Exponential Convergence Rate Research Articles

The initial boundary value problem for the Vlasov–Poisson–Fokker–Planck system

This paper is devoted to the initial boundary value problem of the Vlasov–Poisson–Fokker–Planck system in a bounded domain. The global existence of a unique strong solution and its exponential convergence rate to the equilibrium state are proved under the specular reflection boundary condition. The proof is based on an elementary energy method and the SLp theory developed by Guo et al. [Arch. Ration. Mech. Anal. 236(3), 1389–1454 (2020)] for kinetic models.

Spill-Free Transfer and Stabilization of Viscous Liquid

This article studies the feedback stabilization problem of the motion of a tank that contains an incompressible, Newtonian, viscous liquid. The control input is the force applied on the tank and the overall system consists of two nonlinear partial differential equations and two ordinary differential equations. Moreover, a spill-free condition is required to hold. By applying the control Lyapunov functional methodology, a set of initial conditions (state space) is determined for which spill-free motion of the liquid is possible by applying an appropriate control input. Semi-global stabilization of the liquid and the tank by means of a simple feedback law is achieved, in the sense that for every closed subset of the state space, it is possible to find appropriate controller gains, so that every solution of the closed-loop system initiated from the given closed subset satisfies specific stability estimates. The closed-loop system exhibits an exponential convergence rate to the desired equilibrium point. The proposed stabilizing feedback law does not require measurement of the liquid level and velocity profiles inside the tank and simply requires measurements of 1) the tank position error and tank velocity, 2) the total momentum of the liquid, and 3) the liquid levels at the tank walls. The obtained results allow an algorithmic solution of the problem of the spill-free movement and slosh-free settlement of a liquid in a vessel of limited height (such as water in a glass) by a robot to a prespecified position, no matter how full the vessel is.

A Lyapunov Approach to Consensus of Linear Multi-Agent Systems with Infinite Distributed Communication Delays

This paper considers the leaderless consensus problem of linear time-invariant multi-agent systems with infinite distributed communication delays. A novel distributed low gain controller is proposed based on the solution to a parametric algebraic Riccati equation. It is shown via the newly developed Lyapunov-like method that not only the consensus of linear time-invariant multi-agent systems can be achieved exponentially under some mild assumptions but also an estimate of the exponential convergence rate of consensus is given in this work. The Lyapunov-like method is also extended to handle a special case of linear time-varying multi-agent systems. In addition, the obtained results include the results on the leaderless consensus of linear multi-agent systems with bounded distributed communication delays as special cases. To the best of our knowledge, this is the first work that develops the Lyapunov-like method for the leaderless consensus problems of both time-invariant and time-varying linear multi-agent systems with infinite distributed communication delays. Finally, a numerical example is presented to illustrate the effectiveness of the proposed controller.

Global stability for a SEIQR worm propagation model in mobile internet

Abstract Recently, propagation models of worms in the mobile environment are drawing extensive attention, particularly in the Wi-Fi scenario. Considering that worm-free equilibrium is exponential convergent means that the propagation time and control time of worms are much shorter than for other asymptotic convergence. Besides, the global asymptotic stability of the endemic equilibrium is more important than the local asymptotic stability, which reflects the more global qualitative behavior of the worm propagation. In this paper, we discuss the global dynamics of SEIQR worm propagation model in mobile internet proposed by Xiao et al. [X. Xiao, P. Fu, C. Dou, Q. Li, G. Hu, and S. Xia, “Design and analysis of SEIQR worm propagation model in mobile internet,” Commun. Nonlinear Sci. Numer. Simulat., vol. 43, pp. 341–350, 2017] to improve and complement the related results. Through a series of mathematical derivations, sufficient conditions are derived to ensure the global exponentially stability of worm-free equilibrium, and the exponential convergent rate can be unveiled. Then, by using the classical geometric approach, it is shown that the endemic equilibrium is globally asymptotically stable and the system is persistent when R 0 > 1. Moreover, numerical simulations are given to demonstrate our theoretical results.

Resilient multi-dimensional consensus in adversarial environment

This paper considers the multi-dimensional consensus in networked systems, where some of the agents might be misbehaving (or faulty). Despite the influence of these misbehaviors, the benign agents aim to reach an agreement while avoiding being seriously influenced by the faulty ones. To this end, this paper first considers a general class of consensus algorithms, where each benign agent computes an “auxiliary point” based on the received values and moves its state towards this point. Concerning this generic form, we present conditions for achieving resilient consensus and obtain a lower bound on the exponential convergence rate. Assuming that the number of malicious agents is upper bounded, two specific resilient consensus algorithms are further developed based on the obtained conditions. Particularly, the first solution, based on Helly’s Theorem, achieves the consensus within the convex hull formed by the benign agents’ initial states, where the auxiliary point can be efficiently computed through linear programming. On the other hand, the second algorithm serves as a “built-in” security guarantee for standard average consensus algorithms, in the sense that its performance coincides exactly with that of the standard ones in the absence of faulty nodes while also resisting the serious influence of the misbehaving ones in adversarial environment. Some numerical examples are provided in the end to verify the theoretical results.

