Open Access

Articles

Article ID: 4409

On numerical radius inequalities via McCarthy inequality

by Muhammad Fazeel Anwar, Saira Iqbal, Muhammad Saeed Akram

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper aims to establish new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space. By employing a generalized form of the McCarthy inequality, we derive several upper bounds for the numerical radius of a single operator as well as for expressions involving sums and products of operators. The obtained results extend and refine a number of existing inequalities in the literature. In particular, many known numerical radius inequalities are recovered as special cases of our results, thereby providing a unified framework for their analysis. The refinement is based on the Akkouchi-Ighachane version of the Hölder-McCarthy inequality, which allows the usual McCarthy term to be replaced by a smaller parameter-dependent expression before taking the supremum. This gives a common refinement mechanism for estimates involving a single operator, Cartesian decompositions, finite sums, block matrices and the Euclidean operator radius. We also state explicitly the parameter choices which recover the earlier inequalities and include simple finite-dimensional comparisons showing that, for suitable non-normal matrices, the refined bounds may be strictly sharper than the corresponding classical estimates. These comparisons confirm that the refinement provides a usable quantitative improvement, rather than only a formal parameter extension of existing bounds in concrete cases.

Open Access

Articles

Article ID: 4304

Fractional-order differential models for multiscale transport phenomena in materials and energy systems

by Shuwei Zhang

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Transport in heterogeneous and hierarchical materials often exhibits memory, spatial nonlocality, distributed relaxation, and scale-dependent behavior that cannot be adequately represented by classical local constitutive laws. This review critically examines fractional-order differential models as effective continuum descriptions of multiscale heat, mass, charge, and reactive-species transport in materials and energy systems. It first reviews the principal classes of fractional formulations, including time-, space-, time-space-, variable-, distributed-, tempered-, and selectively fractionalized multiphysics models, and examines how operator definitions, kernel structures, domains, initial and boundary conditions, dimensional consistency, and stochastic or coarse-graining interpretations influence their physical meaning. The review then compares analytical, numerical, data-driven, and hybrid solution strategies while evaluating numerical verification, parameter identification, structural and practical identifiability, uncertainty quantification, model discrimination, and experimental validation. Applications in porous and heterogeneous materials, electrochemical systems, thermal transport, subsurface environments, reactive and degrading media, and engineered functional materials are assessed using three cumulative levels of evidence: phenomenological representation, mechanistic support, and predictive transferability. Classical models are retained as reference baselines throughout. The review shows that fractional models can provide compact descriptions of unresolved multiscale complexity; however, anomalous observations alone do not uniquely identify a fractional mechanism. Their scientific use is warranted only when the selected operator is physically and mathematically defensible, parameters are sufficiently identifiable, credible nonfractional alternatives are outperformed, and predictions remain transferable beyond calibration.

Open Access

Articles

Article ID: 4602

Novel fixed point theorems for order-theoretic enriched contractions with applications to boundary value problems and matrix equations

by Mohammad Akram, Umar Ishtiaq, Muhammad Din, Ioan-Lucian Popa

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This work puts forward a family of operators, which we term enriched ∇-preserving contractions, acting on a normed space carrying an arbitrary binary relation ∇. The family gathers several familiar classes under one roof: Banach contractions, order-theoretic contractions, and certain nonexpansive maps, among them averaged maps whose fixed points lie beyond the reach of ordinary contraction arguments; all arise through suitable choices of the constants and of the relation. Working in this relational framework, we show that any such operator admits a fixed point, even though its defining contractive estimate is demanded only on ∇-related pairs rather than across the whole space. A device used repeatedly is the symmetry of the norm: it forces ∇ and its symmetric closure to act compatibly with the operator, and this is what keeps the fixed-point conclusions meaningful. The fixed points themselves are located through the Krasnoselskii iteration, and a path criterion formulated in the symmetric closure of ∇ then yields their uniqueness; an almost contraction variant of the scheme is treated as well. By way of application, the theory settles the solvability, and under the stated hypotheses the uniqueness, of solutions to a Caputo fractional boundary value problem with an integral condition and to a nonlinear matrix equation. Numerical experiments over several groups of Lipschitz coefficients confirm the convergence threshold predicted by the theory, and the iteration is compared with Riemann-Hilbert analysis, Lie group integrators, physics-informed neural networks, and neural symbolic derivation tools, so that strengths and limits of the framework can be judged objectively.

Open Access

Articles

Article ID: 4605

Enriched polynomial contractions and a neural fixed-point solver for fractional boundary value problems

by Mohammad Akram, Umar Ishtiaq, Muhammad Din, Ioan-Lucian Popa

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

We introduce two classes of operators on normed spaces, the enriched polynomial contractions and the almost enriched polynomial contractions. Both classes contain the enriched contractions of Berinde and Păcurar and the polynomial-type contractions of Jleli, Pacurar and Samet as special cases. In Banach spaces, we prove existence, uniqueness, and convergence theorems for the fixed points, first under continuity of the operator and then under the weaker assumption of Picard continuity, with the fixed points approximated by Krasnoselskii-type iteration. Several classical results, including the Banach contraction principle and the enriched contraction theorem, are recovered as special cases, and several worked examples are given. As the main application, we consider a nonlinear Caputo fractional boundary value problem of order 2 < N ≤ 3. Writing it as a fixed-point equation and assuming an explicit Lipschitz condition, we prove that it has a unique solution. We then examine the constructive side of this result numerically: on a manufactured problem with a known exact solution, we run the associated fractional-quadrature Krasnoselskii iteration, check that its measured geometric rate stays below the certified contraction factor, and recover the same solution with a neural fixed-point solver, a network trained to satisfy the discretized fixed-point equation of the same operator, and we relate the accuracy of both solvers to the underlying quadrature error. The numerical results are consistent with the theory and illustrate its use on a class of nonlinear fractional models of this form.

