Sufficient Conditions For Convergence Research Articles

On the asymptotic stabilization of the nonautonomous evolution equations in Hilbert spaces

This paper deals with the practical feedback stabilization of a class of evolution equations in Hilbert spaces. By using the Lyapunov function method, we give sufficient conditions for the global uniform practical convergence of the closed-loop system. Based on the global null-controllability, we construct an output feedback controller that guarantees global uniform practical stability of uncertain systems. A numerical application is given to show the applicability of our theoretical results.

Convergence analysis of deep residual networks

Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results.

On transforming hybrid nonlinear control problems with model uncertainty and input disturbance to mixed integer-linear programs

Control of multi-input multi-output (MIMO) hybrid nonlinear dynamic systems which are affine in control inputs is studied in this paper. It is assumed that each control input of the system can be continuous or discrete with a bound constraint. It is also assumed that the controller is discrete-time and updates the control(s) with a constant frequency. Based on these assumptions, a theorem which represents the necessary and sufficient condition for uniform convergence of the sequence of samples of the state vector of system towards the desired point is presented and proved. As a result of this theorem, a (mixed integer-) linear programming for calculation of control(s) is proposed. Two well-known nonlinear hybrid control problems with multiple inputs are solved by using the proposed method and the results are presented.

Two Point Iterative Schemes for Nondifferentiable Equations in Banach Space

The local as well as the semi-local convergence analysis is established for a certain single step-two point iterative scheme defined on a Banach space setting. These schemes converge to a locally unique solution of a nonlinear equation. Both types of convergence are based on w-type continuity and majorizing functions and sequences. An auxiliary fixed linear operator is utilized to assure the existence of inverses of the linear operators involved as well as the initial points of the iterative scheme. The local analysis provides the radius of convergence, error estimates and information on the uniqueness of the solution. Moreover, the semi local analysis provides sufficient convergence conditions, error estimates and uniqueness of the solution results. Numerical examples further validate the theoretical results.

A Sufficient Condition for Uniform Convergence of Trigonometric Series with p-Bounded Variation Coefficients

In this paper we consider trigonometric series with p-bounded variation coefficients. We presented a sufficient condition for uniform convergance of such series in case p>1. This condition is significantly weaker than these obtained in the results on this subject known in the literature.

Data-Driven games in computational mechanics

We resort to game theory in order to formulate Data-Driven methods for solid mechanics in which stress and strain players pursue different objectives. The objective of the stress player is to minimize the discrepancy to a material data set, whereas the objective of the strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new non-cooperative Data-Driven games identify an effective material law from the data and reduce to conventional displacement boundary-value problems, which facilitates their practical implementation. However, unlike supervised machine learning methods, the proposed non-cooperative Data-Driven games are unsupervised, ansatz–free and parameter–free. In particular, the effective material law is learned from the data directly, without recourse to regression to a parameterized class of functions such as neural networks. We present analysis that elucidates sufficient conditions for convergence of the Data-Driven solutions with respect to the data. We also present selected examples of implementation and application that demonstrate the range and versatility of the approach.

Adaptive Techniques for the Online Estimation of Spatial Fields With Mobile Sensors

The regression-based reconstruction of spatial fields by static sensors requires the number of measurements be at least equal to the number of unknown parameters in order to provide a unique solution. Improvements include the use of mobile sensor networks to arrive at higher accuracy estimates. An alternate approach utilizes adaptive techniques to update the parameter estimates online. Such adaptive techniques need persistence of excitation to ensure convergence of the parameter and function estimates. It is shown that the motion of just a single sensor is a necessary condition for persistence of excitation and, subsequently, of function convergence. Having a prescribed sensor guidance is also a sufficient condition for function convergence. First, a Lyapunov-based adaptation and guidance is presented and it is shown that persistence of excitation necessitates the time dependence of the outer product of the regressor vector with itself and which in turn requires the motion of the sensor within the spatial domain. Subsequently, a gradient-based adaptive scheme relaxes the Lyapunov-based sensor guidance to any user-defined guidance as long as the outer product of the associated rank-one regressor vector with itself has a uniformly positive definite integral. The latter adaptation is conducive to the inclusion of platform dynamics resulting in a vehicle control design needed to attain parameter convergence. Extension of the centralized solution to distributed estimation using a network of mobile sensors is also examined. These results are demonstrated with simulations in one and two dimensions.

