Digital signal processing (DSP) is an important technology in various research fields. However, mainstream implementations of DSP algorithms either lack flexibility and scalability or have performance saturation and bottleneck, which still have the potential to be optimized. In this work, we proposed DSP-OPU, an FPGA-based overlay processor for digital signal processing. Specifically, we designed an overlay architecture suitable for various DSP algorithms. A unique data path was proposed with multiple computation engines and a reconfigurable pipeline. Additionally, we realized software-hardware co-design for DSP-OPU. On the software side, we achieved excellent scalability with a customized instruction set and a user-friendly compiler. On the hardware side, the reconfigurable data path breaks the dependency between instructions and data, enhancing scheduling ability and transmission efficiency. In addition, we also decreased the performance saturation with increasing engine numbers. Compared to C6678 SoC, our DSP-OPU achieves up to 29 × speedup and 70 × higher energy efficiency for different DSP algorithms. Compared to FPGA-based implementations for a single DSP algorithm, we achieved up to 4.5 × speedup and 4.4 × better energy efficiency.

Through tilting objects, we construct complete cotorsion pairs for specific hereditary abelian categories, such as the category of modules that are finitely generated over a finite-dimensional hereditary algebra as well as the category of coherent sheaves over weighted projective lines. We prove that a complete cotorsion pair exists in the category of coherent sheaves over a weighted projective curve X if and only if X is a weighted projective line. We also characterize the canonical tilting cotorsion pair for any weighted projective line and obtain Hovey triples in the category of vector bundles over a weighted projective line.

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô’s lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value R^e₀ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when R^e₀ > 1, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.

Fractional-order financial systems hold significant importance in practical situations. In this work, a novel fractional-order financial system considering the non-constant elasticity of demand was presented, and the system’s complex dynamics were studied. The results demonstrated that the system can exhibit diverse chaotic dynamics and periodic oscillations, which are influenced by different frac- tional orders and system parameters. Then, to stabilize the proposed chaotic system with uncertainties and achieve predefined-time synchronization of master-slave systems, an effective sliding mode con- trol strategy utilizing the RBF neural network was put forward. In real financial markets, uncertainties and perturbations occur suddenly, and excessive control input can lead to resource inefficiency. There- fore, unlike other papers that rely on conservative estimations using upper bounds, this paper used RBF neural network approximation to design a more flexible and robust controller while reducing the control input. Simulation findings reveal that this approach requires less control input than traditional methods without the RBF neural network and converges more rapidly than finite-time, fixed-time, and other predefined-time sliding mode control strategies with the RBF neural network, which validates the feasibility of this approach. Finally, the proposed chaotic system and control method were successfully applied to secure encryption, demonstrating their practical value.

The boundedness of commutators of Calderón–Zygmund operators in grand variable exponent Morrey spaces is established. The operators and spaces are defined on quasi-metric measure spaces with doubling measure. The obtained results are applied to study regularity properties of solutions of the second-order partial differential equations with discontinuous coefficients in the frame of grand variable exponent Morrey spaces.

Delayed pregnancy of predators and the Beverton-Holt-like alternative food source are key factors in controlling population density. To control the population density of Monochamus alternatus, the vector of pine wilt disease, this paper proposes a control system integrating the Holling II functional response function, Beverton-Holt-like alternative food source, and pregnancy delay. The conditions for the existence of the Hopf bifurcation were analyzed and we derived the normal form of Hopf bifurcation of the system with pregnancy delay and the Beverton-Holt-like alternative food source by using the multiple time scale method. Considering its biological significance, we selected a set of appropriate parameters for numerical simulation. Moreover, we also obtained that Hopf bifurcation can be induced under the effect of pregnancy delay. Finally, we put forward several biological elucidations that are useful for the prevention and treatment of pine wilt disease.

