System Of Stochastic Equations Research Articles

Sampled-Data H∞ Dynamic Output-Feedback Control for NCSs With Successive Packet Losses and Stochastic Sampling.

The problem of sampled-data H∞ dynamic output-feedback control for networked control systems with successive packet losses (SPLs) and stochastic sampling is investigated in this article. The aim of using sampled-data control techniques is to alleviate network congestion. SPLs that occur in the sensor-to-controller (S-C) and controller-to-actuator (C-A) channels are modeled using a packet loss model. Additionally, it is assumed that stochastic sampling follows a Bernoulli distribution. A model is established to capture the stochastic characteristics of both the SPL model and stochastic sampling. This model is crucial as it allows us to determine the probability distribution of the sampling interval between successive update instants, which is essential for stability analysis. An exponential mean-square stability condition for the constructed equivalent discrete-time stochastic system, which also guarantees the prescribed H∞ performance, is established by incorporating probability theory. The desired controller is designed using a step-by-step synthesis approach, which may offer lower design conservatism compared to some existing methods. Finally, our designed approach using a networked F-404 engine system model is validated and its merits relative to existing results are discussed. The proposed method is finally validated by employing a networked model of the F-404 engine system. Furthermore, the advantages of our method are presented in comparison to previous results.

Representing a second-order Ito equation as an equation with a given force structure

The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting the Rayleigh function are obtained. The obtained results are illustrated by an example of motion of a symmetric satellite in a circular orbit, assuming a change in pitch under the action of gravitational and aerodynamic forces.

Adjustment dynamics in network game with stochastic parameters

In this paper, stochastic parameters are introduced into the network games model with production and knowledge's externalities. This model was formulated by V. Matveenko and A. Korolev and generalized two-period Romer model. Agents' productivities have deterministic and Wiener components. We consider the dynamics that occur when two complete networks are combined. Explicit expressions in the form of Brownian random processes are obtained. A qualitative analysis of the solution of a system of stochastic equations is carried out.

Stabilization of Sampled-Data Systems With Noisy Sampling Intervals and Packet Dropouts via a Discrete-Time Approach

In real-world applications, sampled-data systems are often subject to undesirable physical constraints, which results in noisy sampling intervals that fluctuate around an ideal sampling period based on certain probability distribution. In the presence of noisy sampling intervals, this article is concerned with the stabilization problem for a class of sampled-data systems with successive packet dropouts. First, the relationship between two adjacent update times of zero-order hold is established. Then, by discrete-time approach, an equivalent discrete-time stochastic system is introduced. By the law of total expectation, together with confluent Vandermonde matrix, convolution formula, and Kronecker product operation, the mathematical expectation of a product of three matrices, including the system matrix and its transpose, is calculated. Based on this, a stabilization controller is designed such that the closed-loop system is stochastically stable. Finally, an illustrative example is provided to verify the effectiveness of the proposed design approach.

Cluster Synchronization of Boolean Networks Under Probabilistic Function Perturbation

In practical gene regulatory networks, gene expression is often accompanied by stochastic gene mutation. This brief investigates the cluster synchronization problem of Boolean networks (BNs) and Boolean control networks (BCNs) under probabilistic function perturbation, which can better characterize stochastic gene mutation and explore internal synchronization mechanism of the system. By transforming BN into an equivalent stochastic logical system, the algebraic expression of BN (BCN) under probabilistic function perturbation is derived. Then, necessary and sufficient conditions are provided so as to distinguish whether systems can achieve cluster synchronization with probability one under probabilistic function perturbation. Corresponding state feedback controllers are designed as well. The effectiveness is demonstrated by numerical examples.

A Probability Theory Approach to Stability Analysis of Networked Sampled-Data Systems With Consecutive Packet Dropouts

In this brief note, a probability theory approach is proposed to investigate the stability analysis issue for a class of networked sampled-data systems with packet dropouts. The continuous-time linear system under consideration is first transformed into a discrete-time system, where the system matrix of the discrete-time system is characterized by stochastic characteristic. Then, by the matrix exponential computation, an equivalent discrete-time stochastic system is established. Based on this, a controller is designed by the law of total expectation, which guarantees the stochastic stability of the closed-loop stochastic system. Finally, a numerical example and an example using power grid with IEEE 4-bus line are used to verify the effectiveness and applicability of the proposed design algorithm.