Open Access

Articles

Article ID: 4621

Rainfall-induced instability mechanism of high-altitude gravelly soil slopes and machine learning surrogate modeling

by Jianli Jin, Yuhao Zheng, Yongchen Zong, Baoliang Wang, Qiang He

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study investigates the distinctive instability mechanism of high-altitude gravelly-soil slopes driven by particle gradation, weak clayey characteristics and intense rainfall. Soil parameters were obtained from laboratory tests on a typical high-steep bedding gravelly-soil slope in the Parlung Zangbo Basin, southeastern Tibet. A fully coupled stress-pore-pressure finite-element model was built in ABAQUS to simulate seepage-deformation responses under six rainfall intensities and multiple durations. An infiltration threshold of 1.9 × 10⁻⁴ cm/s is determined. This threshold, a near-surface transient saturated zone, develops rapidly and impedes infiltration. Pore-water-pressure at the slope toe responds roughly 12 h earlier than at the crest. Every 10 cm increase in cumulative infiltration produces approximately 15 mm extra toe horizontal displacement (R² = 0.93, 95% confidence interval (CI): 1.43–1.61). A 32-h deformation lag is observed post-rainfall, representing the period required for displacement to attain 90% of its stable value. Random-forest surrogate models for toe displacement and pore-water pressure are trained on 1,296 spatiotemporal finite-element samples, yielding test-set R² > 0.96. Comparisons with Gaussian process regression, support vector regression and eXtreme Gradient Boosting (XGBoost) reveal that XGBoost achieves optimal accuracy (horizontal-displacement root mean square error (RMSE) = 0.42 mm), while random forest provides competitive performance (mean absolute percentage error (MAPE) = 7.8%) and better interpretability. A classification model detects transient saturated zones at 92.3% accuracy. This coupled numerical-simulation and machine-learning framework provides a new tool for rapid early-warning and parameter-sensitivity analysis of similar high-altitude rainfall-induced slope hazards.

Open Access

Articles

Article ID: 4719

Improving the aerodynamic stability and controllability of aircraft, considering the speed of sound and maneuvering oscillations under disturbing atmospheric conditions

by Toghrul Karimli, Aftandil Mammadov, Rauf Guliyev, Elnur Asadov, Vasif Karimli

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

An aircraft's aerodynamic stability and controllability in pitch, roll, and yaw are determined by its flight performance and actual flight conditions. Traditional stability augmentation systems based on gyroscopic and inertial feedback are activated only after a disturbance has been applied, so the aircraft first acquires unwanted angular deflections and kinetic energy that must subsequently be damped. This article proposes an integrated smart aerometric approach that enables early detection of atmospheric disturbances before they affect the aircraft dynamics. Its key element is an atmospheric disturbance monitoring unit based on smart aerometric sensors located on the wingtips and in the tail section. Coupled with advanced feedback structures in the pitch, roll, and yaw channels, these sensors reduce transient response time and improve stability and control in turbulent conditions. The concept is evaluated by comparative modeling in MATLAB/Simulink for classical, modern, and proposed flight control architectures, with true airspeed relative to the speed of sound (Mach number) and load factor taken as the key flight parameters. In stable flight, adaptive modification of the transfer function coefficients increases the stability margin and prevents autopilot disengagement under intense turbulence and overload; in maneuvering mode, automatic coefficient readjustment improves controllability by reducing excess stability. The results of mathematical modeling indicate that the proposed approach reduces the amplitude and duration of damped transient processes during maneuvering. An explicit adaptive gain-scheduling law, a closed-loop stability analysis, and a quantitative case study with performance indices and robustness evaluation are also presented.

Open Access

Article

Article ID: 4390

Not loss of stability but steady-state shift: A systems biology explanation for elevated fasting blood glucose in type 2 diabetes

by Guanyu Wang

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Elevated fasting blood glucose is a hallmark clinical feature of prediabetes and type 2 diabetes, reflecting underlying pathological alterations in the body’s glucose-insulin regulatory system. This study employs a validated mathematical model of the glucose-insulin negative feedback loop to investigate the fundamental mechanisms of this elevation from a systems biology perspective. We specifically analyzed whether hyperglycemia arises from decreased system stability or an upward shift in the steady-state set-point. Our findings demonstrate that the intrinsic stability of the glucose-insulin regulatory system remains largely unchanged from healthy states through early-stage diabetes. Contrary to the hypothesis of stability decay, the essence of elevated fasting glucose is an upward shift of the steady-state level driven primarily by hepatic insulin resistance, which increases basal hepatic glucose output. While system stability is preserved during early progression, the dynamic coupling between glucose and insulin weakens; low-frequency oscillations inherent to the healthy system gradually diminish and eventually disappear as insulin resistance intensifies, signaling a decoupling of dynamics before stability loss. Significant weakening of system stability, characterized by a markedly reduced convergence rate following perturbation, occurs only in late-stage diabetes accompanied by severe pancreatic damage and compromised insulin secretory capacity. These results redefine the pathophysiological understanding of fasting hyperglycemia, suggesting that early therapeutic strategies should target recalibrating the glucose set point and hepatic sensitivity rather than bolstering system stability. This research provides valuable theoretical insights into the pathogenesis of diabetes and highlights potential dynamic biomarkers for monitoring disease progression.

