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• Propose a fully numerical iterative process to optimize the maintenance threshold for systems with non-monotonic degradation behavior subject to competing failures. • Consider lead time before maintenance implementation in the numerical framework. • Assess the impact of shock detection threshold on the expected cost. Shocks have attracted considerable interest in reliability and maintenance engineering because of their impact on vulnerable systems. Most industrial systems suffer from both internal degradation caused by fatigue and wear-out, and external shocks that often occur randomly due to harsh weather conditions, overloading, etc. Developing maintenance optimization models without taking these stochastic shocks into account is often ineffective. This paper develops a model to optimize the maintenance alarm threshold for a single-component continuously monitored system which is exposed to both fatal and non-fatal shocks in the presence of lead time for hard time maintenance. The shocks occur randomly according to a homogeneous Poisson process during the whole degradation process and have a stochastic impact on the degradation level, while the system resistance to shocks decreases as the system approaches failure. We propose a new numerical maintenance optimization model to find the solution without Monte-Carlo simulation and the model is compared to the Wiener process. A numerical example and a real-time experimental case study on roller bearings are used to demonstrate the effectiveness of the model. The results show that the model is capable of improving maintenance decision-making in terms of failure probability and risk perspective. [ABSTRACT FROM AUTHOR]

• A novel hybrid prognostic method for remaining useful life prediction is presented. • A new nonlinear health indicator is generated for the training bearings. • The condition monitoring interval of health state is determined by the 3σ criterion. • The degradation and RUL forecasting is achieved based on the nonlinear Wiener process with random effects method. This paper proposes a novel hybrid method aiming at the fault prognosis of bearings. A nonlinear health indicator (HI) is first constructed using Complete Ensemble Empirical Mode Decomposition with Adaptive Noise and Kernel Principal Component Analysis to reflect the health state of a bearing accurately and convincingly. Subsequently, multi-domain features are extracted from vibration signals and the Dual-Channel Transformer Network with the Convolutional Block Attention Module is applied for constructing HIs of the rest bearings. Moreover, the 3σ criterion is employed to establish the condition monitoring interval of health state and detect the First Prediction Time, with which degradation modeling and probabilistic Remaining Useful Life (RUL) prediction are conducted with the assistance of nonlinear Wiener process with random effects. The superior performance of the proposed hybrid prognostic method confirms that the method contributes to the accurate RUL prediction and uncertainty quantification. [ABSTRACT FROM AUTHOR]

Kernel-type nonparametric estimates of Poisson regression function are considered. We establish the conditions of uniform consistency and the limit theorems for continuous functionals connected with this function on C[a,1-a]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C[a,1-a]$$\end{document}, 0

In this paper, time domain signal and noise behavior of an active CRLH transmission line was investigated in the nonlinear regime at millimeter wave frequencies. In this structure, the transistor is considered as three coupled transmission lines and all three drain,gate and source lines are CRLH transmission lines. The left-handed element values for the active CRLH transmission line are calculated in such a way that the phase cancellation effect and dispersion of the CRLH transmission line are reduced. In order to model this active CRLH transmission line, with the assumption of wave propagation in the form of Quasi-TEM, the distributed modeling method is used to extract stochastic telegraphic equation (STE) in the non-linear regime. To model noise sources in the non-linear regime, the Wiener process and modulation functions have been used to model white noise in the nonlinear regime and in the time domain. By applying the finite integration method to stochastic telegraphic equation, the behavior of the signal and noise of the active structure CRLH has been compared with a line of active structure RH and the obtain results of the numerical method have been verified with the results of the Agilent ADS software. [ABSTRACT FROM AUTHOR]

The stochastic fractional (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation (SFHFSCE), which is driven in the Stratonovich sense by a multiplicative Wiener process, is considered here. The analytical solutions of the SFHFSCE are attained by utilizing the Jacobi elliptic function method. Various kinds of analytical fractional stochastic solutions, for instance, the elliptic functions, are obtained. Physicists can utilize these solutions to understand a variety of important physical phenomena because magnetic solitons have been categorized as one of the interesting groups of non-linear excitations representing spin dynamics in semi-classical continuum Heisenberg systems. To study the impact of the Wiener process on these solutions, the 3D and 2D surfaces of some achieved exact fractional stochastic solutions are plotted. [ABSTRACT FROM AUTHOR]

The accelerated aging process is incorporated into the design and development of intravascular catheters to assess their reliability assuring that this medical device is safe and effective for the intended use during their shelf life. The accelerated aging process is based on a common approach that assumes that the rate of aging increases by a factor of 2A(AT/l0), where AT is the temperature increment. However, with the life data obtained from this empirical method is difficult to do inferences about reliability. This paper presents an accelerated destructive degradation test using thermal stress to obtain degradation data directly relates reliability to critical performance characteristic, which is the tensile strength in the intravascular catheter tip considered as a critical concern in patients' safety. The degradation data model is given by a stochastic Wiener process with the drift parameter being represented as Arrhenius function. The parameters of the Wiener process and Arrhenius function are estimated using maximum likelihood; these parameters are used to estimate the first-passage time (time to failure) distribution when the intravascular catheters degradation path reaches a tensile strength critical value in each thermal stress level. Based on this, a complete product reliability assessment is performed and presented. [ABSTRACT FROM AUTHOR]

Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved. [ABSTRACT FROM AUTHOR]

The Gierer–Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed different activator–inhibitor systems with varying diffusion rates in his pioneering work, which could trigger the emergence of such cell structures. Mathematically, one describes these activator–inhibitor systems as coupled systems of reaction-diffusion equations with different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer–Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer–Meinhardt system with homogeneous Neumann boundary conditions on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer–Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the initial conditions, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question. [ABSTRACT FROM AUTHOR]

In reliability engineering, obtaining lifetime information for highly reliable products is a challenging problem. When a product quality characteristic whose degradation over time can be related to lifetime, then the degradation data can be used to estimate the first-passage (failure) time distribution and the Mean-Time-To-Failure (MTTF) for a given threshold level. To model the degradation data, the commonly used Lévy process modeling approach assumes that the degradation measurements are linearly related to time throughout the lifetime of the product. However, the degradation data may not be linearly related to time in practice. For this reason, trend-renewal-process-type models can be considered for degradation modeling in which a proper trend function is used to transform the degradation data so that the Lévy process approach can be applied. In this article, we study several parametric and semiparametric models and approaches to estimate the first-passage time distribution and MTTF for degradation data that may be not linearly related to time. A Monte Carlo simulation study is used to demonstrate the performance of the proposed methods. In addition, a model selection procedure is proposed to select among different models. Two numerical examples of lithium-ion battery degradation data are applied to illustrate the proposed methodologies. [ABSTRACT FROM AUTHOR]

A multi-objective optimization method for the accelerated degradation test based on Wiener process is proposed in this article in order to solve the problem that a single objective optimization cannot solve the difficulty or even conflicting test configurations caused by different optimization objective functions. An accelerated degradation model is established based on Wiener process, and the unknown parameters are solved by a two-step maximum likelihood estimation method. Considering both the accuracies of life estimation and the model parameter estimation, a multi-objective optimization model is established with the optimization goals of the minimum asymptotic variance of P-quantile of lifetime and the maximum determinant of Fisher information matrix. The Pareto solutions are set by the multi-objective genetic algorithm, and the test configurations for multi-objectives are obtained. Under the step stress accelerated degradation test and the constant stress accelerated degradation test, the effectiveness of the proposed method is verified by an optimization example of LEDs accelerated degradation test. [ABSTRACT FROM AUTHOR]

We present an exact distribution for the bounded range of a Wiener process by using the truncation method of the probability distributions. Rather than deriving some statistical properties of this distribution including reliability properties, moments, stress-strength parameter, order statistics, Bonferroni curve, Lorenz curve, and Gini's index, a dataset is analyzed to clarify the effectiveness of this distribution. [ABSTRACT FROM AUTHOR]

This paper discusses some methods based on the random search problem over the bound domain. The particle moves randomly with a bounded two-dimensional stochastic process in a bounded area. The physical phenomenon determines the boundaries of this area. We have a smart nanosensor tracks randomly for this particle. The necessary condition that maximizes the probability of the first meeting time between the nanosensor and the particle is derived. [ABSTRACT FROM AUTHOR]

We address here the issue of jointly estimating the angle parameters of a single sinusoid with Wiener carrier phase noise and observed in additive, white, Gaussian noise (AWGN). We develop the theoretical foundation for time-domain, phase-based, joint maximum likelihood (ML) estimation of the unknown carrier frequency and the initial carrier phase, with simultaneous maximum a posteriori probability (MAP) estimation of the time-varying carrier phase noise. The derivation is based on the amplitude and phase-form of the noisy received signal model together with the use of the best, linearized, additive observation phase noise model due to AWGN. Our newly derived estimators are closed-form expressions, consisting of both the phase and the magnitude of all the received signal samples. More importantly, they all have a low-complexity, sample-by-sample iterative processing structure, which can be implemented iteratively in real-time. As a basis for comparison, the Cramer-Rao lower bound (CRLB) for the ML estimators and the Bayesian CRLB (BCRLB) for the MAP estimator are derived in the presence of carrier phase noise, and the results simply depend on the signal-to-noise ratio (SNR), the observation length and the phase noise variance. It is theoretically shown that the estimates obtained are unbiased, and the mean-square error (MSE) of the estimators attain the CRLB/BCRLB at high SNR. The MSE performance as a function of the SNR, the observation length and the phase noise variance is verified using Monte Carlo simulation, which shows a remarkable improvement in estimation accuracy in large phase noise. [ABSTRACT FROM AUTHOR]