Non-Euclidean Contraction Theory for Monotone and Positive Systems

In this note we study contractivity of monotone systems and exponential convergence of positive systems using non-Euclidean norms. We first introduce the notion of conic matrix measure as a framework to study stability of monotone and positive systems. We study properties of the conic matrix measures and investigate their connection with weak pairings and standard matrix measures. Using conic matrix measures and weak pairings, we characterize contractivity and incremental stability of monotone systems with respect to non-Euclidean norms. Moreover, we use conic matrix measures to provide sufficient conditions for exponential convergence of positive systems to their equilibria. We show that our framework leads to novel results on (i) the contractivity of excitatory Hopfield neural networks, and (ii) the stability of interconnected systems using non-monotone positive comparison systems.

Complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise extended negatively dependent random variables with statistical applications

In this paper, complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise extended negatively dependent random variables are investigated, and some sufficient conditions for convergence are provided. Three ways of optimising the conditions for convergence, namely reducing the moment condition, narrowing the boundary function, and raising the weight function respectively, are considered under light weights. Besides, rather than assuming stochastic domination, we turn to a weaker assumption of uniformly bounded expectations to describe the moment conditions. The results obtained in the paper extend the corresponding ones for independent random variables and some dependent random variables. In addition, the results are applied to establish strong consistency for estimators in some statistical models.

Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied.However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

Strong Convergence Theorem for Finite Family of Asymptotically Nonexpansive in the Intermediate Sense Nonself Maps

Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P. Let Ti: K → X (i= 1,...,m) be a finite family of uniformly continuous asymptotically nonexpansive in the intermediate sense maps with a nonempty common fixed points set F. Sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F are proved.MSC(2010): 47H10, 47J25

Pinning control of linear systems on hypergraphs

When steering the dynamics of network systems, the control design needs to cope with constraints on actuation and sensing, which often imply that the same control input is injected to each node in a given subset, and this input signal is a function of the state of this node subset. This common situation cannot be modeled in terms of standard pairwise interactions on digraphs, and we propose to use directed hypergraphs as the mathematical object suitable to describe this kind of directed, multibody interactions. We apply this framework to the pinning control problem in networks of coupled linear systems, and derive necessary and sufficient conditions for convergence onto the desired trajectory set by the pinner. Furthermore, we provide a dedicated control algorithm to identify the interconnections that are critical for network control.

A novel fractional-order flocking algorithm for large-scale UAV swarms

The rate of convergence is a vital factor in determining the outcome of the mission execution of unmanned aerial vehicle (UAV) swarms. However, the difficulty of developing a rapid convergence strategy increases dramatically with the growth of swarm scale. In the present work, a novel fractional-order flocking algorithm (FOFA) is proposed for large-scale UAV swarms. First, based on the interaction rules of repulsion, attraction and alignment among swarm individuals, fractional calculus is introduced to replace traditional integer-order velocity updating, which enables UAVs to utilize historical information during flight. Subsequently, the convergence of the algorithm is theoretically analyzed. Some sufficient convergence conditions for the FOFA are presented by exploiting graph theory. Finally, the simulation results validate that our proposed FOFA performs much better than traditional flocking algorithms in terms of convergence rate. Meanwhile, the relationships between the fractional order of the FOFA and the convergence time of the UAV swarm are discussed. We find that under certain conditions, the fractional order is strongly correlated with the convergence rate of the UAV swarm; that is, a small fractional order (more consideration of historical information) leads to better performance. Moreover, the fractional order can be used as an important parameter to control the convergence rate of a large-scale UAV swarm.

Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions

The main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of Henstock-Kurzweil integrable functions. Our approach involves the use of the concept of multiplier functions, where we define a class Φ of multipliers for the Henstock-Kurzweil integral. We consider a sequence (fn) of Henstock-Kurzweil integrable functions on a non-degenerate interval [a, b] and we assume that (fn) converges point wise to a function f. Then we show that f is Henstock-Kurzweil integrable and its integral is equal to the limit of the sequence (∫abfn) if there exists φ∈Φ such that the defined functionals of the type F (φ, fn) satisfy the imposed conditions. Beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas.