Open Access

Article

Article ID: 4327

Channeling paths identification and mitigation by using integrated profile control and flooding based on meshless connection element method

by Wei Yong, Zhijie Wei, Jian Zhang, Wensheng Zhou, Yuyang Liu, Wentao Zhan, Wei Liu, Yixin Zhang, Jiaxu Mei

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Addressing the challenge of quantitatively identifying deep thief zones in mature oilfields during the high water-cut stage, this study proposes a robust quantitative characterization model for thief channels based on the non-Euclidean, meshless Connection Element Method (CEM) to directly guide in-depth fluid diversion and integrated profile control and flooding treatments through automated flow path tracking rooted in a directed-graph depth-first search algorithm. To systematically capture the complex subsurface topological network, a comprehensive multi-parameter connectivity parameter system was constructed by integrating key dynamic indicators, including connection conductivity, connection volume, and path splitting coefficients. By dynamically coupling these parameters with the field-wide Lorentz coefficient, a dimensionless channeling factor was defined to establish a rigorous four-level quantitative standard—ranging from extreme channeling to matrix seepage—thereby successfully advancing preferential pathway evaluation from qualitative inference to spatial grading and precise localization. Quantitative validation against conventional commercial grid-based compositional simulators demonstrates the superior fidelity and performance forecasting efficiency of the proposed method: the CEM framework achieves an exceptionally accurate water-cut prediction Root Mean Square Error (RMSE) of approximately 3.8%. Crucially, by abstracting continuous domains into streamlined networks, it drastically compresses structural degrees of freedom, successfully accelerating the operational execution runtime from 7.3 s to a mere 1.6 s. Ultimately, this work provides a computationally highly efficient, physically sound novel approach for the reliable mapping and graded quantification of deep dominant channeling pathways in mature heterogeneous oilfields.

Open Access

Article

Article ID: 4591

Stability and controllability for fractional damped differential equations with ψ-Hilfer derivatives

by Junjie Xie, Xiaoyue Han, Zhenbin Fan

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study investigates the existence, uniqueness, finite-time stability and controllability for a class of nonlinear ψ-Hilfer fractional differential equations with damping terms, and the main mathematical contributions are achieved through several techniques including parameter selection, reasonable application of weighting methods and transformations. Specifically, by leveraging the generalized Laplace transform method and weighted norm technique, the explicit representation of solutions is derived. The existence and uniqueness of solutions are established via the generalized Banach contraction principle and Schauder fixed point theorem. Then finite-time stability criteria are obtained via inequality estimates and fractional Gronwall inequalities, each providing complementary insights into the behavior of systems within a bounded time interval. Furthermore, we establish controllability criteria for linear and nonlinear systems, the latter relies on the successive approximation method. Finally, several numerical examples are presented to validate our findings. These examples illustrate the necessity of the two existence theorems, since their respective assumptions are not always simultaneously satisfied in practical applications. In particular, we examined the magnitude of the error generated when the controllability conditions for the nonlinear system are not satisfied. These results contribute to the broader understanding of fractional damped systems governed by ψ-Hilfer derivatives and offer practical tools for analyzing stability and controllability in complex damped dynamical systems.

Open Access

Article

Article ID: 4549

On a class of bivalued hybrid cyclic and non-cyclic contractive self-mappings on unions of possibly disjoint subsets in metric spaces

by Manuel De la Sen, Asier Ibeas, Hasanen A. Hammad

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper presents and analyzes a class of self-mappings in metric spaces which are endowed of mixed characteristics between that of contractive cyclic self-mappings on union of subsets and that of contractive mappings within the individual subsets. Stability analysis of dynamic systems can be focused on by formalizing the relationship between cyclic contractive mappings over non-necessarily intersecting subsets and the stability criteria for switched systems with cyclic dynamics. The case of disjoint set structures necessitates the application of best proximity theory. Therefore, this manuscript introduces the mentioned class of hybrid cyclic/non-cyclic bivalued contractive self-mappings which can operate iteration-by-iteration in either a cyclic operation mode (the selected image lies in the next adjacent subset) or in an intra mode operation (the selected image lies in the current subset). Unlike cyclic operators, these proposed self-mappings allow dynamic switches to either the current subset or the next adjacent one in a cyclic disposal, governed by an iteration-dependent image selection sequence. This “modus-operandi” is possible since one of the images of each point at each iteration is activated for the cyclic operation mode while the other one is activated for an intra mode mode within some subset. Under the key assumptions that the subsets are closed, at least one best proximity set is a singleton, and the subset itself is boundedly compact, new boundedness and convergence theorems are established. It is emphasized how cyclic contractive properties are useful in the context of asymptotic stability of hybrid systems, validating the theoretical framework through illustrative examples.