The primary objective of this work is to investigate the existence of mild solutions for some neutral stochastic integrodifferential systems in Hilbert spaces. These systems have a state-dependent delay and noninstantaneous impulses. In order to achieve this objective, the neutral stochastic integrodifferential system that has been proposed with state-dependent delay and noninstantaneous impulses is converted into an equivalent fixed point problem by means of an appropriate integral operator.Together with the theory of stochastic analysis, we develop several existence results that are founded on the theory of resolvent operators in the sense of Grimmer, Hausdorff's measures of noncompactness, and fixed point theorems.In the last part of this paper, specific examples are used to show how the general conclusions reached in this paper are made. [ABSTRACT FROM AUTHOR]

Remaining useful life (RUL) prediction of components is a crucial aspect of mechanical systems prognostic and health management. However, during the online RUL prediction of components, the existing method cannot use the prediction results for feedback and optimize the prediction model. To address this issue, this paper proposes a framework for RUL prediction with adaptive dynamic feedback. First, a feedback multi-feature fusion (FMFF) method is designed. It introduces a series of non-exceeding threshold positive constants, that is fraction thresholds (FT), to perform division of the degradation state dimension. Besides, the first hitting time (FHT) from the current moment to FT is the fraction remaining useful life (FRUL). It is used to construct and update the fusion factors. Then, the feedback multi-model selection (FMMS) method is proposed for adaptive feedback and to select the degradation model based on the FRUL prediction results in different stages. The probability density function of RUL based on the FHT is calculated based on the FMMS method to realize the RUL prediction of the component. Finally, the effectiveness and superiority of the proposed framework are validated based on XJTU-SY dataset and run-to-failure test. [ABSTRACT FROM AUTHOR]

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties. [ABSTRACT FROM AUTHOR]

In this paper we examine the convolution of a compound distribution with another given distribution. For this, new general upper and lower bounds on the tail probabilities of the convolution of a compound geometric distribution are derived. Also, such bounds are obtained under the truncated Cramér-Lundberg condition for the heavy-tailed case. Applications to the ruin probabilities of compound Poisson claims processes perturbed by diffusion as well as some new upper bounds are given. Also, several numerical examples to illustrate the effectiveness of the bounds are given. [ABSTRACT FROM AUTHOR]

• The EMD-based drift increment extraction method is proposed. • The drift increment is predicted based on the LSTM network. • A difference approximation method for the drift function derivative is proposed. • The drift-increment-based representation of RUL's PDF is derived. • Advantages in the improvement of the prediction accuracy are justified. In this paper, a novel remaining useful lifetime (RUL) prediction method that fuses stochastic degradation modeling and machine learning is proposed to improve the fitness of the model and quantify the uncertainty of the prediction results. First, a stochastic degradation model based on the Wiener process is built, and the drift increment is extracted using empirical mode decomposition (EMD). Second, a long short-term memory (LSTM) network is trained to learn the equipment degradation rule and predict the drift increment. The diffusion coefficient of the degradation model is then estimated according to the maximum likelihood principle. The final step is to derive the analytical expression for the probability distribution of remaining useful lifetime (RUL) based on the concept of first hitting time and the difference principle. The lithium battery degradation test confirmed the efficacy of the proposed method, achieving a life cycle average prediction accuracy of up to 97.45%. [ABSTRACT FROM AUTHOR]

This paper is focused on lifetime analysis based on accelerated degradation testing (ADT), considering the influence of stress and measurement errors. The traditional approach of ADT does not consider the random effects due to the influence of stress and measurement errors, and this can lead to inaccurate lifetime predictions. For this reason, we propose an ADT model that accounts for random effects. The model is based on Wiener process, which quantitatively expresses the relationship between the parameters of the ADT model and stress, as well as takes the measurement errors into consideration. Then, the proposed model is extrapolated to normal stress for lifetime analysis and a closed-form expression is deduced. Finally, ADT of the electrical cables is considered as case study to validate the effectiveness of the proposed algorithm. • A novel ADT model based on the Wiener process with random effects is proposed. • Measurement uncertainties are incorporated into the ADT model to account for realistic testing conditions. • The influence of stress is quantified and integrated into the model to enhance its predictive capability. • Lifetime analysis is conducted under normal stress levels, leveraging the proposed model to assess the longevity of the system accurately. [ABSTRACT FROM AUTHOR]

The paper considers the nonparametric Poisson regression problem with a regular equidistant design on the unit interval. The nonparametric estimation of the Poisson regression function is studied. The uniform consistency conditions are established and the limit theorems are proved for continuous functionals on C[a,1-a]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C[a,1-a]$$\end{document}, 0

