One-sided Lipschitz Condition Research Articles

Observer-Based Distributed Adaptive Consensus Tracking of Nonlinear Multi-agent Systems on Directed Graphs

This paper studies the leader-following consensus problem of nonlinear multi-agent systems (MAS) with one-sided Lipschitz conditions under the directed graph, and a novel observer-based distributed tracking protocol that only utilizes the relative output information of adjacent agents is proposed. To avoid the disadvantage of solving Laplace matrix eigenvalues, a fully distributed adaptive consensus scheme is constructed. The sufficient conditions for linear matrix inequalities (LMIs) for the consensus of MAS systems are derived using the Lyapunov function and the properties of one-sided Lipschitz nonlinear systems. Finally, the effectiveness of the proposed control method can be verified by two examples-based simulations with a leader and five followers.

Observer-Based Adaptive Consensus Protocol Design for Multi-Agent Systems with One-Sided Lipschitz Nonlinearity

This paper considers the observer-based leader–follower consensus problem of multi-agent systems with one-sided Lipschitz conditions and quadratic inner-boundedness nonlinearity. Based on the relative outputs of neighboring agents, an adaptive protocol over a directed graph is designed via assigning a time-varying coupling weight to each node. Compared with the existing protocols, the proposed protocol can not only be utilized in the unidirectional flow of information, but it does not require any global graph information for its design. It has been shown that the established criteria can not only make the observer error tendency zero, but one can also achieve the leader–follower consensus of such nonlinear multi-agent systems. Furthermore, the gains of observer and protocol parameters can be constructed by solving the linear matrix inequalities (LMIs), which are conveniently checked. The results are illustrated by numerical simulations.

Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given.

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Cluster Synchronization of a Nonlinear Network With Fixed and Switching Topologies

This article mainly addresses the synchronization problem of leaderless and leader–follower clusters in directed topologically coupled nonlinear systems. The relationship between nodes within each cluster is cooperative, and nodes belonging to different clusters may compete with each other. For each case of leaderless and leader-following, we consider both fixed and switching topologies. The vector field at each node satisfies the one-sided Lipschitz condition. For situations where there is no leader and leader to follow, we do not need to use the in-degree balanced condition that is often required in most existing literature. Under the leaderless framework, the cluster synchronization (CS) problem is transformed into a stability problem through variable transformation. Under the leader-following framework, by equipping each cluster with a virtual leader, we design a new class of leader-following protocols that can be used to achieve CS of coupled nonlinear systems. Finally, two numerical examples are provided to illustrate the validity of the obtained results.

Euler scheme for approximation of solution of nonlinear ODEs under inexact information

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local Hölder continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.

Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H1 stability, higher moment H1 stability, L2 stability, and higher moment L2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

State and output feedback’s finite-time guaranteed cost [formula omitted] control for uncertain nonlinear stochastic systems with time-varying delays

The problems of guaranteed cost control and H∞ control are studied in this paper for nonlinear uncertain Itô’s stochastic systems with time-varying delays in finite-time. It is assumed that the nonlinear function in the systems satisfies the quasi-one-sided Lipschitz nonlinear constraint condition, which is weaker and less conservative than the one-sided Lipschitz condition. Some novel sufficient conditions for the existence of guaranteed cost control strategies are derived by using the Gronwall’s inequality, Dynkin’s formula, and Lyapunov–Krasovskii functional. The state and output feedback’s controllers based on event-triggered control guarantee that the closed-loop system is finite-time stochastically bounded and the given quadratic cost function has an upper bound. Finally, the effectiveness of the suggested method is illustrated by numerical simulation.

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

Euler simulation of interacting particle systems and McKean–Vlasov SDEs with fully super-linear growth drifts in space and interaction

Abstract This work addresses the convergence of a split-step Euler type scheme (SSM) for the numerical simulation of interacting particle Stochastic Differential Equation (SDE) systems and McKean–Vlasov stochastic differential equations (MV-SDEs) with full super-linear growth in the spatial and the interaction component in the drift, and nonconstant Lipschitz diffusion coefficient. Super-linearity is understood in the sense that functions are assumed to behave polynomially, but also satisfy a so-called one-sided Lipschitz condition. The super-linear growth in the interaction (or measure) component stems from convolution operations with super-linear growth functions, allowing in particular application to the granular media equation with multi-well confining potentials. From a methodological point of view, we avoid altogether functional inequality arguments (as we allow for nonconstant nonbounded diffusion maps). The scheme attains, in stepsize, a near-optimal classical (path-space) root mean-square error rate of $1/2-\varepsilon $ for $\varepsilon>0$ and an optimal rate $1/2$ in the nonpath-space (pointwise) mean-square error metric. All findings are illustrated by numerical examples. In particular, the testing raises doubts if taming is a suitable methodology for this type of problem (with convolution terms and nonconstant diffusion coefficients).