In this paper, we proposed a version of the Floquet theory for delay differential equations. We demonstrated that very natural assumptions for control in technical applications can lead us to a one-dimensional fundamental system. This approach allowed researchers to work with classical methods used in the case of ordinary differential equations. On this basis, new original unexpected results on the exponential stability were proposed. For example, in the equation x^′(t)+a(t)x(t−τ(t)) = 0, t ∈ [0, ∞), we avoided the assumption on the smallness of the product $\sup _{t \in[0, \infty)} a(t) \sup _{t \in[0, \infty)} \tau(t)<3 / 2$ for asymptotic stability. We obtained that in the case of ω-periodic coefficient and delay, the fact that the period ω was situated in a corresponding interval can lead to exponential stability. We then applied our new tests of stability to the stabilization of a drone’s flight, where smallness of the noted above product could not be achieved from a technical point of view. For an equation with periodic coefficient and delay, we got a formula of the solution’s representation on the semiaxis.

In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system

where the domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 2) is bounded and smooth,T_max ∈ (0, ∞], and parameters a_i,b_i,c_i,m,l, α, β, χ, ξ > 0 with m,l > 1, i = 1, 2. In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if $m>\max \left\{\frac{\alpha(n+2)}{2}, 1\right\}$ and $l>\max \left\{\frac{\beta(n+2)}{2}, 1\right\}$, then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.

The weighted essentially non-oscillatory (WENO) scheme is widely used in fluid mechanics and other numerical simulation fields because of its high precision and low oscillation characteristics when dealing with hyperbolic conservation law equations containing discontinuities and high-gradient regions. However, the calculation of nonlinear weights in the WENO reconstruction process is complex and entails a high computational cost, especially when addressing two-dimensional and higher-dimensional problems, resulting in a limited overall computational efficiency. To improve computational efficiency, this study introduces a novel neural network-enhanced weighted essentially non-oscillatory method, abbreviated as WENO-NN. This method replaces the reconstruction process in the WENO scheme. Specifically, we used a subset of the data generated by the WENO method to train a neural network that approximates the functionality of WENO. This approach significantly improved the computational efficiency while preserving accuracy. Further, we evaluated the performance of the WENO-NN scheme on both one-dimensional, two-dimensional, and three-dimensional test cases, including scenarios involving the interaction of strong shocks and shock-density waves. The results demonstrated that the WENO-NN scheme exhibits good versatility across all benchmark tests and resolutions. Its accuracy is comparable to that of the classic WENO scheme, while its computational efficiency is improved by 3 times.

This paper proposes an adaptive incremental backstepping control method for stratospheric airship attitude control that combines time delay estimation and incremental backstepping control to enhance robustness under model uncertainties. By integrating incremental control and time delay estimation, a linear time-invariant system relating to the attitude angle tracking error is obtained, where the time-delay estimation error is treated as a disturbance to the system. Meanwhile, an adaptive technique is utilized to reduce the effects of noise and center-of-gravity variations on system robustness. In conclusion, the convergence property of all signals is meticulously examined by employing Lyapunov theory. The proposed scheme is subsequently validated for effectiveness through numerical simulations.

Denoising diffusion probabilistic models (DDPM) have had remarkable success in image generation. Inspired by this, recent medical image segmentation tasks have started to use diffusion-based methods. These methods leverage iterations and sampling to generate smoother and more representative implicit integration. However, current diffusion-based segmentation models mainly rely on traditional neural networks and seldom focus on effectively interacting semantic and noise information. Moreover, they usually use a single network architecture instead of a hybrid one combining CNN and Transformer. To address limitations, we propose a dual-path U-Net segmentation diffusion (DPUSegDiff) model. It comprises two U-shaped networks based on the edge augmented local encoder (EALE) and the mixed transformer global encoder (MTGE). EALE uses the Sobel operator for local feature extraction, and MTGE has a cross-attention mechanism to facilitate information interaction. To integrate information from both paths selectively and adaptively, we design a bilateral gated transformer module (BGTM) to combine deep semantic information effectively. Experiments on three segmentation tasks-skin lesions, polyps, and brain tumors-show that the proposed DPUSegDiff outperforms other state-of-the-art (SOTA) methods in segmentation performance and generalization ability. The code has been released on GitHub (https://github.com/Fanyyz/DPUSegDiff).

