Finite Time Research Articles

Numerical dynamics and optimal control for multi-strain age-structured epidemic model.

In this paper, a novel age-structured epidemiological model that simultaneously considers multiple viral strains is proposed. We develop a numerical framework for the study of the dynamics and optimal control by a linearly implicit Euler method, in which the biological meaning is unconditionally preserved. The first order convergence of numerical solutions in a finite time is derived from a uniform numerical boundedness. Moreover, the numerical dynamics are determined by a numerical basic reproduction number , which reflects the asymptotic stability of the equilibrium points. The abstract framework offers an effective and unified approach to study the long-time behaviour of multi-strain epidemic models that cover a wide variety of well-known models, which also provides a numerical optimal control strategy of the multi-strain age-structured SIR model. Finally, some numerical simulations illustrate the verification and the efficiency of our results.

Finite time mixing and enhanced dissipation for 2D Navier-Stokes equations by Ornstein–Uhlenbeck flow

Finite time mixing and enhanced dissipation for 2D Navier-Stokes equations by Ornstein–Uhlenbeck flow

Discrete adaptive sliding mode control for uncertain fractional-order Hammerstein system

This study explores controlling discrete fractional-order Hammerstein systems through two different approaches. The first method involves a detailed exploration of discrete sliding mode control (SMC), while the second method employs adaptive sliding mode techniques to address challenges posed by unknown time-varying system parameters. The aim is to enhance control strategies for these systems by ensuring stability and robustness amid evolving parameters. Stability analysis using Lyapunov Theory is carried out to establish the stability of the proposed control schemes. Comprehensive reachability analysis is then performed to show that the system reaches a quasi-sliding mode in finite time. The effectiveness of the proposed methods is thoroughly evaluated using simulation cases, covering three distinct scenarios. Finally, a discussion section analyzes the simulation outcomes and assesses the efficiency methodologies in handling complex system dynamics. Through comparative analysis and discussion, this study contributes to advancing control strategies tailored for systems characterized by fractional-order dynamics and varying parameters.

Analysis of Refractive Index Sensing Properties of a Hybrid Hollow Cylindrical Tetramer Array

In recent years, metal surface plasmon resonance sensors and dielectric guided-mode resonance sensors have attracted the attention of researchers. Metal sensors are sensitive to environmental disturbances but have high optical losses, while dielectric sensors have low losses but limited sensitivity. To overcome these limitations, hybrid resonance sensors that combine the advantages of metal and dielectric were proposed to achieve a high sensitivity and a high Q factor at the same time. In this paper, a hybrid hollow cylindrical tetramer array was designed, and the effects of the hole radius, external radius, height, period, incidence angle, and polarization angle of the hollow cylindrical tetramer array on the refractive index sensing properties were quantitatively analyzed using the finite difference time domain method. It is found that the position of the resonance peaks can be freely tuned in the visible and near-infrared regions, and a sensitivity of up to 542.8 nm/RIU can be achieved, with a Q factor of 1495.1 and a figure of merit of 1103.3 RIU−1. The hybrid metal–dielectric nanostructured array provides a possibility for the realization of high-performance sensing devices.

Localized surface plasmon resonance characteristics of gold dimeric nanobipyramids

Dimer is a new type of nanostructure that has attracted much attention due to its ability to achieve more significant field strength enhancement effects through the interaction of two nanoparticles than monomers. This paper applies the dimer structure to gold nanobipyramids particles with unique tip structures. The localized surface plasmon resonance properties of the gold dimeric nanobipyramids are simulated using the finite difference time domain method. The results show that the gold dimeric nanobipyramids achieves several times higher field enhancement in the nanogap compared to the gold monomer nanobipyramids. At the same time, under the same conditions, its electric field intensity is significantly higher than that of the gold dimeric nanospheres and gold dimeric nanorods. Furthermore, by varying the gap, sharpness, and surrounding medium, the changes in the LSPR effects of the gold dimeric nanobipyramids under different conditions were calculated. The research shows that by tuning the structure, morphology, and surrounding medium of the dimeric nanobipyramids, higher extinction intensity, more intensity field enhancement, and greater sensitivity can be achieved, offering significant application prospects in various optoelectronic device fields. This work has significant reference value for studying the localized surface plasmon resonance characteristics of dimeric metal nanoparticles and their related applications.

The Finite Time Blow-up of the Yang-Mills Flow

The Finite Time Blow-up of the Yang-Mills Flow

Stationary Mean Field Games on networks with sticky transition conditions

We study stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent's dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.

