. Full article

This article presents several key findings for fractional-order delay differential equations. First, we establish the existence and uniqueness of solutions using two distinct approaches, the Chebyshev norm and the Bielecki norm, thereby providing a comprehensive understanding of the solution space. Notably, the uniqueness of the solution is rigorously demonstrated using the Lipschitz condition, ensuring a single solution under specific constraints. Additionally, we examine a specific form of constant delay and apply Burton’s method to further confirm the uniqueness of the solution. Furthermore, we conduct an in-depth investigation into the Hyers–Ulam stability of the problem, providing valuable insights into the behavior of solutions under perturbations. Notably, our results eliminate the need for contraction constant conditions that are commonly imposed in the existing literature. Finally, numerical simulations are performed to illustrate and validate the theoretical results obtained in this study. Fractional-order delay differential equations play a crucial role in real-life applications in systems where memory and delayed effects are essential. In biology and epidemiology, they model disease spread with incubation delays and immune memory. In control systems and robotics, they help design stable controllers by accounting for time-lagged responses and past behavior. Full article

This paper presents an efficient algorithm for Explicit Model-Following (EMF) control using an Output-derivative Feedback Control (OFC) scheme within the Reciprocal State Space (RSS) framework, aimed at overcoming the performance limitations associated with state-derivative dependence. For Lipschitz Nonlinear Systems (LNS), two approaches are proposed: a linear EMF (LEMF) strategy, which transforms the system into a Linear Parameter-Varying (LPV) representation via the Differential Mean Value Theorem (DMVT) to facilitate controller design, and a nonlinear EMF (NEMF) scheme, which enables the direct tracking of a nonlinear reference model. The stability of the closed-loop system is ensured by deriving control gains through Linear Quadratic Regulator (LQR) optimization. The proposed algorithms are validated through Real-Time Implementation (RTI) on an Arduino DUE platform, demonstrating their effectiveness and practical feasibility. Full article

This paper addresses the problem of observer-based control (OBC) for nonlinear systems with time delay (TD). A novel hybrid modeling framework for nonlinear TD systems is first introduced by synergistically combining TD Takagi–Sugeno (TDTS) fuzzy and Lipschitz approaches. The proposed methodology broadens the range of representable systems by enabling Lipschitz nonlinearities to fulfill dual functions: they may describe essential dynamic behaviors of the system or represent aggregated uncertainties, depending on the specific application. The proposed TDTS–Lipschitz (TDTSL) model class features measurable premise variables while accommodating Lipschitz nonlinearities that may depend on unmeasurable system states. Then, through the construction of an appropriate Lyapunov–Krasovskii (L-K) functional, we derive sufficient conditions to ensure exponential stability of the augmented closed-loop model. Subsequently, through a decoupling methodology, these stability conditions are reformulated as a set of linear matrix inequalities (LMIs). Finally, the proposed OBC design is validated through application to a continuous stirred tank reactor (CSTR) with lumped uncertainties. Full article

Reinforcement learning (RL), and in particular Proximal Policy Optimization (PPO), has shown promise in high-precision quadrotor unmanned aerial vehicle (QUAV) control. However, the performance of PPO is highly sensitive to the choice of the clipping parameter, and inappropriate settings can lead to unstable training dynamics and excessive policy oscillations, which limit deployment in safety-critical aerial applications. To address this issue, we propose a stability-aware dynamic clipping parameter adjustment strategy, which adapts the clipping threshold ${ϵ}_t$ in real time based on a stability variance metric $S_t$. This adaptive mechanism balances exploration and stability throughout the training process. Furthermore, we provide a Lipschitz continuity interpretation of the clipping mechanism, showing that its adaptation implicitly adjusts a bound on the policy update step, thereby offering a deterministic guarantee on the oscillation magnitude. Extensive simulation results demonstrate that the proposed method reduces policy variance by 45% and accelerates convergence compared to baseline PPO, resulting in smoother control responses and improved robustness under dynamic operating conditions. While developed within the PPO framework, the proposed approach is readily applicable to other on policy policy gradient methods. Full article

We investigate the solvability and stability properties of a class of nonlinear stochastic delay differential equations (SDDEs) driven by Wiener noise and incorporating discrete time delays. The equations are formulated within a Banach space of continuous, adapted sample paths. Under standard Lipschitz and linear growth conditions, we construct a solution operator and prove the existence and uniqueness of strong solutions using a fixed-point argument. Furthermore, we derive exponential mean-square stability via Lyapunov-type techniques and delay-dependent inequalities. This framework provides a unified and flexible approach to SDDE analysis that departs from traditional Hilbert space or semigroup-based methods. We explore a Banach space fixed-point approach to SDDEs with multiplicative noise and discrete delays, providing a novel functional-analytic framework for examining solvability and stability. Full article

In this work, the distributed average consensus for dynamical networks with Lipschitz nonlinear dynamics is studied, where the network communication switches quickly among a set of directed and balanced switching graphs. Differing from existing research concerning uniform constant delay or time-varying delays, this study focuses on consensus problems with mixed delays, equipped with one class of delays embedded within the nonlinear dynamics and another class of delays present in the control input. In order to solve these problems, a proportional and derivative control strategy with time delays is proposed. In this way, by using Lyapunov theory, the stability is analytically established and the conditions required for solving the consensus problems are rigorously derived over switching digraphs. Finally, the effectiveness of the designed algorithm is tested using the MATLAB R2021a platform. Full article

This paper focuses on McKean-Vlasov stochastic differential equations under local Lipschitz conditions. We first introduce the stochastic interacting particle system and prove the propagation of chaos. Then we establish a truncated stochastic theta scheme to approximate the interacting particle system and obtain the strong convergence of the continuous-time truncated stochastic theta scheme to the non-interacting particle system. Furthermore, we study the asymptotical mean square stability of the interacting particle system and the truncated stochastic theta method. Finally, we give one numerical example to verify our theoretical results. Full article