Averaging Theorem Research Articles

Dynamic event-triggered-based adaptive fuzzy optimized control for slowly switched nonlinear system with intermittent state constrains

This article presents a novel fuzzy event-triggered optimized strategy for slowly switched nonlinear system using reinforcement learning technique. For addressing intermittent state constraints problem, some improved shifting functions and barrier functions are designed in the adaptive backstepping process. Subsequently, by designing global performance functions with the discount term, the corresponding Hamilton-Jacobi-Bellman equation is derived and the optimal solution can be obtained under critic-actor structure. Different from the existing event-triggered methods, a novel dynamic event-triggered scheme for subsystem is developed, which not only saves communication resources, but also solves the mismatch behaviour between controller and subsystem without restriction on maximum asynchronous intervals and the number of switch within a triggering interval. Furthermore, by using modified average cycle dwell time method and Lyapunov stability theorem, all signals of system are proved to be bounded. Eventually, numerical simulation results validate the superiority of the proposed approach, and this new scheme is applied to single-link robotic manipulator.

A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case

A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case

Polynomial sequences in discrete nilpotent groups of step 2

Abstract We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

Stochastic time-varying extremum seeking and its applications

We investigate a stochastic time-varying extremum seeking method based on new stochastic averaging theorems for time-varying systems and its application to distributed source seeking with switching topology. Firstly, we present sufficient conditions for the approximation of the time-varying averaging system to the time-varying original system with stochastic perturbations and obtain the solution property of the original system under the stability of invariant sets of the average system. Then, by using our developed stochastic averaging theory, we prove that the proposed stochastic time-varying extremum seeking algorithm converges to the time-varying optimal solution. Furthermore, based on the method of stochastic time-varying extremum seeking, we propose a distributed stochastic source seeking algorithm which navigates multiple vehicles to seek the source of aggregated multiple signal fields under a switching topology. We can prove the exponential convergence of such a distributed algorithm by our stochastic averaging theorems and a Lyapunov-based approach. Finally, we present detailed simulation results to illustrate the effectiveness of our method.

Fixed-Time Average Consensus of a Dynamic Event-Triggered Mechanism in a System of Multiple Underactuated Autonomous Underwater Vehicles Based on a Body Frame Spherical Coordinate System

This paper is concerned with the consensus of a system involving multiple underactuated autonomous underwater vehicles (AUVs). Combined with a dynamic event-triggered mechanism and a fixed-time stability theorem, the backstepping average consensus controllers are designed. Firstly, the new consensus control objective on the system for multiple underactuated AUVs in a body frame (BF) spherical coordinate system is proposed, and the tracking error kinematic equations are established based on the kinematic characteristics of the underactuated AUV. The fixed-time consensus controller is designed by the backstepping method, and the average consensus theorem is proposed to improve the Lyapunov function. Furthermore, the dynamic event-triggered mechanism is adopted to reduce the communication requirements and energy consumption. This is the first solution to the problem of a consensus controller design for a system of multiple underactuated AUVs. Finally, numerical simulation results demonstrate that the proposed method has superior effectiveness over alternatives.

Adaptive parameter-related implementation of the free surface in elastic anisotropic full-waveform modeling

The near-surface anisotropy is one of the factors that lead to errors in static corrections, seismic imaging, and inaccurate surface velocity building. Therefore, it is important to study the propagation features of seismic waves in the near-surface with anisotropy. The finite-difference method is commonly used in seismic wave forward modeling, however, the numerical implementation of the free surface is nontrivial in this method, in particular to its implementation in the anisotropic media which is less studied. Here, we propose an adaptive parameter-related free surface implementation. This method is developed from the physical viewpoint with the average medium theory and limit theorem, in which the stress-free condition is implemented by modifications of the original constitutive relation and density at the free-surface points without additional computational requirements. Numerical experiments with the increasing complexity of anisotropy in both 2D and 3D demonstrate that this method can produce precise simulation results with a spatial sampling of 10 grid points per minimum wavelength. In addition, a time-space discontinues grid strategy can be introduced in this implementation to allow the usage of the coarse grid in the deeper part where the oversampling for the surface waves is not required, which increases the flexibility and the computational efficiency of the proposed method.

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Abstract The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ (z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽 p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽 p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Graphical construction of spatial Gibbs random graphs

We consider a random graph model on Zd that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. Based on a graphical construction of the model as the invariant measure of a birth and death process, we prove the existence and uniqueness of a measure defined on graphs with vertices in Zd, which coincides with the limit along the measures over graphs with the finite vertex set. As a consequence, theoretical properties, such as exponential mixing of the infinite volume measure and central limit theorem for averages of a real-valued function of the graph, are obtained. Moreover, a perfect simulation algorithm based on the clan of ancestors is described in order to sample a finite window of the equilibrium measure defined on Zd.