Open Access

Article

Article ID: 4530

Enhanced PSO with adaptive weights and Gaussian mutation based 3D path planning for UAVs

by Gaofeng Che

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Path planning remains a pivotal metric for evaluating unmanned aerial vehicles (UAVs) autonomy, where intelligent algorithms serve as the cornerstone for generating high-quality trajectories. Path planning is an indispensable part of the autonomous control system for UAVs. In addition, UAV mission planning has been proven to be an NP-hard problem. This paper systematically analyzes the state-of-the-art intelligent optimization algorithms for UAVs' path planning. Considering that traditional particle swarm optimization (PSO) and adaptive weight particle swarm optimization (AWPSO) easily converge to local optima, this work develops an enhanced PSO with adaptive weight and Gaussian mutation (EPSO-AWGM). The introduction of adaptive weight factors and Gaussian mutation operators helps the algorithm escape local optimum solutions. Moreover, the cubic spline method is utilized to smooth the UAVs' flight paths. To verify the performance of the proposed EPSO-AWGM algorithm and the quality of UAV path planning in a 3D environment, this paper sets the traditional PSO and AWPSO as control groups, conducting comparative simulation experiments on 3D terrain path planning. The simulations uniformly set initial parameters, with a particle population size of 20 and a maximum iteration count of 50. All algorithms operate in a 3D environment containing five mountain obstacles, with a unified starting coordinate at (1,1,1) and an ending coordinate at (29,29,17). To ensure objective experimental results and eliminate random error interference, all three algorithms are independently run 20 times. Simulation results show that the proposed EPSO-AWGM can obtain shorter and higher-quality flight paths. It therefore has great application potential in practical UAV path planning scenarios.

Open Access

Article

Article ID: 4372

Nonlinear dynamic response and time-delay feedback vibration control of multilayer beams on compressible clayey foundations subjected to moving concentrated loads

by Roger Eno, Guillaume Hervé Poh'sié, Hermann Joel Ouandji Boutcheng, Ekoum Ewandjo Nkoue, Dianorre Tokoue Ngatcha, Joseph Mousi Bikoun, Fabien Kenmogne, Séverin Nguiya

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study investigates the nonlinear vibration behaviour of multilayer beams resting on a compressible elastic foundation and subjected to a moving concentrated load. The objective is to analyse the combined effects of geometric nonlinearity, interlayer elastic coupling, foundation compressibility, and moving-load excitation on the dynamic response of the structure. An analytical formulation based on the Lagrangian formalism is developed, incorporating kinetic energy, bending energy, foundation stiffness, interlayer coupling, and nonlinear stretching effects. The proposed formulation captures the structural dynamics through coupled Duffing-type nonlinear modal interactions, while accounting for damping and moving-load excitation. A Fourier-Galerkin modal reduction is employed to derive a reduced-order temporal system, whose equilibrium states and stability characteristics are investigated. Numerical simulations performed using Fourier modal decomposition and a fourth-order Runge-Kutta scheme reveal complex dynamic behaviours, including multistability, resonance amplification, quasi-periodic oscillations, and branch-jump phenomena induced by nonlinear interactions and parameter sensitivity. To mitigate excessive vibrations, a time-delay feedback controller is introduced and applied to all modal coordinates. The delayed control law yields a transcendental characteristic equation, from which explicit stability conditions are established in terms of the feedback gain and time delay. Numerical results demonstrate that the proposed controller effectively suppresses nonlinear vibration amplification, reduces oscillation amplitudes, and improves the dynamic stability of the system. The proposed framework provides a comprehensive nonlinear beam-foundation interaction model combined with delay-based vibration control, contributing to the analysis and stabilization of multilayer structural systems subjected to moving loads on deformable foundations.

Open Access

Article

Article ID: 4577

Complex dynamics of a tri-trophic predator-prey system with double fear effects

by Ting Yang, Yue Cai

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

In this paper, we investigate a tri-trophic predator-prey system with double fear effects, where the prey is affected by fear induced by both the intermediate predator and the top predator. The top predator is assumed to feed on both the basal prey and the intermediate predator. We establish the positivity and boundedness of solutions and derive sufficient conditions for the global asymptotic stability of axial equilibria. The possible interior equilibria are characterized algebraically, and their local stability is analyzed. We further derive the Hopf bifurcation conditions and compute the first Lyapunov coefficient to determine the direction of the bifurcation and the stability of the bifurcating periodic orbit. Numerical simulations reveal several forms of complex dynamics, including a period-three orbit in the associated return dynamics, the coexistence of a stable equilibrium and a stable limit cycle, the coexistence of two stable limit cycles, and chaotic behavior. The corresponding multistability is further illustrated by basin-of-attraction diagrams. Numerical evidence for chaotic attractors in parameter regions admitting different numbers of interior equilibria is provided by positive largest Lyapunov exponents, parameter bifurcation diagrams, and bounded long-time trajectories. These results show that, within the parameter regimes examined, the interaction between the two fear mechanisms and the tri-trophic feeding structure can generate pronounced dynamical transitions, multistability, and strong sensitivity to initial population densities.

