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This paper investigates the strong convergence of a fully discrete numerical method for the stochastic partial differential equations driven by multiplicative noise. The fully discrete space–time approximation consists of the symmetric interior penalty discontinuous Galerkin method for the spatial discretization and the implicit Euler method for the temporal discretization. Rather than the usual semi group analysis techniques, in this paper, we present an analysis framework in the variational formulation by introducing new weak variational approximation techniques. Some error estimates in a strong sense are established for the proposed fully discrete scheme. The optimal convergence rates are then obtained in both space and time. Numerical results for the nonlinear stochastic partial differential equations are finally presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes an extension of the model predictive control framework for switching continuous‐time linear systems. The switching times follow a stochastic process with limited statistical information. At each switching time, the controller knows the system state, but it is blind with respect to the switching continuous‐time subsystems. In this setting, the paper's main contribution is to show how to compute the model predictive control gain. The paper also illustrates the implications of our approach for applications. The approach was used in practice to control an electric current source that supplied a switching load. The experimental data support the usefulness of our approach. [ABSTRACT FROM AUTHOR]

Economic convergence has long been a declared objective of the EU and has been considered the fundamental mechanism for achieving socioeconomic cohesion. Even so, the empirical literature finds a lack of real convergence as geographical clusters have emerged. In this paper, we contribute to the literature on income convergence in the EU by analysing convergence in real disposable income across European countries. In contrast to the previous results on real convergence in the EU, we find that most of the countries are converging to the same equilibrium level. [ABSTRACT FROM AUTHOR]

In this paper, some new algorithms based on the iteratively reweighted least squares (IRLS) method are proposed for sparse recovery problem. There are two important parameters in the IRLS method: a weighted parameter and a regularization parameter. On the one hand, in order to improve the performance of IRLS method, a new way is given to update the weight vector. On the other hand, for the regularization parameter, three new update methods are introduced to avoid the phenomenon that the regularization parameter drops too fast. Then, some improved iteratively reweighted least squares (IIRLS) algorithms are proposed, and their convergence and convergence rate are analyzed. The local convergence of our algorithms is superlinear and approaches a quadratic rate in special cases. Finally, a large number of algorithms are compared in solving the sparse recovery problem, including IRLS methods and IIRLS methods with different weighting parameters and regularization parameters, and certain iterative methods. The experimental results demonstrate that the proposed methods are efficient and promising. [ABSTRACT FROM AUTHOR]

In this paper, the leader‐following consensus problem of general linear multi‐agent systems with distributed dynamic event‐triggered control is studied. A distributed controller with estimated states is designed and a dynamic event‐triggered communication protocol with an auxiliary variable is proposed for each agent to update its triggering threshold. Under this control strategy, the systems only need the states in triggering instants as a result of the utilisation of estimated states. The existence of a dynamic threshold guarantees fewer triggering times, in contrast to the conventional static one. Meanwhile, the estimator is designed between the triggering instants to reduce the state shift, the next state of each agent depends on the local information of itself and its neighbours rather than global information. Then, the multi‐agent systems can achieve consensus without Zeno behaviour. Finally, a simulation example is offered to verify the effectiveness and feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

Decentralized optimal control of interval state space systems involves significant complications due to the required sparsity, robustness, and convergence rate. By adding an l1 term to the standard linear quadratic optimization, the obtained composite index, accommodating the sparsity, contains both differentiable and non‐differentiable terms. Knowing that the choice of optimization techniques are problem‐dependent, in this paper, several smart provisions such as "Active set", "Coordinate descent", and separately managing the differentiable part of the cost function are put together to form a relatively fast algorithm. In addition, a rule is introduced to adjust automatically the degree of sparsity and a constraint to confine the interval closed‐loop system eigenvalues to the desired s plane region. As a result, a robust sparse decentralized control (RSDC) approach for interval networks is established. The RSDC method is utilized for the control of a multi‐terminal VSC (Voltage Source Converter) ‐based DC (Direct Current) grid with parametric uncertainty under various external random perturbations. The test results show that the proposed method with just 20 feedback links (out of available 100) performs superiorly to the alternative ones in terms of sparsity, convergence rate, and time‐domain performances, verified through extensive simulations. [ABSTRACT FROM AUTHOR]