• A novel W-DATA method for mechanical degradation performance description and reliability estimation. • ApEn and P-EMD are introduced for degradation feature extraction and quantification. • An ApEn -based Wiener degradation model with a pseudo failure threshold is employed to estimate the reliability. • W-DATA can be effectively used for lubricating oil pump reducer reliability estimation. The Wiener-process-based Degradation Approach with Two-phase Algorithm (W-DATA) is introduced as a multi-dimensional framework explicitly designed for modeling mechanical system reliability. This approach leverages an enhanced Empirical Mode Decomposition, known as P-EMD, for degradation feature extraction, and utilizes Approximate Entropy (ApEn) for quantifying feature complexities. W-DATA seamlessly integrates these components into a time-resolved degradation model. To enhance its robustness, the proposed reliability model synergizes with a Box-Cox transformed Wiener model, allowing for improved adaptability to non-Gaussian data landscapes. The paper outlines a strategy for establishing a reliable threshold for this data-driven degradation model based on non-Gaussian data, a critical aspect for reliability modeling using the Wiener process. This strategy addresses the challenge of Wiener process verification through normality and independent testing of data increments. The integrative methodology is recommended for application in machinery reliability analytics and has undergone rigorous validation through a 770 h real-world test on a marine lubricating oil pump reducer. The results affirm its efficacy and feasibility in accurately estimating mechanical system reliability. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H 1 stability, higher moment H 1 stability, L 2 stability, and higher moment L 2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Due to the changeable internal mechanisms or external working conditions, the degradation trend of products usually presents multiple phase characteristics. However, most existing multi-phase degradation methods treat each phase as linear and rely on artificial designation to directly specify the forms of drift models, which may result in an inaccurate description of degradation characteristics for complex equipment. To this end, we formulate a general nonlinear multi-phase degradation model with three-source variability based on the Wiener process. Meanwhile, a stage division method is developed to automatically determine the degradation phase number, change-point locations, and the forms of drift models. Then, we obtain the expressions of remaining useful life (RUL) by considering the uncertainty of change-point degradation observations. Especially, we derive the approximate analytical solution of RUL based on the linear model. Furthermore, to fully consider the unit-to-unit heterogeneity and utilize the degradation observations of the in-service unit and prior information simultaneously, we propose a parameter estimation method based on variational Bayesian approach, which adaptively updates all parameters as random variables. Finally, two numerical examples and three practical examples are provided to verify the effectiveness of the proposed method. • We first consider a general nonlinear multi-phase model with three-source variability. • Our stage division method has higher model fitting accuracy than existing methods. • Our parameter estimation method fully considers unit variability and it is adaptive. • Identifying linear or nonlinear models ensures both prediction efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Due to the discrepancy of the components themselves, operating conditions and loads, the bearings show different degradation processes, and a single fixed model cannot describe different degradation processes accurately. In order to solve the adaptive prediction model for component degradation, a hybrid prediction technique is proposed in this paper, firstly, the Theil-sen estimator (TSE) is proposed as the health indicator of bearing operation, and it has been validated that the TSE has good trendability, monotonicity, and robustness. Based on the TSE, in order to identify the time to start prediction (TSP) accurately, the idea of deviation degree is proposed to reflect the difference between local and global degradation trend. Then, the RUL prediction is performed by matching the Wiener model suitable for the prediction task through the model adaptive algorithm. Finally, the effectiveness of the proposed method is verified by XJTU-SY bearing dataset and bearing accelerated degradation experiments. [ABSTRACT FROM AUTHOR]

This paper presents two Bayesian hierarchical models—one utilizing the life-time data and other using the structural health monitoring (SHM) data, for degradation modeling and reliability assessment of rolling element bearings. The main advantage of the proposed life-time data based model is that, it accounts for the variability in failure times caused due to the difference in material properties, initial degradation, operating and environmental conditions by introducing Bayesian hierarchy in the model parameters. On the other hand, SHM data (such as vibration and strain) based model focuses on stochastic nature of bearing degradation, and models it using a two-phase Wiener process. In this model, the point of phase-transition is the time when the damage initiates. The detection of such a point is undertaken using Bayesian change point algorithms. For both the models, the model parameters and reliability are updated as more data becomes available. In this manner, the prior domain knowledge and life-time data or SHM data collected from the field can effectively be integrated to get updated reliability. Two case studies for rolling element bearings are presented to demonstrate the applicability to life-time as well as SHM data. [ABSTRACT FROM AUTHOR]

• A framework for degradation modeling under time-varying field operating conditions is proposed. • Offline information and online measurements are incorporated for accurate estimation. • A Bayesian filtering algorithm is developed for online latent degradation state estimation. In practice, degradation modeling and prognostics for a product working in field is of importance for condition-based maintenance, meanwhile a challenging work due to the uncertainty in degradation and the time-varying operating conditions. This study investigates a degradation modeling framework for field working product and its applications by incorporating multiple data sources to associate the operating condition and reduce the uncertainty. The Wiener process is adopted to model the degradation process with nonlinearity, unit-to-unit variation, and condition covariates, which is delineated by state space modeling. Meanwhile, the acceleration factor, which establishes the relationship between operating condition and degradation rate, is integrated to adapt the drift parameter. A recursive filtering algorithm based on Bayesian theorem is employed for online updating of drift. Finally, a simulation study and an application on degradation modeling of turbofan engine are given to demonstrate the feasibility and validity of the proposed model. [ABSTRACT FROM AUTHOR]