In this paper, we focus on the steady-state bifurcation problem of the nonlinear Burgers equation within a bounded domain, considering both homogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions with a mean value constraint. Unlike previous studies, we develop an enhanced turbulence model by incorporating nonlinear higher-order terms (such as u² and u³) and linear source terms λu into the one-dimensional Burgers equation. Our steady-state bifurcation analysis establishes for the first time how the coupled forward energy cascade and inverse energy transfer mechanisms collectively govern the dynamics of initial flow instability. By combining the spectral theorem for a linear compact operator with the normalized Lyapunov–Schmidt reduction method and the implicit function theorem, we derive the complete criterion for the critical bifurcation condition, the explicit form of the bifurcation solution, and its regularity.

With the proposal of the new power system, power supply from renewable energy sources and traditional power supply have emerged as the future development directions of the power grid, while the traditional pricing mechanisms are facing new challenges. Considering the different characteristics of renewable energy power supply and traditional power supply, such as being clean and sustainable, but unstable, for renewable energy power supply,and being stable and technologically mature, but causing significant pollution, for traditional power supply, a multi-price model with the cost of pollution treatment under the multi-energy electricity generation was established in this paper. A distributed algorithm with the non-dominated sorting genetic algorithm (NSGA-II) was also proposed. In the model, the power supply side includes traditional energy generation, renewable energy generation, and the energy storage device. The proposed algorithm was designed using Lagrangian duality theory, and the multi-price is obtained by solving the different lagrange multipliers. Finally, the numerical results show that the model is reasonable when comparing the obtained price and social welfare with that untreated-pollution model, as well as a single supply model. Also, the proposed algorithm always has better computational efficiency, when compared with PSO, HS, and GA algorithms. The proposed model and algorithm provide a new idea and method for the optimal scheduling of a smart grid.

This study centered on employing the physics-informed neural networks (PINNs) approach for resolving the time-dependent Fokker-Planck-Kolmogorov (FPK) equation for the first time, culminating in the derivation of the transient probability density function. First, we derived the FPK equation for a dynamical system driven by fractional Gaussian noise (FGN). Second, a deep learning method based on PINNs was introduced for resolving the corresponding time-dependent FPK equation. Finally, two examples under two different excitation conditions were discussed to determine the effectiveness and feasibility of the PINNs algorithm. The results show that the PINNs algorithm can get the transient solution of the system under additive and multiplicative FGN. Concurrently, the Monte Carlo approach was utilized to evaluate the precision and computational efficiency of the PINNs algorithm. We found that the different comparison results are in good consistency, which proves that the PINNs algorithm is not only efficient, but also effective and interpretable.

This paper focused on a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with standard incidence and transfer from infected individuals to susceptible individuals. We assumed that the incidence rate satisfied the log-normal Ornstein-Uhlenbeck process. First, by using stochastic Lyapunov analysis method, the sufficient condition for the existence of stationary distribution was obtained. After that, we established the sufficient criteria for the extinction of the infectious disease. It was worth noting that the dynamical behavior of the considered model was governed by a threshold. In addition, we derived the exact expression of probability density function near the positive equilibrium point of the corresponding deterministic system. Finally, some numerical simulations were carried out to confirm theoretical results.

In this paper, we investigated the generalized lower triangular matrix algebra, and gave the sufficient and necessary condition for the generalized lower triangular matrix algebra to be quasi-tilted.

In this paper, we considered an eco-epidemiological model including toxicity, treatment, time-varying incubation, and Holling II functional response. First, for the model without time delay, the positivity and boundedness of solutions were investigated, and some conditions for local asymptotic stability of all possible equilibriums and condition for global stability of the positive equilibrium were also given. Then, the local stability around the positive equilibrium and conditions for the existence of Hopf bifurcation of such model with time delay were explored. Furthermore, by using the method of multiple scales, a control strategy based on time delay was obtained to suppress oscillation. Moreover, the global stability of the positive equilibrium of the model with time-varying delay was investigated. Finally, the theoretical findings were validated through numerical simulations. Those results demonstrated that time-varying delays can induce more complex dynamics in systems. By combining the method of multiple scale with periodic perturbations, the oscillatory behavior induced by Hopf bifurcation can be effectively suppressed, providing a novel approach for stability control in time-varying delay systems.