Event‐Triggered Integral Sliding Mode Control for Nonlinear Networked Cascade Control Systems With Uncertain Delay

ABSTRACTThis article investigates the problem of event‐triggered integral sliding mode control (ISMC) for a class of nonlinear networked cascade control systems (NCCSs). Due to the limited communication bandwidth between network nodes, uncertain delay can occur during transmission. This may result in a deterioration of the capabilities of NCCS and system instability in certain instances. Thus, an event‐triggered mechanism (ETM) and ISMC are introduced into the system, and a new model is constructed to increase the network bandwidth and reduce the effect of nonlinear perturbation. After that, the design of two controllers and ETM is accomplished through the use of the Lyapunov generalized method and linear matrix inequality (LMI) techniques. Finally, a simulation example of a power plant model is given to verify the effectiveness of the method. The results show that, for the first time, the event‐triggered ISMC is applied to NCCSs, which greatly saves the network bandwidth resources. The states of the system can reach the sliding mode surface (SMS) in finite time and remain stabilized there afterward, which ensures the stable operation of the system.

Online Learning and Pricing for Multiple Products With Reference Price Effects

ABSTRACTWe consider the dynamic pricing problem of a monopolist seller who sells a set of mutually substitutable products over a finite time horizon. Customer demand is sensitive to the price of each individual product and the reference price which is formed from a comparison among the prices of all products. To maximize the total expected profit, the seller needs to determine the selling price of each product and also select a reference product (to be displayed) that affects the consumer's reference price. However, the seller initially knows neither the demand function nor the optimal reference product, but can learn them from past observations on the fly. As such, the seller faces the classical trade‐off between exploration (learning the demand function and reference price) and exploitation (using what has been learned thus far to maximize revenue). We propose a rate‐optimal dynamic learning‐and‐pricing algorithm that integrates iterative least squares estimation and bandit control techniques in a seamless fashion. We show that the cumulative regret, that is, the expected revenue loss caused by not using the optimal policy over periods, is upper bounded by where hides any logarithmic factors. We also establish the regret lower bound (for any learning policies) to be . We then generalize our analysis to a more general demand model. Our algorithm performs consistently well numerically, outperforming an exploration‐exploitation benchmark. The use of price experimentation and estimation techniques could be readily applied in real retail management.

Islands on codim-2 branes in Gauss-Bonnet Gravity

Abstract We study the black hole information problem on codim-2 branes in Gauss-Bonnet gravity.
Thanks to the island surface ending on the brane, the Page curve of eternal black holes can
be recovered for all of the GB couplings within the causal constraints. Our results strongly
support the universality of the island mechanism. Similar to Einstein’s gravity, the HM surface
can exist only in a finite time in GB gravity. Remarkably, for various parameters, the maximum
times of HM surface are always larger than the Page times. As a result, the strange behavior of
HM surfaces does not affect the Page curves for general GB gravity. Finally, we establish the
correlation between the Page time, GB couplings, and brane tension, revealing that the Page
time increases with these factors.

Nonlinear Rayleigh–Taylor instability in an accelerated soap film

Rupture and fragmentation of thin liquid films are of great importance in manufacturing, agriculture, disease transmission, and other fields. An individual rupture event typically results from the nonlinear growth of small perturbations excited by the classical Rayleigh–Taylor mechanism and opposed by surface tension on the film interfaces. We study an initially uniform liquid film accelerated in the direction perpendicular to its interfaces, constructing a new reduced description of the flow in the limit of large surface tension consisting of three coupled (nonlinear) partial differential equations. This system is linearly unstable to disturbances of small wavenumber; we derive a simplified asymptotic description to gain analytical understanding of the nonlinear development until rupture. An initial phase of exponential film thinning leads to the accumulation of liquid in a central bulge. At a well-defined time, a localised pinch-off mechanism, driven by a mismatch in the interfacial curvature at the edge of the bulge, results in a self-similar thinning that breaks the film in finite time. Across a wide parameter range, the resultant droplet formed from the central bulge contains a significant fraction of the initial film volume.

Re-evaluation of the impact of wall retention on Tritium-self-sufficiency in fusion reactors

Abstract In recent years several publications have claimed that the retention of tritium (T) in the first wall armor material poses a limitation on when a fusion reactor becomes T-self-sufficient. While the prediction on wall retention therein are correct, the model applied to assess the T-self-sufficiency is based on wrong assumptions and thus, the importance of T-retention in the first wall is greatly overestimated. In this publication the predictions on T-retention in the first wall armor components are included in state of the art residence time models of the entire T-cycle. These calculations show that T-wall-retention only affects T-self-sufficiency when the T-breeding ratio (TBR) is so low that the power plant is only marginally T-self-sufficient. For TBR values that allow for a finite T-doubling time and maintain a T-reserve inventory, the T-production must be so high, that the T-wall-retention is insignificant. The paper will explain the error in the previous T-self-sufficiency estimates, how to correct them and what this means for the conclusions drawn by previous publications. Then an augmented residence time model will be presented, that can include diffusion/trapping model based, time dependent wall loss data. This augmented model results in orders of magnitude higher T-retention in the first wall compared to current assumptions in residence time models. Still, it is negligible compared to the T-content in the entire T-cycle. Thus it can be concluded that based on current material data T-retention in the first wall armor will not affect T-self-sufficiency.