An ergodic theorem for the weighted ensemble method

AbstractWe study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.

A computational homogenization framework for non-ordinary state-based peridynamics

Peridynamic theory has been shown to possess the capabilities of describing phenomena that theories based on partial differential equations are not capable of describing. These phenomena include nonlocal interactions and presence of singularities in system responses. To exploit the capabilities offered by peridynamics in the homogenization of heterogenous media, a nonlocal computational homogenization theory based on peridynamic correspondence model (non-ordinary state-based peridynamics) is proposed. To set the development of the theory on a rigorous mathematical framework and to ensure consistency with the nonlocal nature of the peridynamic theory, a nonlocal vector calculus was used in the analysis of the nonlocal homogenization theory. The proposed theory is a two-scale micro–macro-homogenization strategy in which the constitutive relation at the macroscale is derived from explicit solution of a nonlocal volume constraint problem at the microscale. To justify the coupling between the two scales, nonlocal analogues of the stress and strain average theorems as well as the Hill–Mandel macrohomogeneity condition were derived. Validation of the proposed theory is achieved via numerical solution of Representative Volume Elements (RVE) from composite materials and comparing the results with those obtained by means of established methodologies.

Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Suppose that \(\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}\) is the solution to a d-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form \(N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x\), as \(N\rightarrow \infty \), where g is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf–Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

Efficient prediction of the effective nonlinear properties of porous material by FEM-Cluster based Analysis (FCA)

The present paper extends the FEM-Cluster based Analysis (FCA) method to the efficient prediction of effective properties of porous materials in the nonlinear range. Different from the traditional methods of homogenization or mean field theory, this new approach does not fill the pore with low modulus materials which causes numerical inaccuracy and convergence difficulties as reported in several literature. With no material filling in pores, the average strain theorem and the Hill–Mandel condition which is the basis of the Representative Volume Element (RVE) method for the prediction of effective properties of heterogeneous materials is almost impossible to implement. Furthermore, because of the lack of pore strain information, the compatibility condition of cluster strains cannot be applied in the incremental analysis of reduced order model in the nonlinear stage. In the present study we have overcome these difficulties by several approaches. First, it is found that the energy equivalence principle provides a more suitable basis for effective properties prediction of porous materials. Second, the self-equilibrium cluster stress field obtained from the interaction matrix and the cluster-based minimum complementary energy in FCA can be successfully applied to the incremental analysis of reduced order model in the nonlinear stage instead of the strain compatibility condition. However, due to the errors introduced by averaging stress and strain over each cluster, the interaction matrix becomes non-singular and the self-equilibrium properties of the cluster stress field generated from the interaction matrix become very poor. Fortunately, the most exciting discovery of the present study is that we have found a way to recover the singularity of the interaction matrix by minimizing its rank on the premise that the elastic energy norm of RVE remains unchanged within the given tolerance. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed approach.

An almost sure central limit theorem for the stochastic heat equation

Let u(t,x) be the solution to the stochastic heat equation on R+×Rd driven by a Gaussian noise that is white in time and has a spatially covariance that satisfies Dalang’s condition. In this paper, we prove an almost sure central limit theorem for spatial averages of the form ∫[0,N]dg(u(t,x))dx as N→∞ for fixed t>0, where g is a globally Lipschitz function or belongs to a class of locally Lipschitz functions.

On good universality and the Riemann Hypothesis

We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelöf Hypothesis and the Riemann Hypothesis. These ideas are then used to study the value distribution of Dirichlet L-functions, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. Adler and B. Weiss, M. Lifshits and M. Weber, J. Steuding, J. Lee and A. I. Suriajaya using Birkhoff's ergodic theorem and probability theory.

Electromechanical response of multilayered piezoelectric BaTiO3/PZT-7A composites with wavy architecture

Electromechanical laminated composites with piezoelectric phases are increasingly being explored as multifunctional materials providing energy conversion between electric and mechanical energies. The current work explores thus-far undocumented combined microstructural effects of amplitude-to-wavelength ratio, volume fraction, poling direction of piezoelectric phases on both the homogenized properties and localized stress/electric field distributions in multilayered configurations under fully coupled electro-mechanical loading. In particular, the Multiphysics Finite-Volume Direct Averaging Micromechanics (FVDAM) and its counterpart, an in-house micromechanical multiphysics finite-element model, are utilized to investigate the homogenized and localized responses of wavy multilayered piezoelectric BaTiO3/PZT-7A architectures. These two methods generate highly agreeable results. Moreover, we critically examine the convergence of the finite-volume and finite element-based approaches via the Average Stress Theorem and Average Electric Displacement Theorem. The comparison shows the finite volume-based approach possesses a better numerical convergence. This study illustrates the FVDAM’s ability toward the analysis and design of engineered multilayered piezoelectric materials with wavy architecture.