Open Access

Article

Article ID: 4607

Stochastic irregular linear-quadratic optimal control with dual controllers and two-layer asymmetric partial observations

by Yangyang Shi, Tianfu Ma

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Linear-quadratic (LQ) optimal control problems have been widely studied under symmetric information and regularity assumptions. However, in many practical systems, multiple controllers operate with different information sets, and the standard regularity condition may fail due to singular weighting matrices. This paper investigates a stochastic irregular LQ optimal control problem with dual controllers and two-layer asymmetric partial observations, in which one controller has strictly less information than the other. The main contribution is the derivation of both the solvability conditions and the explicit feedback-form solutions for this problem. To this end, we reformulate the optimization problem as a system of forward-backward stochastic differential equations (FBSDEs) that naturally captures the asymmetric information structure. By solving these FBSDEs via four interconnected Riccati equations, we obtain explicit optimal controllers for both the regular and irregular cases. In the irregular case, where the standard Riccati equation is not directly solvable, we introduce auxiliary Riccati equations and derive additional existence conditions involving singular matrix decompositions. A detailed technical comparison with existing results confirms that our solution is not a straightforward extension, as prior works either addressed irregular LQ control under full information or asymmetric information under regular conditions. This work provides a unified framework for optimal control problems where information asymmetry and singularity arise simultaneously.

Open Access

Article

Article ID: 4582

GRF-based trajectory planning and application for bipedal motion

by Qianying Zhu, Aliyu Hamza, Yanan Gao, Jiaqi Gao, Shouhua Liu, Enping Guo, Qiuyue Cheng, Yanxiang Chen, Yuanyang Chen, Yue Wang

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Trajectory planning for the center of mass (CoM) constitutes a fundamental challenge in bipedal locomotion. Conventional approaches, which predominantly rely on experimental measurements or simplified analytical models, are often constrained by discretization errors, parameter uncertainty, and prohibitive experimental costs. To address these limitations, this paper proposes a numerical framework for generating CoM trajectories directly from ground reaction force (GRF) dynamics. The methodology first determines key gait event positions and formulates a coupled system of governing equations that integrate force and motion. The method then establishes a unified spatio-temporal-GRF coordinate system and incorporates boundary conditions, along with load-sharing mechanisms between the trailing and leading legs, to derive the force and geometric constraint equations. The resulting reference trajectory is represented using Fourier coefficients, which are determined via an Adam-based constrained residual minimization solver. This Fourier-based representation offers notable advantages, including a compact parameter set, high computational efficiency, and the capacity to capture the essential kinematic features of bipedal walking. To validate the proposed reference trajectory, a sliding mode control (SMC) framework is employed for bipedal walking control, in which the walker successfully tracks the reference trajectory with stable gait execution. The results confirm that the generated trajectory faithfully reproduces the characteristic CoM response of bipedal walking, thereby providing a high-fidelity reference for robust motion control. Furthermore, a systematic analysis of the control parameters is conducted to evaluate their influence on walking stability. This work offers a new paradigm that synergistically combines physical interpretability with computational efficiency, holding significant promise for applications in humanoid robot motion control and biomechanical analysis.

Open Access

Article

Article ID: 4535

Online non-intrusive grid impedance estimation method for energy storage converters

by Yunkun Xiang, Kai Liu, Cheng Wan, Hao Zhou, Zipeng Hu

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Grid impedance has an important influence on the synchronization stability of grid-connected energy storage converters and their adaptability to weak grids. Conventional intrusive estimation methods require additional perturbations to be injected, which reduces power quality and complicates practical applications. This paper proposes a non-intrusive online grid impedance estimation method. The excitation source is the power regulation dynamics of the energy storage converter itself. The sliding-window discrete Fourier transform is used to extract the fundamental components of the point of common coupling (PCC) voltage and the converter output current from two adjacent single-cycle windows. By combining the voltage equations of the two windows, the unknown equivalent grid voltage components are eliminated, and the grid resistance and reactance can be calculated without injecting additional perturbations or using external phasor measurement equipment. Under constant-current conditions, a simulation with a sudden change in grid impedance is used to verify the effectiveness of the method. Experiments are carried out on a three-phase three-wire energy storage converter under current-step and impedance-switching conditions. The results show that the estimated grid resistance and reactance have small differences from the preset values. The estimation errors of the resistance and reactance on the α-axis are 5.78% and 5.27%, respectively, while those on the β-axis are 0.27% and 1.60%, respectively. This shows that the proposed method can achieve accurate online grid impedance estimation without interfering with the normal operation of the converter, and can improve the stability and adaptive control capability of the energy storage converter.

Open Access

Article

Article ID: 4667

Wellbore-reservoir coupled simulation study for CO₂ flooding in oil reservoirs

by Bujie Ding, Huanying Ma, Jincang Li, Kaifang Jin, Tao Zou, Hui Zhao, Jiaxu Liu, Siyuan Chen, Xi Ouyang, Xiang Rao