This paper addresses the exploitation of vector based contraction theory to synchronize the motion of agents in multiagent systems. The convergence analysis of the agents using this theory provides a flexible frame as the components of a vector distance norm follow less stringent conditions than classical contraction analysis. The key feature is that it encompasses the formulation of some comparison system, as a result, the problem of convergence analysis of a complex multiagent system reduces to the convergence analysis of a much simpler comparison system. Hence, despite the nonlinearities, the methods of designing consensus controls remain simple as compared to the existing literature. We address four different cases: 1) consensus of third‐order dynamics multiagent systems subjected to acceleration and input constraints; 2) consensus of multiagent systems with heterogeneous nodes under communication imperfections; 3) synchronization of multiagent systems with connected and disconnected switching topologies; 4) synchronization in networked systems with time‐varying couplings. Moreover, an effective synchronization control scheme for a class of multiagent systems with underactuated agent dynamics is developed using the method of aggregation of comparison systems. Finally, simulation results are illustrated that show the validity of the mathematical results. [ABSTRACT FROM AUTHOR]

Stochastic gradient descent algorithm is a classical and useful method for stochastic optimisation. While stochastic gradient descent has been theoretically investigated for decades and successfully applied in machine learning such as training of deep neural networks, it essentially relies on obtaining the unbiased estimates of gradients/subgradients of the objective functions. In this paper, by constructing the randomised differences of the objective function, a gradient‐free algorithm, named the stochastic randomised‐difference descent algorithm, is proposed for stochastic convex optimisation. Under the strongly convex assumption of the objective function, it is proved that the estimates generated from stochastic randomised‐difference descent converge to the optimal value with probability one, and the convergence rates of both the mean square error of estimates and the regret functions are established. Finally, some numerical examples are prsented. [ABSTRACT FROM AUTHOR]

There is a growing literature on the importance of persistent regional inequalities in developing countries, with evidence that the lack of convergence in incomes across regions is associated with a rise in conflict and social unrest. However, there is a relative paucity of literature on the stochastic characteristics of gross domestic product at the sub‐national level. In this paper we identify new stochastic properties of Indian states' gross domestic product between 1960 and 2019 using a fractional stochastic convergence approach. We test for fractional stochastic convergence in relative incomes in order to identify high persistence and mean reversion. Interval estimates of the largest autoregressive coefficient for the relative incomes of Indian states are wide, thus including many alternatives that are persistent. Interval estimates of the half‐life of relative income shocks suggest that in several cases shocks die out within 0–10 years. Finally, we estimate a fractionally integrated model and obtain mixed evidence of mean reversion and non‐stationarity, with six out of 16 states experiencing mean reversion. These results are highly encouraging and contradict earlier studies which show long‐term divergence and polarization of regional incomes across Indian states, and are thus of great relevance to policy‐makers. [ABSTRACT FROM AUTHOR]

This study tests for convergence of the three main sub‐components of economic freedom (government spending, taxation, and labor market freedom) for the 50 US states from 1981 to 2019. In particular, we test for stochastic convergence using unit root tests that account for structural breaks and bounded processes. By controlling for these sources of parameter instability, we find limited evidence in favor of stochastic convergence across the three sub‐components of economic freedom. However, we find more evidence of stochastic convergence with respect to the sub‐components than in the case of the overall economic freedom at the US state level. Our results highlight the distinct nature of the institutional quality across states as represented by government spending, taxation, and labor market freedom. [ABSTRACT FROM AUTHOR]

This study tests for convergence in economic freedom across the 50 U.S. states from 1981 to 2019. In particular, we test for stochastic convergence in overall economic freedom using unit root tests that account for structural breaks and bounded processes, two sources of parameter instability. We find limited evidence of stochastic convergence. Further analysis of relative (club) convergence and weak σ‐convergence rejects the presence of overall convergence in the panel of the U.S. states, but the emergence of two convergence clubs with respect to state‐level economic freedom. The logit analysis of the determinants of the convergence clubs reveals that more prosperous states and states with a higher initial level of economic freedom have a higher probability of being in the club with more economic freedom. However, more racially diverse states have a lower probability of being in the club with more economic freedom. [ABSTRACT FROM AUTHOR]

A stochastic SIR (susceptible → infective → recovered) model is considered for the spread of an epidemic on a network, described initially by an Erdős–Rényi random graph, in which susceptible individuals connected to infectious neighbors may drop or rewire such connections. A novel construction of the model is used to derive a deterministic model for epidemics started with a positive fraction initially infected and prove convergence of the scaled stochastic model to that deterministic model as the population size n→∞. For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as n→∞, the final fraction infected τ(λ) is discontinuous in the infection rate λ at its threshold λc, thus not converging to 0 as λ↓λc. The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that τ(λ) jumps from 0 to 1 as λ passes through λc. [ABSTRACT FROM AUTHOR]