On weak convergence of stochastic differential equations with irregular coefficients

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter ε and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at ε → 0 were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

Sufficient Conditions for the Linear Convergence of an Algorithm for Finding the Metric Projection of a Point onto a Convex Compact Set

Sufficient Conditions for the Linear Convergence of an Algorithm for Finding the Metric Projection of a Point onto a Convex Compact Set

Adaptive observer design for uncertain hyperbolic PDEs coupled with uncertain LTV ODEs; Application to refrigeration systems

The problem of estimating the temperatures and the heat transfer coefficient of a concentric tube heat exchanger coupled with a heater is considered in this work. Measurements collected from the extremities of the exchanger tube are used to estimate the heat distribution over the length of the exchanger, which induces a boundary estimation problem. This system, which is part of any standard cooling plant, is particularly challenging due to the distributed nature of its variables. It is modeled by a system of (2 × 2) hyperbolic PDEs, coupled with an ODE at the boundary. To solve the estimation problem, we consider a general class of systems consisting of a (2 × 2) hyperbolic system coupled with a set of nX linear time-varying (LTV) ODEs at the boundary. Both the PDE and the ODEs have uncertain parameters to be estimated. The objective is to estimate the PDE states, the ODE states, and the parameters simultaneously with no assumption on the ODEs stability. We design a Luenberger state observer, and our method is mainly based on the decoupling of the PDE estimation error states from that of the ODEs via swapping design. We then derive the observer gains from the Lyapunov analysis of the decoupled system after proving the boundedness of the swapping filters. We give sufficient conditions of the exponential convergence of the adaptive observer through differential Lyapunov inequalities (DLIs). Finally, we apply the developed theory on the coupled heat exchanger–heater model to evaluate the performance of the observer in numerical simulations.

Convergence properties concerning Lebesgue‐p norm of iterative learning control for a class of fractional differential systems

AbstractThis paper investigates the monotonic convergence and speed comparison of first‐ and second‐order proportional‐α‐order‐integral‐derivative‐type ( type) iterative learning control (ILC) schemes for a linear time‐invariant (LTI) system, which is governed by the fractional differential equation with order . By introducing the Lebesgue‐p ( ) norm and utilizing the property of the Mittag‐Leffler function and the boundedness feature of the fractional integration operator, the sufficient condition for the monotonic convergence of the first‐order updating law is strictly analyzed. Therewith, the sufficient condition of the second‐order learning law is established using the same means as the first one. The obtained results objectively reveal the impact of the inherent attributes of system dynamics and the constructive mode of the ILC rule on convergence. Based on the sufficient condition of first/second‐order updating law, the convergence speed of first‐ and second‐order schemes is determined quantitatively. The quantitative result demonstrates that the convergence speed of second‐order law can be faster than the first one when the learning gains and weighting coefficients are properly selected. Finally, the effectiveness of the proposed methods is illustrated by the numerical simulations.

Fast Fixed-Time Output Multi-Formation Tracking of Networked Autonomous Surface Vehicles: A Mathematical Induction Method

In this paper, we aim to exploit an effective way to solve the output multi-formation tracking problem of the networked autonomous surface vehicles (ASVs) in a fast fixed time manner. Specifically, addressing the output multi-formation tracking problem implies that 1) the networked ASVs are divided into multiple interconnected subnetworks with respect to multiple targets; 2) for each subnetwork, the outputs of the networked ASVs form a desired geometric formation with exchanging the local interactions. Besides, solving the fast fixed-time tracking problem in this paper implies that 1) the convergence time is independent of the initial conditions; 2) the system states are forced to reach the employed nonsingular fixed-time sliding surface in a prescribed time, which thus called fast fixed-time control. Then, based on a time-related function and a nonsingular fixed-time sliding surface, a hierarchical fast fixed-time control algorithm is proposed to solve the aforementioned problem within a fast fixed time being independent of the initial conditions. Furthermore, by employing the Lyapunov argument and mathematical induction, we present the sufficient conditions for fast fixed-time convergence of the tracking errors with respect to multiple targets. Finally, numerous simulation examples are presented to demonstrate the effectiveness of the proposed control algorithm.

Exponential stabilization of stochastic complex networks with Markovian switching topologies via intermittent discrete-time state observation control

This paper is concerned with the exponential stability of the stochastic complex networks with Markovian switching topologies. It is worth emphasizing that the topological structure of the stochastic complex networks is Markovian switching. Not all switching subnetworks must contain a spanning tree or be strongly connected. Moreover, a new type of aperiodically intermittent discrete-time state observation control is proposed, which is a significant extension of discrete-time state observation control and intermittent control. Based on the M-matrix, the Lyapunov method, and the graph theory, some sufficient conditions for exponential convergence. Moreover, we obtain that the average control rate in this paper is greater than the control rate proposed in the existing literature, which is less conservative. In addition, the theoretical results are used to discuss the exponential stability of stochastic coupled oscillators and a communication network model, respectively. Finally, two numerical examples are given to verify the effectiveness of the results.