The first-passage probabilities in stochastic processes are important for the evaluation of structural reliability. In this work, Wiener process's first-passage probability was analysed using different methods. First, the Poisson process method was used to analyse the first-passage probability of the Wiener process, and it was found this method cannot calculate the first-passage probability of the Wiener process. Then, the symmetry and Markov property of the Wiener process were used to derive equations for calculating the first-passage probability of a Wiener process. The results indicate analytical expressions for the first-passage time probability distribution function of the Wiener process can be derived either through the symmetry or Markov property of the process. Neither of the analytical expressions requires the use of assumptions in their derivations. The analytical method based on the symmetry of the Wiener process yields fully exact solution, whereas the analytical method based on the Markov property of the Wiener process produces highly accurate solution. Finally, the Wiener process is used to represent the time-varying amount of structural degradation at any point in time, leading to a non-stationary stochastic model for the degradation process, which will provide a basis for an estimation of the durability and reliability of concrete structures. [ABSTRACT FROM AUTHOR]

In this work, we deal with the existence, uniqueness and controllability outcomes for stochastic partial integro-differential equations with nonlocal conditions. Our analysis uses Kuratowski's measure of noncompactness, generalized Darbo's fixed point Theorem, Mönch fixed point Theorem and Grimmer's resolvent operator theory. Some appropriate conditions for existence, uniqueness and controllability are obtained. Some examples are provided for illustration. [ABSTRACT FROM AUTHOR]

This paper proposes a systematic and general approach to model correlated stochastic processes in power systems by means of stochastic differential-algebraic equations. The paper discusses the theoretical background of stochastic differential-algebraic equations and provides a variety of examples of correlated stochastic models for power system applications. With this aim, stochastic processes with Normal and Weibull distributions are considered. The case study utilizes the well-known two-area system to demonstrate that the presence of correlation between the stochastic processes can cause instability, despite the fact that the same system is stable if the same processes are uncorrelated. The case study also considers a 1479-bus dynamic model of the all-island Irish transmission system to show the scalability of the proposed technique, and to compare scenarios with different levels of correlation among stochastic processes. Results indicate that correlation has a non-negligible impact on short-term dynamics. A high level of correlation among the processes, in fact, can give raise to instability. [ABSTRACT FROM AUTHOR]

Degradation-model-based remaining useful life (RUL) estimation is essential for effective prognostic and health management; this method can provide information for enabling effective maintenance decisions pertaining to degrading systems to avoid or mitigate loss due to impending failures. According to existing studies that estimate degradation-model-based RUL, stochastic-process-based methods are widely advocated by many researchers because they can capture stochastic dynamics within the degradation processes of systems. However, most existing studies primarily utilize the first passage time (FPT) to define the lifetime/RUL. The definition is generally conservative; thus, the lifetime/RUL estimation may be underestimated. This is particularly true for nonmonotonic stochastic degradation processes. In some extreme cases, with strong fluctuations within the degradation processes, the estimated lifetime/RUL under the FPT can be significantly less than the actual lifetime/RUL. To address this limitation, this study investigates prognostic issues, based on the stochastic degradation process, from the last exit time (LET) perspective. In contrast to the FPT, the lifetime/RUL of the degrading system is defined as the LET of its degradation process, i.e., the instant at which the degradation process passes the failure threshold for the last time. Given the new definition, we consider the most widely used degradation process model (i.e., Wiener-process-based model) as an example to demonstrate how the lifetime/RUL is estimated. Two general methods of lifetime estimation for the Wiener-process-based model are given, and some examples with associated exact and closed-form solutions are also provided for better illustration. Finally, numerical examples and a practical case study are presented to substantiate the theoretical results and illustrate the superiority of the proposed method. The results imply that the proposed method exhibits the potential to prevent premature maintenance and resource wastage because the lifetime/RUL estimation from the LET perspective can help avoid conservative results from being obtained. [ABSTRACT FROM AUTHOR]

Residual current device (RCD), the critical equipment to protect the safety of electrical users, are known to experience performance degrade over time during their use lifetime. Therefore, in order to promptly replace the RCD before failure, it is very important to predict its lifetime. In order to predict the remaining life of the RCD, a constant‐stress accelerated degradation test (CSADT) and a combination prediction model are proposed. First, the CSADT is conducted with temperature as the acceleration stress and the residual operating current as degradation characteristic. Then, based on the prediction result of pseudo failure life, a combination model of Wiener process and GM (1, 1) model is given. Finally, the remaining life of the RCD under normal use environment is extrapolated. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In this article, we study the stochastic fractional optimal control problem for a system governed by a class of non-instantaneous impulsive stochastic fractional differential equation in the infinite-dimensional spaces. We utilize the fractional calculus, stochastic analysis theory and a suitable fixed point approach to present the solvability of the stochastic fractional system. Then, the existence of optimal state-control pairs of the Lagrange problem is derived without uniqueness of solutions of the stochastic system. Finally, the main results are validated with the aid of an example. [ABSTRACT FROM AUTHOR]