Summary: Motivation: Empirical studies on income convergence for India indicate that Indian states exhibit divergence in income. The literature, however, has neglected the convergence dynamics of the main sources of output. A pertinent question is whether divergence in income is due to divergence in factor inputs (capital and labour) and/or in total factor productivity (TFP) in India, a low middle‐income country with high dispersion of per capita income across its states. Purpose: This article examines the sources of income divergence in Indian states by analysing convergence in TFP, labour, and capital. Methods and approach: We test for stochastic convergence hypothesis by employing state‐level data for 19 Indian states during the period 2001–2015. Given the small sample dimension of the Indian state‐level income data, we benefit from the recent developments in non‐stationary panel data literature; and conduct novel panel stationarity and unit root tests with a fixed time dimension. We further carry out a robustness analysis to account for cross‐section dependency across the states. Findings: TFP as well as the factor inputs (labour and capital) exhibit divergence, implying persistence in income inequality across Indian states. Divergence in labour across the states reflects the fact that migration is primarily an intra‐district phenomenon. Poor investment climate in low‐income states acts as a barrier for convergence in capital. TFP and capital stock are found to be correlated, and thereby lower investment in poorer states may be responsible for divergence in TFP across the states. Policy implications: Migration in India is primarily an intra‐district phenomenon. There is a need to study the reasons which are discouraging inter‐district and inter‐state migration for better utilization of the labour resource. Government intervention has not been adequate to improve the infrastructure position and encourage capital inflows to low‐income states. Low‐income states should improve their business climate and create support infrastructure to earn the confidence of investors. Increasing investment in low‐income states would help to increase their TFP and catch up with the high‐income states. [ABSTRACT FROM AUTHOR]

A design method of neural network adaptive interval observer for the continuous‐time unknown nonlinear system is proposed. The bounds of unknown nonlinear functions are described by the state and the input, so that the bounds of the nonlinear function vector are unmeasurable. Therefore, this method can be used to observe systems with higher degree of nonlinearity and any priori knowledge about systems. In this work, the hyper basis function neural network (HBFNN) is presented to estimate the boundary (including lower and upper bounds) of the unknown nonlinear function vector. Furthermore, the non‐negativity and stability of the error dynamic system are given, and the convergence and non‐negativity are also guaranteed. Compared with the traditional interval observer, the efficiency of the proposed approach is verified through a numerical simulation example. [ABSTRACT FROM AUTHOR]

Input saturation, which brings input non‐linearity to systems and limits the closed‐loop systems' performances severely, is an ubiquitous problem in practice. Here, triggered finite‐time consensus problem of disturbed leader‐follower multi‐agent systems is addressed under input saturation constraint. To overcome the coupling of the non‐linearity and finite‐time convergence together with input saturation, non‐linear triggered control protocol has been proposed, which is confirmed through a two‐step stabilization procedure. First, the saturated closed‐loop system is designed to be asymptotically stable, under which the systems converge into a desaturated region in finite time. Then in the second step, with the proposed control protocol, the desaturated closed‐loop multi‐agent systems can achieve practical finite‐time consensus. In addition, to get rid of continuous communication in the monitoring of event‐triggered condition, a self‐triggered approach is further proposed. With rigorous analyses, practical finite‐time consensus can be achieved for the overall closed‐loop systems and Zeno behavior is also excluded. Simulations are carried out in detail to illustrate the theoretical results in this study. [ABSTRACT FROM AUTHOR]

In the present article the two‐dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi‐incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high‐order hybrid equilibrium formulation is developed for dynamic analysis of elastic solids in the variational framework of the Toupin principle, which is the complementary form of the Hamilton principle. The Newmark time integration scheme is introduced for discretization of the stress fields in the time domain and dynamic analysis of both the compressible solid and quasi‐incompressible ones. The hybrid equilibrium element formulation provides very accurate solutions with a high‐order stress field and the results of the static and dynamic analyses are compared with the solution of the classic displacement‐based quadratic formulation, showing the convergence of the two formulations to the exact solution and the very satisfying performance of the proposed formulation, especially for analysis of quasi‐incompressible elastic solids. [ABSTRACT FROM AUTHOR]

Block‐sparse signals, whose non‐zero entries appear in clusters, have received much attention recently. An unfolded network, named Ada‐BlockLISTA, was proposed to recover a block‐sparse signal at a small computational cost, which learns an individual weight matrix for each block. However, as the number of network parameters is increasingly associated with the number of blocks, the demand for parameter reduction becomes very significant, especially for large‐scale multidimensional harmonic retrieval (MHR) problems. Based on the dictionary characteristics in two‐dimensional (2D) harmonic retrieve problems, the authors introduce a weight coupling structure to shrink Ada‐BlockLISTA, which significantly reduces the number of weights without performance degradation. In simulations, the proposed block‐sparse reconstruction network, named AdaBLISTA‐CP, shows excellent recovery performance with a smaller number of learned parameters. [ABSTRACT FROM AUTHOR]