Blow-up for a reaction–diffusion system with gradient terms under Robin boundary conditions

This paper is concerned with a reaction–diffusion system with gradient terms under Robin boundary conditions { ( h ( u ) ) t = ∇ ⋅ ( | ∇u | p − 2 ∇u ) + f ( u , v , | ∇u | 2 , t ) in Ω × ( 0 , t ∗ ) , ( r ( v ) ) t = ∇ ⋅ ( | ∇v | q − 2 ∇v ) + g ( u , v , | ∇v | 2 , t ) in Ω × ( 0 , t ∗ ) , ∂ u ∂ ν + γu = 0 , ∂ v ∂ ν + σv = 0 on ∂Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 in Ω ¯ , where p, q>2 and Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with smooth boundary. Under some suitable assumptions, a sufficient condition that ensures the positive solution blows up in a finite time is obtained. Meanwhile, the upper bound for the blow-up time and the upper estimate for the blow-up rate are derived. The key to the research is to use the maximum principles and differential inequality technique. In addition, as an application of the abstract results obtained in this paper, an example is given.

High sensitivity tapered fiber refractive index biosensor using hollow gold nanoparticles

A localized surface plasmon resonance (LSPR) sensor based on tapered optical fiber (TOF) using hollow gold nanoparticles (HAuNPs) for measuring the refractive index (RI) is presented. This optical fiber sensor is a good candidate for a label-free RI biosensor. In practical biosensors, bioreceptors are immobilized on nanoparticles (NPs) that only absorb specific biomolecules. The binding of these biomolecules to the receptors changes the local RI around the sensor and this change is detected by the transmittance spectrum of the fiber. Fast, accurate, easy and low-cost disease diagnosis are the advantages of optical fiber biosensors. In this paper, the structure theory is reviewed and the sensor is simulated by the finite difference time domain (FDTD) method and the finite element method (FEM) and the effect of the thickness and diameter of the HAuNPs and the waist diameter of the TOF is investigated. For the structure with HAuNPs thickness (2.5 nm), diameter (50 nm), and the fiber waist diameter of 10 μm, the wavelength sensitivity of 489.8 nm/RIU and full width at half maximum (FWHM) of 50 nm are obtained, which are better than those specifications in some other LSPR fiber sensors. In addition, the sensitivity of the sensor increases about 2–3 times compared to those of sensors with the same structure. Although there are many parameters in human blood that can change its RI, in practical work, the special bioreceptors on the sensor can deactivate other markers except the specific cancer markers, which changes the effective RI. Therefore, this optical fiber sensor is used for label-free detecting the RI of cancer cells and can be used as a biosensor for the detection of early stages of cancers in a non-invasive way, just using human blood samples.

Robust trajectory tracking of an uncertain quadrotor with almost global finite-time convergence

This paper addresses the three-dimensional trajectory tracking control problem of quadrotor unmanned aerial vehicles (UAVs) in the presence of modeling uncertainties and external disturbances. We propose two control strategies that steer the quadrotor onto its desired path within a finite time and maintain it thereafter for all future times. The proposed strategies use a cascaded control architecture, where the inner loop consists of the attitude dynamics and the outer loop consists of the translational dynamics of the quadrotor. The proposed strategies guarantee the global finite-time convergence of the tracking errors. Consequently, the quadrotor tracks its desired trajectory globally regardless of its initial engagement geometry with respect to the path. The first strategy draws motivation from the terminal sliding mode control. Meanwhile, in the second strategy, we modify the reaching law using an exponential function, which dynamically adapts to the variation of system states and guarantees a shorter reaching time with improved tracking performance. We demonstrate the robust performance of the proposed strategy in the presence of system lag and bounded external disturbances, which may emulate the effect of wind waves and other environmental imperfections neglected during the controller design.