Consensus of multi-Agent systems with one-Sided lipschitz nonlinearity via nonidentical double event-Triggered control subject to deception attacks

As for the multi-agent systems (MASs) with time-varying switching subject to deception attacks, the leader-following consensus problem is studied in this article. The one-sided Lipschitz (OSL) condition is utilized for the nonlinear functions, which makes the results more general and relaxed than those obtained by Lipschitz condition. The nonidentical double event-triggering mechanisms (ETMs) are adopted for only a fraction of agents, and each agent transmits the data according to its own necessity. Semi-Markov process modeling with time-varying switching probability is adopted for switching topology and deception attacks occurring in transmission channel are considered. By using the cumulative distribution function (CDF) and the linear matrix inequality (LMI) technology, sufficient conditions for MASs to achieve consensus in mean square are obtained. An effective algorithm is presented to obtain the event-based control gains. The merits of the proposed control scheme are demonstrated via a simulation example.

The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero

Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.

Observer-based H∞ control for uncertain one-sided Lipschitz Markovian jump-delayed systems with partially unknown transition rates

This paper addresses the observer-based control design for uncertain one-sided Lipschitz Markovian jump-delayed systems. By constructing the Lyapunov functional of the closed-loop augmented system, combining with the one-sided Lipschitz condition and quadratically inner-bounded condition, it is derived that the closed-loop augmented system is stochastic stable with an [Formula: see text] performance level [Formula: see text]. With the help of some special derivations, bilinear matrix inequalities are successfully transformed into a group of linear matrix inequalities. The robust controller and observer gains are solved by linear matrix inequality (LMI) toolbox in MATLAB.

Nontrivial solutions for a fourth-order Riemann-Stieltjes integral boundary value problem

<abstract><p>In this paper we study a fourth-order differential equation with Riemann-Stieltjes integral boundary conditions. We consider two cases, namely when the nonlinearity satisfies superlinear growth conditions (we use topological degree to obtain an existence theorem on nontrivial solutions), when the nonlinearity satisfies a one-sided Lipschitz condition (we use the method of upper-lower solutions to obtain extremal solutions).</p></abstract>

The tamed Euler–Maruyama approximation of Mckean–Vlasov stochastic differential equations and asymptotic error analysis

By a semi-discrete tamed Euler approximation as a bridge, this paper examines the tamed Euler scheme for the Mckean-Vlasov stochastic differential equations (SDEs) and obtains the convergence rate under the one-sided Lipschitz condition for the drift coefficient. Under the local Lipschitz condition for the drift coefficient, the strong convergence of the tamed scheme is also examined. When the coefficients satisfy the Lipschitz condition, this paper also obtains the asymptotic error distribution.

The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

We consider the follow-the-leader particle approximation scheme for a [Formula: see text] scalar conservation law with non-negative compactly supported [Formula: see text] initial datum and with a [Formula: see text] concave flux, which is known to provide convergence towards the entropy solution [Formula: see text] to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density [Formula: see text] is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on [Formula: see text] (which imply concavity of [Formula: see text]) we prove that the continuum version [Formula: see text] of said condition allows to select a unique weak solution, despite [Formula: see text] is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition [Formula: see text]. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case [Formula: see text] the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

In this article, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _\infty$</tex-math></inline-formula> norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input-state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.

On consensus control of nonlinear multiagent systems satisfying incremental quadratic constraints

In this paper, we investigate the problem of leaderless consensus control for the multiagent systems whose nonlinear dynamics satisfying incremental quadratic constraints. A distributed dynamic consensus protocol, decided by communication among neighboring agents, is presented to render nonlinear agent consensus with appropriate coupling weights. Next, an observer-based distributed protocol is considered to ensure consensus of nonlinear system without knowing full state information. Further, extensions to consensus strategies with nonlinear dynamics for the leader-following fashion are also addressed. By comparison to the traditional nonlinear consensus control methodologies, the proposed approach generalizes the Lipschitz nonlinearity as well as the combined nonlinearity of one-sided Lipschitz condition and quadratic inner-boundness condition towards a more generalized type of nonlinearity, which shows us a less conservative result in the Lyapunov proof. Finally, the numerical simulations for six agents are illustrated to show the feasibility and performance of the proposed control protocol with or without the presence of the observer.

The numerical algorithm for stochastic age-dependent population system with Lévy noise in a polluted environment

Environmental pollution has significant effects on population systems worldwide. In this study, we established a numerical algorithm for modelling stochastic age-dependent population systems driven by Lévy noise in a polluted environment. A modified explicit Euler–Maruyama (EM) discretization scheme was employed for ensuring the positiveness of the numerical solution used for modelling the stochastic age-dependent population hybrid system with Lévy noise in a polluted environment. The convergence of the numerical solution was confirmed by It's formula and a projection operator under the condition that the drift coefficient satisfies the local Lipschitz and global one-sided Lipschitz conditions. The theoretical results were finally verified with a numerical example.

Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps

The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.