For the accelerated degradation test (ADT) of lithium-ion battery, the degradation path models are usually applied with the fixed parameters. To optimize the degradation path models, so many battery cells are required for the ADT models based on the degradation path. However, for some lithium-ion batteries with long cycle life and high reliability, the samples are limited due to experimental cost constraints. Meanwhile, owing to the differences in raw materials and manufacturing processes, it is easy to observe variations among the degradation path of lithium-ion battery cells. Fortunately, Wiener process model considering the individual difference can express the variations and be modeled with limited samples for lithium-ion battery. Motivated by this issue, an improved method with random effects Wiener process (REWP) is proposed in this article. First, to verify the degradation process following the Wiener process, the Shapiro–Wilk test is carried out on the basis of a significance level of 0.05. Second, we analyze the varieties of every lithium-ion battery cell. To select a suitable Wiener process model, we apply the coefficient of variation as a measure of random parameters selection for Wiener process. Meanwhile, the maximum likelihood estimation (MLE) is used to optimize parameters of the proposed model. Next, the three different REWP models and general Wiener process model are compared. Finally, to demonstrate the proposed method, the lithium-ion battery cells used for a spacecraft in low Earth orbit (LEO) are conducted for ADTs. The results suggest that the REWP model has a better performance for assessing reliability of lithium-ion battery with long cycle life. [ABSTRACT FROM AUTHOR]

We construct and study a test to detect possible change points in the regression parameters of a linear model when the model errors and covariates may exhibit heteroscedasticity. Being based on a new trimming scheme for the CUSUM process introduced in Horváth et al. (2020), this test is particularly well suited to detect changes that might occur near the endpoints of the sample. A complete asymptotic theory for the test is developed under the null hypothesis of no change in the regression parameter, and consistency of the test is also established in the presence of a parameter change. Monte Carlo simulations show that our test is comparable to existing methods when the errors are homoscedastic. In contrast, existing methods developed for homoscedastic data are demonstrated to be ill‐sized and poorly performing in the presence of heteroscedasticity, while the proposed test continues to perform well in heteroscedastic environments. These results are further demonstrated in a study of the linear connection between the price of crude oil and the U.S. dollar, and in detecting changes points in asset pricing models surrounding the COVID‐19 pandemic. [ABSTRACT FROM AUTHOR]

Many products degrade over time and their degradation processes could be affected by instantaneous shocks during field usage. Instantaneous shocks can cause incremental increases to the degradation signals through shock damages, and can also increase the degradation rates of products. In practice, some kinds of products can recover fully or partially from shock damages in a certain period of time. In this article, a degradation model for soft failure is proposed considering continuous degradation processes with recoverable shock damages for reliability assessment and lifetime prediction of products. The random component of the degradation processes is characterized by a Wiener process and the effect of instantaneous shocks on the degradation process is expressed by an exponential function with a residual effect for either partial or full recovery. The impact of shocks affecting the degradation rate is established to be proportional to the shock size. The resulting model includes some existing models as its special cases and can easily be extended to cases where the degradation process follows other random processes with independent increments. Numerical examples are presented to illustrate the applications of the proposed model. Sensitivity analysis and validation of the two-stage parameter estimation approach are conducted based on the simulation. [ABSTRACT FROM AUTHOR]

The degradation and damage of concrete material and structure during their service life are significantly characterized by randomness, so it is of great importance to establish a stochastic process model for the structural durability degradation. This study was based on Wiener process to identify the durability degradation of concrete structures. Firstly, two identification methods for Wiener process were introduced, including autocorrelation function method and probability statistics verification method. The former is a rough method for preliminary identification, and the latter is a strict method for accurate identification. According to the degradation processes and damage mechanisms of dynamic elastic modulus of concrete and section stiffness of reinforced concrete girder, and based on relevant data, autocorrelation function method and probability statistics verification method were used to identify whether the degradation processes of concrete dynamic elastic modulus and girder section stiffness conform to Wiener process. The results showed that the degradation process of concrete dynamic elastic modulus and the degradation process of section stiffness of reinforced concrete girder can be modeled with Wiener process. The conclusions contribute to the prediction of performance degradation and the maintenance for concrete structures in service. [ABSTRACT FROM AUTHOR]

Accelerated degradation test (ADT) is generally used to accelerate degradation processes in products to estimate their lifespan and to assess their reliability in a short period of time. How to perform the failure mechanism consistency test is crucial in the application of the ADT method. Existing failure mechanism consistency test methods assume that degradation rates among individual products are the same. However, these methods do not take degradation dispersions caused by manufacturing technologies into consideration. To address this issue, a failure mechanism consistency test method for ADT based on the activation energy invariant method and the likelihood ratio test is proposed. First, a degradation modeling method for ADT is introduced. Then, the logarithmic maximum likelihood function values of the degradation models are estimated based on the two‐step maximum likelihood estimation (MLE) method. Finally, the decision rule is proposed based on the likelihood ratio test. The method mentioned above is, then, used on the real degradation data of carbon‐film resistors and bullet O‐rings, and its effectiveness is verified. Furthermore, based on the failure mechanism change point in RTV5370 siloxane rubbers, the simulated degradation data are degenerated to compare the proposed method with the method not considering individual differences in different ADT programs and degradation dispersions. [ABSTRACT FROM AUTHOR]