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

CO₂ flooding can simultaneously enhance oil recovery and enable geological carbon storage. This study develops a tNavigator-based coupled wellbore-reservoir CO₂ flooding model to examine tubing-head-to-downhole pressure conversion and early gas breakthrough through a high-permeability channel. A three-dimensional isothermal compositional model represents reservoir flow, while vertical flow performance (VFP) tables constructed using the Beggs-Brill correlation describe wellbore multiphase flow and pressure conversion. The coupled VFP relationships, well flow equations, and reservoir mass-conservation equations are solved using the adaptive implicit method (AIM) implemented in tNavigator, enabling bidirectional interactions among wellbore pressure variation, downhole boundary conditions, and reservoir dynamic parameters. A continuous high-permeability channel between injectors and producers is incorporated to characterize reservoir heterogeneity. Simulations under fixed injection rate and fixed tubing-head pressure conditions are conducted to analyze the influences of injection and production rates, porosity and streak permeability on CO₂ migration and flooding performance. The results show that injection and liquid production rates dominate the inter-well pressure difference and control gas breakthrough. Reservoir porosity primarily governs reservoir storage and pressure buffering capacity, while streak permeability determines CO₂ preferential migration velocity. Under tubing-head pressure constraints, actual injection and production performances are co-regulated by tubing-head pressure limits, wellbore pressure loss and reservoir injectivity/deliverability. The proposed model provides a numerical framework for investigating wellbore-reservoir interactions and gas-channeling risks in heterogeneous reservoirs with high-permeability channels. Quantitative validation against field measurements or controlled experimental data will be undertaken in future work.

Open Access

Article

Article ID: 4643

Computational modeling of turbulent nanofluid heat transfer over a heated moving surface with local thermal non-equilibrium and machine-learned eddy viscosity

by Muhammad Shoaib Arif, Yasir Nawaz

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study presents an adaptive modified Runge-Kutta compact scheme for the numerical simulation of unsteady k − ω turbulent nanofluid flow over a heated moving surface under local thermal non-equilibrium conditions. The surface-interfacial model incorporates mixed convection, viscous dissipation, turbulence transport, and separate energy equations for the base fluid and nanoparticle phases, with the effective thermal conductivity described by Xue’s formulation. The proposed time-integration method is explicit and combined with a compact finite-difference discretization that provides fourth-order spatial accuracy. The temporal coefficients are selected to achieve second-order accuracy, and the method is further enhanced through adaptive time stepping based on local error control. Stability analysis for the scalar convection–diffusion problem and conditional convergence analysis for the corresponding system formulation are also established. Numerical comparisons show that the proposed adaptive scheme yields lower error than existing adaptive Euler- and Runge-Kutta-based schemes. The computed results further demonstrate that thermal buoyancy increases the mean velocity, whereas larger Prandtl numbers reduce the thermal boundary-layer thickness of the fluid and nanoparticle phases. In addition, stronger interphase coupling modifies the two-temperature fields in a manner consistent with local thermal nonequilibrium. A machine-learning model is also employed to predict eddy viscosity, and its reliability is confirmed through profile comparisons, contour analyses, sensitivity assessments, and Taylor diagram evaluations. Overall, the proposed framework provides an accurate and efficient computational tool for surface-associated turbulent nanofluid transport with interfacial thermal nonequilibrium.

Open Access

Article

Article ID: 4603

On (θ, α)-metric space and fixed point results with application

by Mohammad Akram, Umar Ishtiaq, Zainab Bibi, Tayyab Kamran, Ioan-Lucian Popa

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

The notion of a double controlled metric type space was introduced by Abdeljawad et al. as a broadening of the controlled metric type spaces of Mlaiki et al., and shortly afterwards Aydi et al. formulated the new extended b-metric spaces to widen the extended b-metric spaces of Kamran et al. Building on these developments, the present work advances the class of (θ, α)-metric spaces, a structure that simultaneously subsumes b-metric spaces, new extended b-metric spaces, controlled metric type spaces and double controlled metric type spaces. This is accomplished by attaching two control functions θ(ζ, µ, s) and α(ζ, µ, s) to the terms situated on the right-hand side of the triangle inequality in the axioms defining a (θ, α)-metric space. We first prove, through explicit propositions, that each of the aforementioned structures is recovered as a special case under an appropriate choice of θ and α, and we supply non-trivial examples confirming that the new class is strictly broader than a b-metric space. Within this setting, a collection of fixed point theorems is derived for φ-contractive and Reich–Ćirić–Rus–type operators, together with their corollaries. Finally, two applications are presented: the principal theorem is first used to guarantee that a nonlinear algebraic equation possesses a unique solution, and it is then applied to establish the existence and uniqueness of a solution to a nonlinear Caputo fractional boundary value problem, thereby strengthening the connection of the theory with differential equations.

Open Access

Article

Article ID: 4670

Joint estimation of leverage thresholds in state-dependent capital structure dynamics: An (S,s) target zone framework for African firms

by Oluseun Paseda, Peter Ashade, Charles Manasseh, Felicia Olokoyo, J. Abiodun Oladimeji, Fatimah Abdulazeez, Idowu Bosede Fasola