An iterative framework to solve nonlinear optimal control with proportional delay using successive convexification and symplectic multi-interval pseudospectral scheme

In this paper, we propose an iterative framework to solve optimal control for nonlinear proportional state-delay systems. The successive convexification technique is first implemented to convert the original nonlinear problem into a sequence of linear-quadratic problems. And a symplectic pseudospectral method, where the multi-interval pseudospectral scheme is applied with a proportional mesh, to solve the transformed problems is then developed based on the first-order necessary conditions. Each linear-quadratic problem is finally transformed into a system of linear algebraic equations with a sparse coefficient matrix. Due to the benefit of the successive convexification technique and the multi-interval pseudospectral method, initial guess on costate variables is avoided and converged solutions can be obtained with an exponential convergent rate. The proposed iterative framework is validated by four examples with distinct features, highlighting its numerical precision and efficiency. And either exponential or linear convergence property can be exhibited by tuning the approximation degree or the mesh number.

Birth and death processes in interactive random environments

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results a coupling method turns out to be useful.

The pressureless damped Euler–Riesz equations

In this paper, we analyze the pressureless damped Euler–Riesz equations posed in either \mathbb{R}^d or \mathbb{T}^d . We construct the global-in-time existence and uniqueness of classical solutions for the system around a constant background state.We also establish large-time behaviors of classical solutions showing the solutions towards the equilibrium as time goes to infinity. For the whole space case, we first show an algebraic decay rate of solutions under additional assumptions on the initial data compared to the existence theory. We then refine the argument to have an exponential decay rate of convergence even in the whole space. In the case of the periodic domain, without any further regularity assumptions on the initial data, we provide the exponential convergence of solutions.

Body-Ordered Approximations of Atomic Properties

We show that the local density of states (LDOS) of a wide class of tight-binding models has a weak body-order expansion. Specifically, we prove that the resulting body-order expansion for analytic observables such as the electron density or the energy has an exponential rate of convergence both at finite Fermi-temperature as well as for insulators at zero Fermi-temperature. We discuss potential consequences of this observation for modelling the potential energy landscape, as well as for solving the electronic structure problem.

Error estimates for discrete generalized FEMs with locally optimal spectral approximations

This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation spaces are constructed using eigenvectors of local eigenvalue problems solved by the finite element method on some sufficiently fine mesh with mesh size h h . The error bound of the discrete GFEM approximation is proved to converge as h → 0 h\rightarrow 0 towards that of the continuous GFEM approximation, which was shown to decay nearly exponentially in previous works. Moreover, even for fixed mesh size h h , a nearly exponential rate of convergence of the local approximation errors with respect to the dimension of the local spaces is established. An efficient and accurate method for solving the discrete eigenvalue problems is proposed by incorporating the discrete A A -harmonic constraint directly into the eigensolver. Numerical experiments are carried out to confirm the theoretical results and to demonstrate the effectiveness of the method.

Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

Yet Another Computation-Oriented Necessary and Sufficient Condition for Stabilizability of Switched Linear Systems

This article presents a computational method to test the stabilizability of discrete-time switched linear systems. The existence of a conic cover of the space on whose elements a convex condition holds is proved to be necessary and sufficient for stabilizability. An algorithm for computing a conic partition that satisfies the new necessary and sufficient condition is given. The algorithm, which allows us to determine bounds on the exponential convergence rate, is proved to overcome the conservatism of conditions equivalent to periodic stabilizability and is applied to a 4-D system.

Multirobot System Formation Control With Multiple Performance and Feasibility Constraints

In this work, we propose a novel framework to address the formation control problem for a class of multirobot systems with two types of constraints, namely the performance constraints and the feasibility constraints. For the performance constraints, we consider the constraint requirements on the distance tracking errors between the real and the desired trajectories for each robot, so that to ensure <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">precise</i> tracking of the robot without deviating too much from its desired trajectory, as well as the constraints on the interrobot distance, so that to ensure the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safe</i> operation of the team. For the feasibility constraints, we consider the constraints on the heading angle, so that the controllers designed in the brief are feasible. Universal barrier functions are adopted in the controller design and analysis, which is a generic framework that can address systems with different types of constraints in a unified controller architecture. Through rigorous analysis, exponential convergence rate can be guaranteed on the distance tracking errors, while all constraints are satisfied during the operation. A simulation example and an experiment using three AmigoBot mobile robots further demonstrate the efficacy of the proposed control framework.

Distributed resource allocation algorithm for second-order multi-agent systems with external disturbances

This paper focuses on the distributed resource allocation problem of a second-order multi-agent system with external disturbances. The global network resource equality constraint is considered. A distributed algorithm with an exponential convergence rate is proposed based on an internal model for a known bounded disturbance. By using the active disturbance approach to design a reduced-order extended state observer, the proposed algorithm is modified to reject an unknown external disturbance. Moreover, exponential convergence is achieved through the Lyapunov theory when the disturbance is constant. When the differential of the disturbance is bounded, by adjusting the control parameter of the observer, the proposed algorithm converges to an arbitrarily small neighbourhood of the optimal solution. Finally, our simulation results demonstrate the effectiveness of our proposed algorithm.