• A new model for setting an optimal price for a product is suggested. • Price-sensitive demand influences both revenue from sales and operational costs. • A closed-form solution for the infinite-horizon variety of the problem is found. • A numerical method for dynamic pricing over a finite horizon is proposed. • Insights into the effect of operational factors on the optimal solution are gained. This paper deals with the problem of setting an optimal price for a product. The product is priced by maximizing the objective function, which accounts for both income from sales and operational costs. Since the price-sensitive demand is the common factor that determines the two components of the objective, it is used as an independent decision variable. We develop a closed-form solution for the discounted infinite-horizon variety of the problem. For a finite horizon, we suggest an approximate numerical procedure that sets the price dynamically as the horizon-length decreases, such as may occur, for example, towards the end of the sales period. When modeling the operational cost component, we consider a stochastic production-inventory problem and solve it using optimal control methods. In particular, we show that the optimal production policy is of a threshold type and we calculate the threshold value. [ABSTRACT FROM AUTHOR]

We prove that the set of laws of stochastic integrals H ·W, where W is a multidimensional Wiener process and H2 takes values in a compact convex subset D of the set of symmetric positive semidefinite matrices, is weakly dense in the set of laws of martingales M with d⟨M⟩/dt taking values in D. [ABSTRACT FROM AUTHOR]

Considering the large‐scale development of renewable energy resources globally, control optimization of coal‐fired power plants is becoming increasingly crucial. The dynamics of the coordinated control system (CCS) must be studied prior to designing a controller. These dynamics of the subcritical units have been expressed through various modeling methods. However, in previous studies, ordinary differential equations (ODEs) have been primarily employed, which cannot reflect the uncertainties in the system. Therefore, a model with a greater accuracy should be observed to comprise uncorrelated residuals. In this study, the uncertainties in the calorific value of fired coal and combustion process were analyzed first. A normal distribution of disturbance was assumed in this process. The dynamics of the CCS were then described with stochastic differential equations (SDEs). Furthermore, a parameter estimation procedure was designed. The residual evaluation is employed to improve the model evaluation. After simulations, the SDE‐based model‐3 in this study elucidates the dynamics of the subcritical boiler‐turbine system better than the ODE‐based model and other SDE‐based models. The measured values can be regarded as a possible result of this model, rendering it a potential platform to employ stochastic model predictive control in CCS. [ABSTRACT FROM AUTHOR]

This article concerns the optimal control problem for internal gravitational waves in a model with additive "white noise". This mathematical models based on the stochastic Sobolev equation, Dirichlet boundary conditions, and a Cauchy initial condition. The inhomogeneity describes random heterogeneities of the medium and fluctuations. By white noise we realize the Nelson-Gliklikh derivative of the Wiener process. The study was carried out within the framework of the theory of relatively bounded operators and the theory of Sobolev-type stochastic equations of higher order and the theory of (semi) groups of operators. We show the existence and uniqueness of a strong solutions, and obtain sufficient conditions for the existence of an optimal control of such solutions. The theorem about the existence and uniqueness of the optimal control is based on the works of J.-L. Lyons. [ABSTRACT FROM AUTHOR]

As one of the most important aspects in prognostics and system health management (PHM), spare parts play significant roles in enhancing reliability, shortening downtime, and reducing operating cost. However, the performance degradation of spare parts, though frequently encountered in industrial practice, has been rarely concerned during lifetime estimation and the according inventory optimization. Based on an amenable Wiener-process-based degradation model, this paper integrates the performance degradation of spare parts into lifetime estimation for stochastic multi-component degrading systems with congeneric key parts. Under two prevalent replacement policies, i.e., unit-by-unit replacement policy and batch-by-batch replacement policy, the lifetime distribution of the system considering the performance degradation of spare parts in storage, has been derived and the corresponding cost function for inventory management optimization has been constructed. Simulations and case study of an inertial navigation platform with three gyros are conducted to illustrate and validate the proposed method. The results show that the performance degradation of spare parts should be considered during inventory management, and the presented method can achieve an accurate lifetime estimation of the concerned system. • Lifetime PDF of multi-component systems with degrading spare parts is derived. • Influence of different replacement policies on the lifetime PDF is investigated. • The expected cost function for inventory management optimization is constructed. [ABSTRACT FROM AUTHOR]

