Exponential Convergence Speed Research Articles

Equilibrium States of Endomorphisms of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k}$\\end{document}: Spectral Stability and Limit Theorems

We establish the existence of a spectral gap for the transfer operator induced on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k} = \\mathbb{P}^{k} (\\mathbb{C})$\\end{document} by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Distributed Dynamic Event-Triggered Control to Leader-Following Consensus of Nonlinear Multi-Agent Systems with Directed Graphs.

This paper investigates achieving leader-following consensus in a class of multi-agent systems with nonlinear dynamics. Initially, it introduces a dynamic event-triggered strategy designed to effectively alleviate the strain on the system's communication resources. Subsequently, a distributed control strategy is proposed and implemented in the nonlinear leader-follower system using the dynamic event-triggered mechanism, aiming to ensure synchronization across all nodes at an exponential convergence speed. Thirdly, the research shows that under the dynamic event-triggered strategy the minimum event interval of any two consecutive triggers guarantees the elimination of Zeno behavior. Lastly, the validity of the calculation results is verified by a simulation example.

Concurrent Learning-Based Adaptive Control of Underactuated Robotic Systems With Guaranteed Transient Performance for Both Actuated and Unactuated Motions.

With the wide applications of underactuated robotic systems, more complex tasks and higher safety demands are put forward. However, it is still an open issue to utilize "fewer" control inputs to satisfy control accuracy and transient performance with theoretical and practical guarantee, especially for unactuated variables. To this end, for underactuated robotic systems, this article designs an adaptive tracking controller to realize exponential convergence results, rather than only asymptotic stability or boundedness; meanwhile, unactuated states exponentially converge to a small enough bound, which is adjustable by control gains. The maximum motion ranges and convergence speed of all variables both exhibit satisfactory performance with higher safety and efficiency. Here, a data-driven concurrent learning (CL) method is proposed to compensate for unknown dynamics/disturbances and improve the estimate accuracy of parameters/weights, without the need for persistency of excitation or linear parametrization (LP) conditions. Then, a disturbance judgment mechanism is utilized to eliminate the detrimental impacts of external disturbances. As far as we know, for general underactuated systems with uncertainties/disturbances, it is the first time to theoretically and practically ensure transient performance and exponential convergence speed for unactuated states, and simultaneously obtain the exponential tracking result of actuated motions. Both theoretical analysis and hardware experiment results illustrate the effectiveness of the designed controller.

The Fractional Laplacian with Reflections

AbstractMotivated by the notion of isotropic $$\alpha $$ α -stable Lévy processes confined, by reflections, to a bounded open Lipschitz set $$D\subset \mathbb {R}^d$$ D ⊂ R d , we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time.

Cooperative concurrent targeting for planar arrays of point sources

Motivated by practical applications in satellite formations and directional antenna arrays, the problem of targeting a planar array of point sources at one common object of interest is proposed and then solved using novel distributed coordination strategies. These point sources are arbitrarily located on the x–y plane and the boundary point sources, whose either x or y coordinate is extremal, already orient to the non-coplanar target point. The only global information shared among the remaining point sources is the positive directions of the global coordinate axes x and y, and hence, they have to rely on sensing the changes of the orientation angles of their nearest neighbors to adjust their own orientation. We prove that under our control law, the orientation lines will asymptotically intersect at the same point of concurrency as the boundary point sources. The main idea behind the designed control law is the intuitive argument from Euclidean geometry that reducing the differences between the distances to the x–y plane of the pairwise intersection points of the orientation lines leads to realizing concurrent targeting. We further show the boundedness and exponential convergence speed of the orientation angles.