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study develops a joint-estimation framework for analysing state-dependent capital structure dynamics in African listed firms within an (S,s) target-zone setting. Unlike conventional speed-of-adjustment models that assume continuous convergence towards an optimal leverage ratio, the proposed framework allows firms to tolerate leverage drift within an inaction band and undertake refinancing only when upper or lower leverage thresholds are breached. The model jointly estimates target leverage, refinancing thresholds, and inaction-band width using dynamic panel techniques, constrained nonlinear estimation, and a state-dependent refinancing hazard with bootstrap-robust inference. The African setting provides a particularly suitable environment for investigating such dynamics because firms face episodic access to capital markets, exchange-rate and inflation shocks, shallow debt markets, and heterogeneous institutional environments that elevate refinancing frictions and encourage discontinuous adjustment behaviour. Empirical analysis is conducted using an unbalanced panel of listed non-financial firms from fifteen African stock exchanges over 1999–2024. The results reveal pronounced (S,s) behaviour characterised by wide periods of leverage inertia punctuated by discrete refinancing interventions. Inaction bands widen with firm-level volatility, financing deficits, and market-timing opportunities, but narrow significantly with liquidity, creditor-rights protection, governance quality, and market depth. Additional evidence indicates that upper-brink adjustments occur substantially more frequently than lower-brink adjustments, suggesting that deleveraging pressures dominate leverage-increasing episodes in African markets. Extensive Monte Carlo experiments, including misspecification tests against linear adjustment processes, demonstrate superior bias, RMSE (Root Mean Square Error), coverage, and classification performance relative to conventional alternatives while avoiding spurious threshold detection. The findings highlight the importance of institutional quality and financial-market development in shaping corporate refinancing behaviour and provide new evidence that leverage adjustment in African firms is inherently nonlinear, state dependent, and threshold driven.

Open Access

Article

Article ID: 4510

Nonlinear hybrid impedance control for hydrodynamically coupled dual-arm underwater manipulators

by Sandeep Yadav, Sunil Kumar, Manoj Goyal, Pravin Kumar

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Underwater robotic manipulators require precise force–position interaction control to perform reliable operations in uncertain and dynamically coupled underwater environments. However, the performance of conventional impedance controllers is significantly degraded by nonlinear hydrodynamic disturbances, structural flexibility, and dynamic coupling effects. This paper proposes a hydrodynamic coupling-aware hybrid impedance control framework for a dual-arm with two-link underwater robotic manipulator (DT–URM). The proposed strategy integrates a passive impedance formulation with a compensation gain and a proportional–integral–derivative (PID)-bas–ed auxiliary loop to enhance interaction stability, trajectory-tracking accuracy, and disturbance-rejection capability. A DT–URM model is developed in SolidWorks and validated in MATLAB/Simulink under nonlinear underwater operating conditions. To evaluate controller robustness under realistic dynamics, the framework is further extended to a flexible two-link underwater manipulator by incorporating hydrodynamic and buoyancy effects into the system model. In addition, interaction constraints are introduced to maintain safe distances from virtual boundaries during manipulation tasks. Comparative simulation results demonstrate that the proposed controller reduces trajectory tracking and force-tracking errors by 18.5%, respectively, compared with conventional impedance control. Furthermore, disturbance rejection capability and settling time are improved by 19%, respectively, while maintaining stable operation under varying environmental conditions. The results confirm that the proposed control framework improves dynamic interaction performance and robustness, making it suitable for underwater inspection, maintenance, and cooperative manipulation applications.

Open Access

Article

Article ID: 4619

Fiberwise Rossi extension domains for rank-one Matsuki orbits in complex Grassmannians

by Irfan Ullah, Khurram Shabbir

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Let G0 = SU(p, q) act on the complex Grassmannian Z = Grk(C p+q ), and let M be a Matsuki orbit, i.e., the intersection of a G0-orbit with its Matsuki-dual K-orbit. Such intersections are compact homogeneous real-analytic Cauchy–Riemann (CR) manifolds. This paper studies the holomorphic extension problem for rank-one Matsuki orbits with residual signature s ≥ 2. The fibers of the Matsuki fibration are isotropic CR spheres in projective spaces of signature (1, s) or (s, 1), whose one-sided Rossi envelope is the corresponding projective ball. We construct the resulting fiberwise Rossi extension domain Dℓ,m,1 in a naturally associated projective residual bundle Xℓ,m → Grℓ(E+) × Grm(E−), where the Matsuki orbit embeds as the boundary. A key clarification is that the evaluation image of Dℓ,m,1 in the ambient Grassmannian Z is generally not open, so extended holomorphic functions need not descend. Our main theorem establishes a precise fiberwise Rossi extension: every real-analytic CR function extends holomorphically to Dℓ,m,1, and this domain is maximal among those obtained by the fiberwise Rossi-envelope construction. The proof uses horizontal CR vector fields, joint holomorphicity, and fiberwise patching. We also relate the construction to rank-one cycle domains via the evaluation map and formulate the higher-rank problem.

Open Access

Article

Article ID: 4733

Control-Theoretic Framework for Autonomous UAV Navigation in GNSS-Denied Environments via Adaptive Multi-Sensor Fusion

by Jiafeng Li

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper develops a control-theoretic framework for autonomous unmanned aerial vehicle (UAV) navigation in environments where Global Navigation Satellite System (GNSS) signals are degraded or denied. The navigation problem is posed as a stochastic dynamical system: vehicle kinematics and inertial error states are modelled by a nonlinear Itô stochastic differential equation, and heterogeneous sensor observations enter through a family of nonlinear output maps. State estimation is formulated as a matrix Riccati differential equation in which measurement confidence is modulated by a bounded, continuously differentiable weighting law governed by its own first-order differential equation. We prove that, under uniform observability and boundedness of the weighting law, the estimation error is exponentially bounded in mean square, and we obtain an explicit convergence rate that degrades continuously rather than discontinuously as individual sensors lose reliability — a formal statement of graceful degradation. Learning-based feature extraction is incorporated as a bounded exogenous perturbation, and the estimator is shown to be input-to-state stable with respect to it. Trajectory generation is cast as a constrained optimal control problem solved in receding-horizon form, and the estimator–controller interconnection is established as stable by a small-gain argument. A schedulability analysis bounds the sensing-to-actuation latency, making the real-time claim a proved property of the task set. Experiments across five scenarios yield 0.38 m RMSE indoors and 0.72 m in urban canyons, improving on visual-inertial odometry by 41.5% and conventional extended Kalman filtering by 66.1% in indoor flight.