Density function and stationary distribution of a stochastic SIR model with distributed delay

In this paper, we first transfer a stochastic SIR model with strong kernel into an equivalent high-dimensional stochastic system. Then the stationary distribution of the model with the white noises is obtained by constructing a suitable Lyapunov function, which determines a critical value R∗ corresponding to the control reproduction number R0 of the corresponding determined system. Particularly, we obtain an exact local expression of the density function of the stochastic model around the unique quasi-equilibrium.

Exact solutions for the system of stochastic equations for the ion sound and Langmuir waves

We consider the system of stochastic equations of the ion sound and Langmuir waves with multiplicative noise in the Itô sense. To get new exact stochastic solutions, we apply three distinct methods such as: He’s semi-inverse method, the Riccati–Bernoulli sub-ODE method and the sine–cosine method. Such solutions are crucial to understanding some exciting complicated physical phenomena. Also, we improve and generalize some previously obtained results. Moreover, we study the influence of multiplicative noise on the obtained solutions of the system of stochastic equations for the ion sound and Langmuir waves.

Stochastic response of an energy harvesting system with viscoelastic element under Gaussian white noise excitation

Energy harvesting system is a facility that converts waste kinetic energy into serviceable electric energy. Vibration energy harvesting (VEH)) technology can be used in many different domains, such as controllers, medical implants, wireless sensors. Therefore, for the past few years, energy harvesting has become a prominent and growing field of research. There were few works focused on the vibration energy harvesting with viscoelastic property under random excitation. The crucial difference of the present study and previous studies is that we consider the viscoelastic property and the random excitation simultaneously: the stochastic response of energy harvesting and viscoelastic system subjected to Gaussian white noise excitation is investigated in this paper. First, the term associated with the viscoelasticity can be approximately replaced by stiffness and damping forces via variables transformation. Thus, an equivalent stochastic system without viscoelastic element can be obtained. Then, the approximately analytical solutions of the equivalent stochastic system can be given by the stochastic averaging approach. Furthermore, the theoretical analysis results are compared with the numerical results obtained by Monte Carlo method. The numerical results indicate that the method presented in this paper has a satisfactory level of accuracy. The last, the effects of system parameters on the mean square value and mean output power are discussed. The results show that the noise intensity, electro-magnetic coupling term and the ratio of the resistance and inductance has a positive effect on the mean output power.

Continuous Time Mixed State Branching Processes and Stochastic Equations

A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

Stochastic dynamics of a piezoelectric energy harvester with fractional damping under Gaussian colored noise excitation

This paper introduces the fractional damping for a piezoelectric energy harvester subjected to Gaussian colored noise and investigates the dynamic response by the stochastic averaging method. To successfully achieve this goal, the original system is decoupled into the equivalent stochastic system through variable transformation at first. Then the stochastic averaging approach is employed to construct the theoretical results. Furthermore, the feasibility and performance of the stochastic averaging method are evaluated with the assistance of the numerical results from Monte Carlo simulation. Finally, we concentrate on discussing the effect of fractional orders and system parameters on the mean square voltage and analyzing the effectiveness of the stochastic averaging method for the piezoelectric energy harvesting system by heat maps of the relative error in a quantitative perspective.

Square-Mean Pseudo Almost Periodic Solutions for Quaternion-Valued Stochastic Neural Networks with Time-Varying Delays

In this paper, we are concerned with a class of quaternion-valued stochastic neural networks with time-varying delays. Firstly, we cannot explicitly decompose the quaternion-valued stochastic systems into equivalent real-valued stochastic systems; by using the Banach fixed point theorem and stochastic analysis techniques, we obtain some sufficient conditions for the existence of square-mean pseudo almost periodic solutions for this class of neural networks. Then, by constructing an appropriate Lyapunov functional and stochastic analysis techniques, we can also obtain sufficient conditions for square-mean exponential stability of the considered neural networks. All of these results are new. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.

ПРОИЗВОДНЫЕ В СРЕДНЕМ СЛУЧАЙНЫХ ПРОЦЕССОВ И ДИФФУЗИОННЫЕ МОДЕЛИ В ЭКОНОМИКЕ

The article is devoted to diffusion models. The authors discuss the theoretical and methodological foundations of diffusion models in financial mathematics. Like the economic system, the modern world is developing rapidly. It seems impossible to predict what will happen tomorrow, how the emergence of new technologies will affect the market, and how changes in random factors will affect the product and the market as a whole. Diffusion models are one of the main methods for studying economic objects and processes. This is why it is so important to develop a diffusion model. The authors propose extending the applicability of the models by passing from Itô type stochastic equations to equations with so-called derivatives in the mean. For this, following E. Nelson, the authors introduce the concept of derivatives in the mean on the right and on the left. The equation with the derivative in the mean does not involve the Wiener process, therefore, it is not assumed in advance that the solution is diffusional. The article describes some well-known diffusion models, in which the transition from equations like an Itô type stochastic differential equation to equations satisfying a system of equations with derivatives in the mean leads to an expansion of the set of possible solutions. The authors also consider a generalization of geometric Brownian motion that satisfies a system of stochastic equations with derivatives in the mean and can cover a wider class of problems.