Spatial averages for the parabolic Anderson model driven by rough noise

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

Random belief system dynamics in complex networks under time-varying logic constraints

People change their beliefs randomly such that both the structure and the properties of the belief system vary with time under several conditions, which is difficult to describe mathematically due to the complexity of this randomness. A precise belief system dynamics can not only help us to identify which properties the belief system has and forecast how it moves, but it can also help the belief system to move in a reasonable direction. To find this dynamics, a generalized Friedkin model with time-varying parameters was constructed to analyze the following events: (1) inter-dependent issues, (2) heterogeneous systems of issue dependency constraints, and (3) inter-dependent issues under three types of systems. These systems are characterized with: (1) parameter variations and distributions being only assumed to be bounded, (2) zero mean random parameter variations and disturbances being, in general, a correlated process, and (3) the parameter process {Δk} being a random walk. The dynamics is then obtained by invoking LMS algorithms due to the property of the time-varying parameter. Parameters of the dynamic statistical model produce large enough random matrices with random time-varying structure. The random average theorem was then introduced to transform these matrices to deterministic ones under four strict conditions such that the dynamics could be estimated accurately. The stability conditions corresponding to the belief system dynamics are then provided, in case the belief system is not complex. By contrast, if the belief system is more complex, a corresponding multi-agent computational model is constructed to explain how a more unstable belief system would move. The results show that, the parameter estimated via LMS is relatively deterministic if the issues are inter-independent and the parameter variation is smooth, and vice versa. It is thus possible to conclude that there exists a critical point of the parameter variation such that the analytic solution of the dynamics could be obtained if the complexity is simpler than the critical point. Otherwise, the belief system is unstable and uncontrollable. In the latter case, the dynamics just relies on the parameters statistical property if the beliefs or issues are inter-independent and the dynamics of belief system is more complex. Furthermore, there would exist a phase transition for the belief system such that the minority view would become the dominant view under certain conditions, if the beliefs or issues are inter-dependent. It is concluded that both the system structure and the belief configuration are important to the belief dynamics. The two most important findings of this work are: (1) the stochastic time-varying model matches the property of the belief system and can thus be used to discover more interesting results and (2) the estimation method of the LMS driven by the random average theorem can be generalized to almost all social systems, if the parameters changes are not too violent, while the multi-agent simulation can be used if the parameters change more strongly. These results reveal the law of the essence of economic and management complex adaptive systems.

Robust hybrid zero-order optimization algorithms with acceleration via averaging in time

This paper presents a new class of robust zero-order algorithms for the solution of real-time optimization problems with acceleration. In particular, we propose a family of extremum seeking (ES) dynamics that can be universally modeled as singularly perturbed hybrid dynamical systems with restarting mechanisms. From this family of dynamics, we synthesize four fast algorithms for the solution of convex, strongly convex, constrained, and unconstrained optimization problems. In each case, we establish robust semi-global practical asymptotic or exponential stability results, and we also show how to obtain well-posed discretized algorithms that retain the main properties of the original dynamics. Given that existing averaging theorems for singularly perturbed hybrid systems are not directly applicable to our setting, we derive an extended averaging theorem that relaxes some of the assumptions made in the literature, allowing us to make a clear link between the KL bounds that characterize the rates of convergence of the hybrid dynamics and their average dynamics. We also show that our results are applicable to non-hybrid algorithms, thus providing a general framework for accelerated ES dynamics based on averaging theory. We present different numerical examples to illustrate our results.

AN AVERAGE THEOREM FOR TUPLES OF k ‐FREE NUMBERS IN ARITHMETIC PROGRESSIONS

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.

Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulses

This paper aims at addressing the issue of a periodic averaging method for neutral stochastic functional differential equations with delayed impulses. Two periodic averaging theorems are presented and the approximate equivalence between the solutions to the original systems and those to the reduced averaged systems without impulses is proved. Further, we show a brief framework of extending our main results to Lévy case. At last, an example is given to demonstrate the procedure and validity of the proposed periodic averaging method.