• A novel deterioration-integrated failure model with varying environments. • A matrix-based approximation method for health measures calculation. • An adaptive reliability-based maintenance policy with a switchable control chart. • An efficient computational algorithm to find the optimal decision variables. • Considerable better maintenance performance than published results. The performance of mechanical systems is greatly influenced by varying operational environments such as load, temperature, and humidity, which can result in accelerated degradation and failure. To address this issue, this article considers an adaptive maintenance policy based on updated conditional reliability for mechanical systems under variable environments. A model of deterioration-integrated failure process is presented, which integrates degradation process, failure events, and environmental variables of the system. The degradation process is modeled by a random effect Wiener process, with its drift parameter following a general distribution to represent the environmental random effects. The failure events are modeled by a proportional hazards (PH) model with environment-varying covariate. The health of the system is evaluated using a matrix-based approximation method that provides estimates of the system's conditional reliability and mean residual life. Using the prognostic information, an adaptive maintenance model with a switchable control chart is then developed to monitor and maintain system based on the status of system reliability. The decision variables of the adaptive reliability-based maintenance are determined by a computational algorithm formulated in a semi-Markov decision process (SMDP) framework, with the objective of minimizing the long-run expected average cost per unit time. Vibrational data of bearings under varying operational environments are presented to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The integration of renewable energy sources into the power systems and the growth of electricity consumption leads to a considerable increase in the power fluctuations. In the first part of this study, the control of the rotor angle stability of single machine infinite bus system in the presence of Wiener type power fluctuations has been achieved by minimizing the corresponding stochastic sensitivity function. In the second part, the power fluctuations have been modeled by alpha-stable Levy processes and since stochastic sensitivity function is not available for alpha-stable Levy processes, then the control of the rotor angle stability has been numerically achieved by minimizing the corresponding rotor angle dispersion for the first time in the literature. [ABSTRACT FROM AUTHOR]

In 1956 Mark Kac proposed a process related to the telegrapher equation where the particle travels at constant speed (say the speed of light c) and randomly inverts its velocity. This process had important applications concerning the path-integral solution and the probabilistic interpretation of the 1 + 1 dimensions Dirac equation. The extension to 3 + 1 dimensions requires that the particle only moves at light-speed, which implies that velocity can be represented as a point on the surface of a sphere of radius c. The realizations of the process for the velocity only may connect these points, and, by strict analogy with the Kac model, it can be assumed that the velocity jumps from one value to another. In this paper we follow a new and different strategy assuming that the velocity performs continuous trajectories (velocity changes direction in a continuous way) which are the realization of a Wiener process on the surface. The processes which emerge transform one in the other by Lorentz boost. The associate Forward Kolmogorov Equation for the joint probability density of position and velocity, which is the (3 + 1) dimensional analogous of the telegrapher equation, is examined and a simplification is performed by means of variables separation. [ABSTRACT FROM AUTHOR]

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the L 2 -norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere ℳ in a Hilbert space L 2 . [ABSTRACT FROM AUTHOR]

A single component nonrepairable system suffering from both an internal stochastic degradation process and external random shocks is investigated in this paper. More specifically, the Wiener process with a positive drift coefficient is introduced to describe the gradual deterioration and the arrival number of external shocks is counted with a nonhomogeneous Poisson process (NHPP). Meanwhile, fault tolerant design is incorporated into the stochastically deterioration system so as to protect it from shock failures to some extent and is consummately addressed via a generalized m − δ shock model. From the actual engineering point of view, external shocks are typically classified into two distinct categories in this current research, that is, a minor shock (Type I shock) increasing the damage load on current degradation level and a traumatic shock (Type II shock) resulting in system catastrophic failure immediately. The closed‐form expression of system survival function is derived analytically and is viewed as the generalization of existing reliability function for systems subject to dependent and competing failure processes. Based on which, two time‐based maintenance (TBM) policies including an age replacement model and a block replacement model are scheduled, where the expected long‐run cost rate (ELRCR) in each model is, respectively, optimized to seek the optimal replacement interval. In the illustrative example part, a subsea blowout preventer (BOP) control system is arranged to validate the theoretical results numerically. To compare which policy is more profitable under different conditions, the relative gain on optimal maintenance cost rate of the two TBM policies is presented. [ABSTRACT FROM AUTHOR]

We study optimal control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L 2 regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrödinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U 2 and V 2 , adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

A degradation model is used to analyze degradation data and product reliability. Owing to model uncertainty, the assumed degradation model may be improper, which can create a difference between the evaluated and actual product reliabilities. Thus, model misspecification analysis is essential. Recent studies on misspecification analysis have mainly considered single‐phase degradation models; two‐phase degradation models have not been investigated intensively. In this study, an approach to model misspecification analysis was developed for a two‐phase gamma‐Wiener process, where the change point was assumed to be known. The gamma and Wiener processes in a two‐phase degradation model may be incorrectly assumed, so five misspecified degradation models were considered: the two‐phase Wiener process, two‐phase gamma process, two‐phase Wiener‐gamma process, one‐phase Wiener process, and one‐phase gamma process. A Monte Carlo simulation was performed to examine the effects of model misspecification on the relative bias and relative variability. A case study was performed to examine the effects of model misspecification on the evaluated mean time to failure and median lifetime of a product for different sample sizes, termination time, and change points. The results clearly demonstrated the effects of model misspecification on the evaluated reliability of a product. [ABSTRACT FROM AUTHOR]