Generalized Max-Weight Policies in Stochastic Matching

We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, that is, leave the system before being matched. Using specific properties of the max-weight policy, we construct a simple quadratic Lyapunov function. This allows us to establish stability results, to prove exponential convergence speed toward the stationary measure, and to give explicit bounds for the stationary mean of the largest queue size in the system. We finally illustrate some of these results using simulations on toy examples. Funding: N. Soprano-Loto is partially supported by Aristas, MathAmSud [Grant RSPSM (Random Structures and Processes in Statistical Mechanics)], and ANR (Agence Nationale de la Recherche) [Grant PRC (Projets de Recherche Collaborative) MATCHES (CE40)].

A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

An exponentially convergent distributed algorithm for resource allocation problem

AbstractIn this paper, the distributed resource allocation problem over undirected networks, considering box constraints and demand–supply constraints, is studied. Furthermore, each agent has one private cost function that is only known by itself. The aim is to minimize the sum of all local cost functions in a distributed manner while meeting the constraints. To solve the dispatch problem, a continuous‐time distributed algorithm with fixed step‐size is proposed. After that, a modified distributed algorithm of discrete‐time multi‐agent system is provided. Moreover, the proposed algorithms obtain the global optimal solution with an exponential convergence speed by the conventional Lyapunov methods. Finally, several simulations of IEEE‐14 bus are presented to validate and illustrate the proposed designed algorithm.

On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors

This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [6]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [6], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases that were not covered previously in the literature.

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely $$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$ with $0 < a_{{+}} < a_{{-}}$. We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

Hyperbolic components of rational maps: Quantitative equidistribution and counting

Let $\Lambda$ be a quasi-projective variety and assume that, either $\Lambda$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $\Lambda$ parametrizes an algebraic family $(f_\lambda)_{\lambda\in\Lambda}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $\log^+$ of the modulus of the multipliers of periodic points.

A distributed cooperative learning algorithm based on Zero-Gradient-Sum strategy using Radial Basis Function Network

Abstract This paper aims to propose a distributed cooperative learning (DCL) algorithm for Radial Basis Function Network (RBFN), where training samples are often extremely large-scale, high-dimensional and located on distributed nodes over undirected and connected networks. The algorithm is investigated for RBFN with different basic functions, such as Gaussian function, multi-quadric function, thin plate spline function and so on. The proposed DCL–RBFN algorithm is based on Zero-Gradient-Sum (ZGS) distributed optimization strategy and works in a fully distributed fashion, which has no requirement on central node and communication of raw data from each node during the training process. Moreover, in order to avoid the waste of unnecessary network resources, an event-triggered communication mechanism is applied to our algorithm. Each node exchanges learned information with its neighboring nodes using the ZGS strategy under event-triggered mechanism. The convergence of the proposed DCL–RBFN and event-triggered DCL–RBFN algorithms are presented by using the Lyapunov method. It is proved that the proposed algorithms have exponential convergence speed and require less communication data over networks. Finally, a series of numerical simulations are given to verify the advantages of the proposed algorithms.

Repeated Bidding Games with Incomplete Information and Bounded Values: On the Exponential Speed of Convergence

We consider the repeated zero-sum bidding game with incomplete information on one side with non-normalized total payoff. De Meyer and Marino [(2005) Continuous versus discrete market game, Cowles Foundation Discussion Paper 1535] and Domansky and Kreps [(2005) Repeated games with asymmetric information and random price fluctuations at finance markets, Proc. Appl. Ind. Math. 12(4), 950–952 (in Russian)] investigated a game [Formula: see text] modeling multistage bidding with asymmetrically informed agents and proved that for this game [Formula: see text] converges to a finite limit [Formula: see text], i.e., the error term is [Formula: see text]. In this paper, we show that for this example [Formula: see text] converges to the limit exponentially fast. For this purpose we apply the optimal strategy [Formula: see text] of insider in the infinite-stage game obtained by Domansky [(2007) Repeated games with asymmetric information and random price fluctuations at finance markets, Int. J. Game Theor. 36(2), 241–257] to the [Formula: see text]-stage game and deduce that it is [Formula: see text]-optimal with [Formula: see text] exponentially small.