Differentially private distributed algorithms for stochastic aggregative games

Designing privacy-preserving distributed algorithms for stochastic aggregative games is urgent due to the privacy issues caused by information exchange between players. This paper proposes two differentially private distributed algorithms seeking the Nash equilibrium in stochastic aggregative games. By adding time-varying random noises, the input and output-perturbation methods are given to protect each player’s sensitive information. For the case of output-perturbation, utilizing mini-batch methods, the algorithm’s mean square error is inversely proportional to the privacy level ε and the number of samples. For the case of input-perturbation, a differentially private distributed stochastic approximation-type algorithm is developed to achieve almost sure convergence and (ε,δ)-differential privacy. Under suitable consensus time conditions, the algorithm’s convergence rate is rigorously presented for the first time, where the optimal convergence rate O(1/k) in a mean square sense is obtained. Then, utilizing mini-batch methods, the influence of added privacy noise on the algorithm’s performance is reduced, and the convergence rate of the algorithm is improved. Specifically, when the batch sizes and the number of consensus times at each iteration grow at a suitable rate, an exponential rate of convergence can be achieved with the same privacy level. Finally, a simulation example demonstrates the algorithms’ effectiveness.

Coordinate-Free Circumnavigation of a Moving Target Via a PD-Like Controller

This article proposes a coordinate-free controller for a nonholonomic vehicle to circumnavigate a fully- actuated moving target by using range-only measurements. If the range rate is available, our proportional derivative (PD) like controller has a simple structure as the standard PD controller, except the design of an additive constant bias and a saturation function in the error feedback. We show that if the target is stationary, the vehicle asymptotically encloses the target with a predefined radius at an exponential convergence rate, i.e., an exact circumnavigation pattern can be completed. For a moving target, the circumnavigation error converges to a small region whose size is shown proportional to the maneuverability of the target, e.g., the maximum linear speed and acceleration. Moreover, we design a second-order sliding mode (SOSM) filter to estimate the range rate and show that the SOSM filter can recover the range rate in a finite time. Finally, the effectiveness and advantages of our controller are validated via both numerical simulations and real experiments.

Continuous-Time Convergence Rates in Potential and Monotone Games

In this paper, we provide exponential rates of convergence to the interior Nash equilibrium for continuous-time dual-space game dynamics such as mirror descent (MD) and actor-critic (AC). We perform our analysis in $N$-player continuous concave games that satisfy certain monotonicity assumptions while possibly also admitting potential functions. In the first part of this paper, we provide a novel relative characterization of monotone games and show that MD and its discounted version converge with $\mathcal{O}(e^{-\beta t})$ in relatively strongly and relatively hypomonotone games, respectively. In the second part of this paper, we specialize our results to games that admit a relatively strongly concave potential and show that AC converges with $\mathcal{O}(e^{-\beta t})$. These rates extend their known convergence conditions. Simulations are performed which empirically back up our results.

Exponential Continuous Non-Parametric Neural Identifier With Predefined Convergence Velocity

This paper addresses the design of an exponential function-based learning law for artificial neural networks (ANNs) with continuous dynamics. The ANN structure is used to obtain a non-parametric model of systems with uncertainties, which are described by a set of nonlinear ordinary differential equations. Two novel adaptive algorithms with predefined exponential convergence rate adjust the weights of the ANN. The first algorithm includes an adaptive gain depending on the identification error which accelerated the convergence of the weights and promotes a faster convergence between the states of the uncertain system and the trajectories of the neural identifier. The second approach uses a time-dependent sigmoidal gain that forces the convergence of the identification error to an invariant set characterized by an ellipsoid. The generalized volume of this ellipsoid depends on the upper bounds of uncertainties, perturbations and modeling errors. The application of the invariant ellipsoid method yields to obtain an algorithm to reduce the volume of the convergence region for the identification error. Both adaptive algorithms are derived from the application of a non-standard exponential dependent function and an associated controlled Lyapunov function. Numerical examples demonstrate the improvements enforced by the algorithms introduced in this study by comparing the convergence settings concerning classical schemes with non-exponential continuous learning methods. The proposed identifiers overcome the results of the classical identifier achieving a faster convergence to an invariant set of smaller dimensions.

Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation

The present work deals with the global solvability as well as asymptotic analysis of the stochastic generalized Burgers–Huxley (SGBH) equation perturbed by a white-in-time and correlated-in-space noise defined in a bounded interval of $${\mathbb {R}}$$ . We first prove the existence of a unique mild as well as strong solution to the SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of the SGBH equation, that is, irreducibility and the strong Feller property. These two properties guarantee the uniqueness of invariant measures and ergodicity also. Then, under further assumptions on the noise coefficient, we discuss the ergodic behavior of the solution of the SGBH equation by providing a large deviation principle for the occupation measure for large time (Donsker–Varadhan), which describes the exact rate of exponential convergence.

Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.