Gene regulatory networks (GRNs) play a crucial role in biological processes, with their dynamic behaviors heavily influenced by the spatial organization of genes. In particular, reaction- diffusion mechanisms govern the coupling between adjacent spatial locations in continuous time GRNs. However, traditional models often ignore the spatial coupling and reaction-diffusion properties of these networks, especially in discrete-time settings. In order to solve this problem, a new discrete- time gene regulatory network model is proposed in this paper, which explicitly considers the mutual coupling between adjacent spatial positions. To ensure the passivity of the proposed model, delay- dependent stability criteria are established by constructing appropriate Lyapunov-Krasovskii functions formulated in terms of linear matrix inequalities. To showcase the effectiveness and validity of this approach, a numerical example is presented in this paper. The results reveal that the model accurately captures the spatial coupling and reaction-diffusion nature of gene regulatory networks in discrete time settings.

In this study, we focused on developing a new viscous contact force model by incorporating a novel viscous fluid damping factor derived from the principles of incompressible Newtonian fluid dynamics. This new viscous fluid damping factor was formulated based on the Navier-Stokes equations, providing a robust framework for accurately modeling energy dissipation during impacts. Building on this foundation, a new coefficient of restitution (CoR) model was proposed by integrating the Hertz contact law with the newly developed viscous fluid damping factor. To validate the correctness of the proposed viscous fluid damping factor, a series of experimental data was collected to examine the relationship between the CoR and the impact velocity. The results confirmed the validity of the proposed CoR model and demonstrated that using fluid-based concepts to describe energy dissipation during impacts between solid bodies is feasible and effective. To highlight the advantages of the new viscous contact model, a comparative analysis was conducted against existing viscous contact models. This comparison demonstrated that the new model achieves the highest accuracy in calculating impact behavior, outperforming conventional models. The superior performance of the new model underscores its potential as a powerful tool for studying dynamic interactions in particle systems. The new viscous contact model was further applied to analyze impact behavior in two scenarios: A bouncing ball and a vertical granular chain. In both cases, the results obtained from the model were validated against experimental data, showcasing its ability to capture complex collision dynamics with remarkable precision. In conclusion, our findings establish the new viscous contact force model as an effective and reliable approach for analyzing impact behavior in particle systems. This investigation paves the way for novel approaches to studying energy dissipation during impacts in particle systems by utilizing fluid damping factors instead of the internal damping of particles. The proposed contact force model is applicable to a wide range of dynamic systems involving contact interactions. It is particularly effective for simulating contact events in both flexible and rigid multibody systems. Moreover, the model demonstrates strong performance in granular systems, where frequent collisions, adhesion, and nonlinear contact behaviors are prevalent.

Considering the distributed time delay and the stochastic change of environment, this research primarily studied the dynamical behavior of the gamma-distributed delay SIR model driven by Lévy noise and Markov chain. First, it is proved that there is a unique global positive solution for stochastic model. Second, by constructing appropriate stochastic Lyapunov functions that account for regime switching, we derive a sufficient condition, R_s > 1, indicating the existence of a stationary distribution. This suggests the disease will persist over a long period. Finally, numerical simulations confirm the theoretical findings.

Malaria continues to pose a considerable threat to global health. This study investigates the use of releasing Wolbachia-infected male mosquitoes as a method to mitigate the spread of malaria. We have formulated a reaction-diffusion model with nonlocal delays that includes the Wolbachia release strategy. The basic reproduction number R₀ is defined within our model framework, serving as a critical threshold parameter that dictates the dynamic behavior of the model. A thorough dynamic analysis of the model reveals that when R₀ < 1, a globally attractive infection-free steady state is established. In contrast, if R₀ > 1, the disease persists uniformly. Numerical simulations are conducted to validate the theoretical results and to further illustrate the effectiveness of the Wolbachia release strategy on transmission and control of malaria. These simulations underscore the potential of using Wolbachia- infected male mosquitoes to significantly reduce spread of malaria.

We provide generalizations of the classical Wigner’s theorem as well as Uhlhorn’s version of Wigner’s theorem by considering maps that send rank-1 projections to rank-n projections. Namely, we describe the general form of maps ϕ :P₁(H) → P_n(K) multiplying n times the transition probability and maps ϕ : P₁(H) → P_n(K) sending each complete orthogonal system of rank-1 projections to some complete orthogonal system of rank-n projections.