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d − 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d − 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d − 1) multipliers outside a pluripolar set.

Distribution of postcritically finite polynomials II: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d -- 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d -- 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d -- 1) multipliers outside a pluripolar set.

A Distributed Gauss-Newton Methodfor Power System State Estimation

We propose a fully distributed Gauss-Newton algorithm for state estimation of electric power systems. At each Gauss-Newton iteration, matrix-splitting techniques are utilized to carry out the matrix inversion needed for calculating the Gauss-Newton step in a distributed fashion. In order to reduce the communication burden as well as increase robustness of state estimation, the proposed distributed scheme relies only on local information and a limited amount of information from neighboring areas. The matrix-splitting scheme is designed to calculate the Gauss-Newton step with exponential convergence speed. The effectiveness of the method is demonstrated in various numerical experiments.

Distributed impulsive control for islanded microgrids with variable communication delays

This study develops a new droop-based secondary distributed control strategy with hybrid dynamical form for the islanded microgrids (MGs). Two impulsive control algorithms containing time-varying delays are designed, under which the frequency and voltage restoration and active power sharing accuracy of all distributed generators can be realised with an exponential convergence speed. Compared to the traditional continuous or discrete-time distributed algorithms, the designed impulsive controllers formulate the controlled MG to be a hybrid dynamics, which are more general and applicable for practical power systems. Moreover, some delay-dependent sufficient conditions on the requirements for sparse communication network and impulsive time intervals are derived, in which the time delays do not need to be less than the length of the impulsive intervals. Simulation results on the control parameter analysis, and communication network topologies validates the developed theories.

Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak–Keller–Segel model

This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic–elliptic semilinear Patlak–Keller–Segel system in a ball of RN, exhibiting a critical mass phenomenon for N≥2. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case. We stress that this covers in particular the classical Keller–Segel system with N=2, well known for its critical mass 8π, and that the result improves on the known results even for this most studied problem. A key tool is an associated one-dimensional degenerate parabolic problem. The proof exploits its formal gradient flow structure ut=−∇F[u(t)] on an “infinite dimensional Riemannian manifold”. In particular, we show a new Hardy type inequality, equivalent to the strict convexity of F at any steady state of subcritical mass, which heuristically explains the exponential speed of convergence.

A reduced basis method for microwave semiconductor devices with geometric variations

Purpose – The Reduced Basis Method (RBM) generates low-order models of parametrized PDEs to allow for efficient evaluation of parametrized models in many-query and real-time contexts. The purpose of this paper is to investigate the performance of the RBM in microwave semiconductor devices, governed by Maxwell's equations. Design/methodology/approach – The paper shows the theoretical framework in which the RBM is applied to Maxwell's equations and present numerical results for model reduction under geometry variation. Findings – The RBM reduces model order by a factor of $1,000 and more with guaranteed error bounds. Originality/value – Exponential convergence speed can be observed by numerical experiments, which makes the RBM a suitable method for parametric model reduction (PMOR).

Distributed Concurrent Targeting for Linear Arrays of Point Sources

Motivated by practical applications in satellite formations and antenna arrays etc., the problem of targeting a linear array of point sources at one common point of interest is formulated and then solved using a novel distributed control strategy. These point sources are located collinearly and the two at the two ends orient readily to the target point. The other point sources have to rely on sensing the changes of the relative orientation angles of their nearest neighbors to adjust their own orientations; we rigorously prove that under our control law using only local information, their orientation lines will intersect at the same point of concurrency as the two point sources at the two ends. The crucial idea behind our designed control law is the intuitive argument from plane geometry that reducing the differences between the distances to the baseline of the pairwise intersection points of the orientation lines helps realizing concurrent targeting. This idea is further utilized to construct the key argument in our analysis about the boundedness and the exponential convergence speed of the orientation angles. Numerical simulations are used to validate the effectiveness of our control strategy.