PARALLEL PERFORMANCE OF THE EXPLICIT FDTD METHODS APPLIED ON 3D ACOUSTIC WAVE EQUATION

The 3D acoustic wave equation is a second-order linear hyperbolic partial differential equation. It is widely studied in acoustics, fluid dynamics, electromagnetism, and seismology within both science and engineering fields. Among the various numerical methods, the finite difference method is considered to be the simpler to understand and easy to implement. Based on the discretize schemes finite difference time domain method described by the explicit scheme in which spatial second order derivatives are evaluated at the previous time step. The solution of 3D acoustic wave equation may be required on the high-resolution mesh points consequently more computational resources are required. In this study parallel algorithm for the numerical solution of 3D acoustic wave equation is proposed and designed by using data parallel approach and message passing schemes. The algorithm is implemented on shared memory parallel system using MATLAB parallel computing mode. The parallel performance of the designed algorithm is analyzed on different mesh sizes and time steps. It is revealed that the proposed algorithm may reduce computational time up to 3 times as compared to sequential solution algorithm. The proposed parallel algorithm remains more efficient on workers while for the efficiency of the algorithm drops because of the high communication time. The results of the proposed research could be utilized in the large-scale simulations of 3D acoustic wave equations and may enable to simulate the acoustic wave pressure at more refined meshes on high performance computing systems.

Fractional-order Sliding Mode Control of Manipulator Combined with Disturbance and State Observer

A fractional-order sliding mode control (FSMC) method for a manipulator based on disturbance and state observers is proposed. First, a state estimator is designed that can estimate the velocity and acceleration, and only joint position feedback and the mathematical model of the manipulator are needed. The state estimator converges in finite time. Then, the disturbance observer is designed. By designing the nominal system model of the manipulator, a disturbance observation error is introduced into the closed-loop control so that the performance of the manipulator can track the nominal system. Finally, a sliding mode controller (SMC) based on the fractional differential operator theory is also designed. The value of the sliding mode variable in the derivation of the controller is composed of the fractional derivative of the trajectory tracking error of the manipulator, whereas the fractional differentiation operation uses integration in its realization, and the integration is a low-pass filter, thus, high-frequency noise is suppressed. In the experimental section, the method designed is compared with the conventional sliding mode, which further reveals the rapidity and control accuracy of FSMC.

Global practical tracking control via output feedback for more general nonlinear systems.

This paper focuses on the issue of global practical tracking control by output feedback for uncertain nonlinear systems with unknown control coefficients and unknown reference signal. Unlike other tracking works, the upper and lower bounds of the unknown control coefficients in the studied nonlinear system are not required to be known, while the nonlinearities are bounded by the unmeasured states multiplying an unknown constant, the polynomial-of-output and the polynomial-of-input. Inspired by related works, an adaptive tracking controller based on a new dynamic high gain has been successfully constructed by combining the universal control idea and the concept of dead-zone with backstepping technique, which effectively handles the impacts of multiple uncertainties. The designed adaptive controller ensures that the state of the closed-loop system is globally bounded, and the tracking error of the system converges to any arbitrarily small range of the origin after a finite time. Finally, two numerical examples are provided to illustrate the effectiveness of the theoretical results.

Robust finite-time stabilization of linear systems with limited state quantization

This paper investigates the robust asymptotic stabilization of a linear time-invariant (LTI) system by static feedback with a static state quantization. It is shown that the controllable LTI system can be stabilized to zero in a finite time by means of nonlinear feedback with a quantizer having a limited (finite) number of values (quantization seeds) even when all parameters of the controller and the quantizer are time-invariant. The control design is based on generalized homogeneity. A homogeneous spherical quantizer is introduced. The static homogeneous feedback is shown to be a local (or global) finite-time stabilizer for any controllable linear system (depending on the system matrix). The tuning rules for both the quantizer and the feedback law are obtained in the form of Linear Matrix Inequalities (LMIs). The closed-loop system is proven to be robust with respect to some bounded matched or vanishing mismatched perturbations. Theoretical results are supported by numerical simulations.

Lifted time stable inversion based feedforward control for linear non-minimum phase systems

The feedforward control strategy exhibits substantial capability and high-precision control for output tracking tasks. However, the feedforward control action obtained through solving the inverse problem is unstable for non-minimum phase systems. In this paper, a novel stable inversion method is presented, termed lifted time stable inversion. Compared to the existing method, the proposed method does not necessitate infinite window to accomplish the tracking tasks. A comprehensive analysis of the developed method is provided, focusing on analysis of finite time stability and input–output finite time stability, aspects that have garnered limited attention in the literature on feedforward control. Furthermore, the relationship with existing stable inversion method is illustrated by constructing a linear transformation of the initial conditions for both inversions. Simulation results substantiate the validity of the finite time bounds and demonstrate the superior tracking advantage of the proposed method relative to the existing method. The performance of the proposed method is further displayed experimentally on a piezoelectric ceramic positioning platform.