In the era of big data, interval-valued data is quite common in real life and can be used to describe the uncertainty of variables. In this paper, we introduced random effects panel interval-valued data models based on the center and range method and constructed a Bayesian method for the models, including estimation and prediction. Some simulation studies indicate that the proposed Bayesian method performs well. Finally, our proposed panel interval-valued data Bayesian models were applied in forecasting of the Air Quality Index, and the experimental evaluation of actual data sets shows the advantages and the performance of our proposed models.

In this paper, we explored the existence and uniqueness of solutions for a boundary value problem involving (p, q)-difference equations with integral conditions. By employing well-established fixed-point theorems, we established new and significant results in this area. To further illustrate the applicability of our findings, we presented three concrete examples that demonstrate the validity of the theoretical results.

The traffic assignment problem (TAP) is essential to efficient road network operation and significantly influences urban mobility and development. Traditional optimization algorithms typically rely on strict assumptions and iterative optimization methods, making them computationally intensive and inflexible. Deep learning methods, conversely, offer a promising alternative by effectively capturing heterogeneous and nonlinear traffic flow characteristics from diverse datasets. This study introduced a graph convolutional network (GCN)-based framework for the user equilibrium traffic assignment problem (UE-TAP). Specifically, the proposed GCN model learned the implicit relationships between origin-destination (OD) demand matrices and the resulting equilibrium traffic flows, providing efficient and reliable traffic flow estimations without iterative computations. Furthermore, to accommodate variations in network topology, an innovative deep learning approach based on network partitioning and subgraph training was introduced, significantly enhancing the scalability and adaptability of the model. Numerical experiments conducted on the Sioux-Falls and Eastern Massachusetts networks demonstrated that the proposed model achieved robust and high-accuracy estimations across diverse scenarios. In fixed-topology scenarios with random variations in OD demands and link capacities, the proposed model achieved R² of approximately 0.90. Even in scenarios with random link failures coupled with varying OD demands and capacities, the model maintained R² of around 0.84. Overall, the proposed methodology represented a significant advancement in solving UE-TAP, particularly in dynamic environments with evolving road network structures.

Deep learning has recently gained prominence in fluid dynamics due to advances in computational power, algorithm development, and data availability. While most applications have focused on modeling and control, its potential for design and optimization remains relatively unexplored. In this study, a conditional generative adversarial network (CGAN) was developed for inverse design of airfoils with flow control devices. This CGAN receives the characteristics of the flow (Reynolds number and angle of attack) and the desired aerodynamic characteristics (drag and lift coefficients). Based on those inputs, the CGAN generates an airfoil that fulfills the defined specifications. With this objective, numerical simulations of 4-digit NACA airfoils with variable flap configurations and flow conditions were conducted, in order to obtain the necessary aerodynamic data for training and testing the CGAN. The results demonstrate that the proposed CGAN generates airfoil geometries with high accuracy and efficiency, showing minor deviations from real geometries and achieving aerodynamic performances that approach the desired ones for all the flow conditions considered. Additionally, the model is able to generalize to extreme cases not seen during training, which significantly broadens its application range. This approach offers a significant reduction in computational time compared to traditional iterative optimization methods, making it suitable for rapid design exploration and real-time applications.

Nonlinear tuned mass dampers (TMDs) are widely recognized for their superior damping performance, which stems from their broad damping bandwidth. In this study, we propose a single-pendulum TMD for application in vibration absorption within train compressors to protect critical components. However, the inherent nonlinear vibration characteristics of the TMD may undermine its damping effectiveness. To address this challenge, we develop a three-degree-of-freedom mechanical model of a train compressor integrated with a single-pendulum TMD. The equations of motion for the system are derived using the second kind of Lagrange formula. By employing the fourth-order Runge-Kutta numerical integration method and using the excitation frequency as the bifurcation parameter, we analyze the bifurcation behavior and the transition to chaos via the Poincaré cross-section method. Our results demonstrate that the system can transition to chaos through multiple routes, including multiplicative bifurcation, Hopf bifurcation, and residual-dimension bifurcation, depending on the excitation frequency. These findings provide critical insights for the dynamic design and chaotic motion prediction of single-pendulum TMD systems, offering practical guidance to enhance their reliability and performance in real-world applications.