Open Access

Review

Article ID: 4514

Frontier advances in solving fractional differential equations via physics-informed neural networks: A comprehensive review

by Yi Tian, Xiuqing Peng, Ji-huan He

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Fractional differential equations (FDEs) serve as indispensable mathematical instruments for modeling complex dynamical systems characterized by inherent memory retention, hereditary evolution, and anomalous diffusion—behaviors that elude accurate description by conventional integer-order differential models. Traditional numerical algorithms for FDEs are plagued by substantial mesh discretization overhead, prohibitive computational costs, and poor compatibility with various inverse parameter identification tasks. As a groundbreaking computational framework, physics-informed neural networks (PINNs) synergistically combine data-driven fitting capabilities with rigorous physical governing laws, offering transformative solutions to general differential equations. Cutting-edge advancements in fractional PINNs (fPINNs) are systematically synthesized in this comprehensive review. Physical motivations for adopting FDEs are elaborated, the fundamental architecture of vanilla PINNs is delineated, and core obstacles originating from the non-local integral characteristics of fractional derivative operators are dissected to reveal critical restrictions on direct application of standard PINNs to FDEs. All mainstream fPINN approaches are taxonomically grouped into three distinct technical branches, with focused mathematical derivations and in-depth discussions dedicated to fractional-order inverse identification. A homotopy-augmented simultaneous training scheme is formulated to mitigate unstable joint optimization of fractional orders and neural network weights, enabling a smooth transition of optimization targets from simplistic integer-order dynamics toward complex fractional non-local responses. Relative advantages and disadvantages of simultaneous and two-stage sequential identification workflows are thoroughly contrasted, and clear, actionable criteria are delivered to facilitate selection of appropriate identification pipelines for diverse fractional modeling scenarios. Systematic theoretical references and operable technical guidance for fractional modeling and scientific machine learning are provided throughout the review

Open Access

Review

Article ID: 4689

Survey of research over the past fifty years in stochastic mathematics (S.M.) exploring interdisciplinary frontiers

by Mu-Fa Chen

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This article provides an overview of the explorations over the past fifty years or more, as indicated in the title. The main new areas explored are the first three terms below. The fourth item is a related exploration result. The interaction of stochastic mathematics (S.M.) and statistical physics opens up new frontiers. Interaction of S.M. with other branches of mathematics: a nearly new theory for various stability and its speed estimation. Interaction of S.M. and economy: the first precise theory of economic optimization. Optimization method in deep mathematical research. It is hoped that these experiences would be helpful to future generations.

Open Access

Article (This article belongs to the Special Issue "Generalized Fuzzy Structures in Decision Analysis and in Differential Equations")

Article ID: 4690

Temporal-aware decision modeling for future energy planning based on complex q-rung orthopair fuzzy Aczel–Alsina CRITIC-WASPAS framework

by Khurram Ali, Muhammad Shoaib Arif, Tahir Mahmood, Hafiz Muhammad Waqas, Kamaleldin Abodayeah

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

The increasing global demand for energy, environmental concerns, and rapid technological advancements have made future energy planning a challenging multi-criteria decision-making (MCDM) problem. The selection of appropriate energy strategies requires simultaneously considering economic, environmental, technological, and policy-related factors under uncertain and evolving conditions. Conventional MCDM approaches are often inadequate for handling complex uncertainty and temporal variations in decision information. To address these challenges, this study proposes a Temporal-Aware Complex q-Rung Orthopair Fuzzy Aczel–Alsina CRITIC-WASPAS (Cq-ROF-AA-CRITIC-WASPAS) framework for future energy planning. The proposed framework employs Cq-ROFSs to represent uncertain decision information, while the proposed Aczel–Alsina aggregation operators effectively fuse temporal decision matrices by assigning greater importance to recent evaluations. The Criteria Importance Through Intercriteria Correlation (CRITIC) method objectively determines attribute weights by considering contrast intensity and inter-criteria correlation, whereas the Weighted Aggregated Sum Product Assessment (WASPAS) method ranks the energy alternatives through a hybrid additive-multiplicative evaluation strategy. A hypothetical future energy planning case study is presented to demonstrate the applicability of the proposed mathematical model using evaluation criteria related to economic viability, environmental sustainability, technological preparedness, energy security, and policy flexibility. Furthermore, comparative and sensitivity analyses are conducted to evaluate the effectiveness and stability of the proposed framework. The obtained results demonstrate that the proposed framework provides a flexible and systematic approach for handling complex uncertainty and temporal decision information, offering reliable decision support for future energy planning and other uncertain MCDM applications.