Analytic description of anomalous diffusion in heterogeneous environments: Fokker-Planck equation without fractional derivatives.

This article presents a one-dimensional model of diffusion in a heterogeneous environment, which qualitatively reflects the transport properties of a polymeric membrane with carbon nanotube areas. We derive the Fokker-Planck equation from a system of stochastic equations using a diffusion regime in polymers and a ballistic diffusion regime in nanotube areas. We demonstrate how the probability density function changes in the presence of nanotubes. The mean-square displacement of nonlinear time dependence is observed, indicating anomalous diffusion. This model explains the mechanism of anomalous diffusion in a ballistic-diffusion regime. Our approach does not suppose any type of distribution and does not use a fractional differentiate apparatus.

Stochastic Response of SDOF Self-Centering System

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises

Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.

Quantum Vacuum: The Structure of Empty Space–Time and Quintessence with Gauge Symmetry Group SU(2) ⊗ U(1)

We consider the formation of structured and massless particles with spin 1, by using the Yang–Mills-like stochastic equations system for the group symmetry S U ( 2 ) ⊗ U ( 1 ) without taking into account the nonlinear term characterizing self-action. We prove that, in the first phase of relaxation, as a result of multi-scale random fluctuations of quantum fields, massless particles with spin 1, further referred as hions, are generated in the form of statistically stable quantized structures, which are localized on 2D topological manifolds. We also study the wave state and the geometrical structure of the hion when as a free particle and, accordingly, while it interacts with a random environment becoming a quasi-particle with a finite lifetime. In the second phase of relaxation, the vector boson makes spontaneous transitions to other massless and mass states. The problem of entanglement of two hions with opposite projections of the spins + 1 and − 1 and the formation of a scalar zero-spin boson are also thoroughly studied. We analyze the properties of the scalar field and show that it corresponds to the Bose–Einstein (BE) condensate. The scalar boson decay problems, as well as a number of features characterizing the stability of BE condensate, are also discussed. Then, we report on the structure of empty space–time in the context of new properties of the quantum vacuum, implying on the existence of a natural quantum computer with complicated logic, which manifests in the form of dark energy. The possibilities of space–time engineering are also discussed.

Optimal control of random evolutionary Boolean games

This paper studies the optimal control of n-person random evolutionary Boolean games (REBGs). Firstly, the considered n-person REBG is converted to an equivalent stochastic control system by using the semi-tensor product of matrices. Secondly, based on the equivalent stochastic control system, an algorithm is proposed to solve the finite horizon optimal control problem of n-person REBGs, which is applied to the Mayer-type optimal control problem. Thirdly, using the receding horizon control method, the infinite horizon optimal control problem is considered, and an algorithm is established. Finally, an illustrate example is given to support the obtained results.

<inline-formula> <tex-math notation="LaTeX">$H_{\infty}$ </tex-math> </inline-formula> Fault Estimation for 2-D Linear Discrete Time-Varying Systems Based on Krein Space Method

This paper addresses the finite horizon ${H_\infty }$ fault estimation problem for 2-D linear discrete time-varying systems with bounded unknown input and measurement noise. The main contribution of this paper is the ${H_\infty }$ fault estimator for 2-D systems with a necessary and sufficient existence condition. By introducing a partially equivalent stochastic dynamic system in Krein space, the necessary and sufficient condition for the existence of the ${H_\infty }$ fault estimator is derived based on innovation analysis and projection formula in Krein space. Then, the solution of the estimator is achieved by means of a Riccati-like difference equation for 2-D systems. Finally, a thermal process example is given to demonstrate the effectiveness of the proposed method.

Stability of Random Processes with the 1/f α Spectrum

Extreme fluctuations have been modeled based on point nonpotential and potential systems of stochastic equations, in which white noise induces random processes with a power-law frequency dependence of the power spectra. The distribution of extreme fluctuations corresponds to the maximum of statistical entropy, which indicates their stability. The stability of fluctuation processes with 1/f α power spectra has been analyzed based on the principle of maximum information entropy.