This paper investigated a numerical method for a two-dimensional variable coefficient evolution equation (VCEE), utilizing the alternating direction implicit (ADI) method and an extrapolation formula. The time derivative was discretised by the backward Euler (BE) scheme on a uniform mesh and the finite difference method (FDM) was applied to spatial discretization. We proved an priori estimate and the error bound of the solution to the difference scheme using the energy analysis method, and verified the uniqueness, stability, and convergence of the proposed scheme. To further improve numerical accuracy, we introduced a Richardson extrapolation method, which enhances the global accuracy to fourth order. Finally, some numerical examples were provided to demonstrate the validity of the theoretical analysis.

Covering arrays on graphs can be used to generate test suites in component-based systems but they cannot identify and determine faulty interactions from the outcome of the test. To address this problem, the notion of detecting arrays on graphs (DAGs) was proposed in this paper. Then, the equivalence and the existence of DAGs were intensively studied. We established a general criterion for measuring the optimality of DAGs in terms of their size. Based on this optimality criterion, the equivalence between optimal DAGs and orthogonal arrays with prescribed properties was established. With this equivalence property, a great number of optimal detecting arrays on cycles were produced by constructing the equivalent combinatorial configurations. In particular, the existence of optimal detecting arrays on cycles with few vertices was almost completely determined.

This paper conducts a study on the alternating direction implicit (ADI) difference schemes for a class of three dimensional hyperbolic equations with constant coefficients. The central difference methods are employed in the temporal and spatial direction. The solvability, stability, and convergence of the proposed ADI schemes are proven. Moreover, the Richardson extrapolation method is established to enhance the accuracy of the algorithm. Numerical examples are presented for the errors and convergence orders of the established ADI schemes and extrapolation schemes. By comparing the results of numerical examples , it can be concluded that the proposed Richardson extrapolation method can effectively improve the accuracy of the numerical solutions and reduce the errors.

This paper investigates a class of dissipative boundary value problems arising from a third-order differential equation with distributional potentials and eigenparameter-dependent boundary conditions. Initially, we transform the boundary value problem into the corresponding operator problem. We then demonstrate that the operator is dissipative and examine certain eigenvalue properties of the operator. Furthermore, by applying Krein’s theorem, we establish the completeness theorems for both the boundary value problem and the corresponding operator.

Synthetic aperture radar (SAR) is an advanced microwave sensor widely used in ocean monitoring because of its resilience to light and weather conditions. However, SAR ship detection tends to have relatively low accuracy due to the prevalence of complex backgrounds and small targets in the detection process. To address these issues, we proposed ECF-YOLO, an improved ship detection algorithm based on YOLOv8. The algorithm enhanced the feature extraction ability of the model and reduced the number of parameters and computational cost by developing a novel C2f-EMSCP module, which replaced the original C2f module in the backbone network. Additionally, we proposed the CGFM module in the neck network, which was designed to improve the detection accuracy of small ship targets by selecting features after combining shallow and deep feature maps. Furthermore, the Inner-SIoU loss function was introduced to replace the CIoU, providing a more precise overlap calculation between the target and anchor boxes, thus further improving detection accuracy. The experimental results for the SAR ship detection dataset showed that compared to YOLOv8n, ECF-YOLO improved AP₇₅ by 2.8% and AP_50:95 by 0.9%. Compared to other mainstream algorithms like YOLOv9t, YOLOv10n , and YOLO11n, ECF-YOLO achieved improvements of 3.4%, 4.6% , and 4.9% for AP₇₅, and 3.4%,1.9%, 3.0% for AP_50:95, respectively, demonstrating its effectiveness for detecting small targets.

In this paper, we propose a finite-time flocking particle model with nonlinear velocity coupling and inter-driving forces. Initially, we demonstrate that under specific initial conditions, the system achieves flocking within a finite timeframe, with all particles’ velocities converging to the average of their initial velocities. Our results include some results in the literature. Furthermore, a special case of mutual driving is given. Finally, we validate the obtained results through numerical simulations to confirm their accuracy.