One-sided Lipschitz Condition Research Articles

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.

A Design Framework of Nonlinear H∞PD Observer for One-Sided Lipschitz Singular Systems With Disturbances

This brief addresses the nonlinear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> proportional derivative (PD) observer design problem of one-sided Lipschitz (OSL) singular systems with disturbances. In general, the studied nonlinearity can be regarded as an extension of the traditional Lipschitz restriction and has inherent merits with respect to conservativeness. A novel sufficient condition for the existence of the present observer is given by integrating the concept of quadratic inner-bounded, free-weighting matrix method and OSL condition. For the purpose of observer synthesis, the achieved condition is further converted into the form of linear matrix inequalities (LMIs), by which the gains of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> PD observer can be solved simultaneously. Finally, the performance of theoretical results is shown through an application example of DC motor.

Backward Euler–Maruyama Method for the Random Periodic Solution of a Stochastic Differential Equation with a Monotone Drift

In this paper, we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler–Maruyama method. The existence of the random periodic solution is shown as the limit of the pull-back flows of the SDE and the discretized SDE, respectively. We establish a convergence rate of the strong error for the backward Euler–Maruyama method with order of convergence 1/2.

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.

Robust sliding mode control for stochastic singular Markovian jump systems with unmatched one-sided Lipschitz nonlinearities

This paper investigates sliding mode control of stochastic singular Markovian jump systems with nonlinearity. The unmatched nonlinearity satisfies one-sided Lipschitz condition and quadratically inner-boundedness. In term of a new technical variable transformation, sufficient conditions are developed for nonlinear stochastic singular Markovian jump systems constrained on sliding manifold to guarantee stochastic admissibility and uniqueness of solution based on implicit function theorem. The sliding mode control law by which the trajectories of system can be compelled to the predefined sliding surface in finite time no matter what initial state value is, is synthesized. The derivative singular matrix is fully considered in the whole design process such that the derived conditions can be checked easily.The technical treatment of the nonlinear matrix term avoids the classification discussion of sliding mode controller design. Convex optimization problems subject to linear matrix inequalities are formulated to optimize the desired indexes of interest. Finally, the effectiveness of the proposed approach is illustrated by a numerical example and a practical example.

Cluster synchronization in a network of nonlinear systems with directed topology and competitive relationships

This article studies the cluster synchronization problem of coupled nonlinear systems with directed topology and competitive relationships. We assume that nodes within the same cluster have the same intrinsic dynamics, while the dynamics of nodes belonging to different clusters are different. In the same cluster, there only exist cooperative relationships, and there may have competitive relationships among nodes belonging to different clusters. Under the assumptions that the dynamic of each node satisfies one-sided Lipschitz condition, the digraph of each cluster is strongly connected, and the digraph between different clusters satisfies the cluster-input-equivalence condition, some sufficient conditions for cluster synchronization in the cases of linear coupling and nonlinear coupling are obtained respectively. The obtained conditions are presented as some algebraic conditions which are easy to solve. Finally, the obtained results are validated by two numerical simulations.

Pinning bipartite synchronization for coupled nonlinear systems with antagonistic interactions and time delay

This paper investigates the synchronization problem for coupled nonlinear systems in which antagonistic interactions and bounded time delay coexist. In order to make the network achieve bipartite synchronization, we design a pinning scheme to pin a part of agents such that the network is connected. Under the assumptions that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz and quadratic inner-boundedness conditions, we derive some bipartite synchronization conditions for non-differentiable and differentiable coupling delays respectively. All criteria are presented as linear matrix inequalities. Finally, we present two numerical examples about neural network to illustrate the effectiveness of the obtained results.

Newton’s second law with a semiconvex potential

We make the elementary observation that the differential equation associated with Newton’s second law \(m\ddot{\gamma }(t)=-D V(\gamma (t))\) always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if \(-D V\) satisfies a one-sided Lipschitz condition. We will then build upon this idea to verify the existence of solutions for the Jeans-Vlasov equation, the pressureless Euler equations in one spatial dimension, and the equations of elastodynamics under appropriate semiconvexity assumptions.

One Sided Lipschitz Evolution Inclusions in Banach Spaces

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

Existence and pathwise uniqueness of solutions for stochastic differential equations involving the local time at point zero

In this article, we aim to obtain the existence and pathwise uniqueness of the solution to the one-dimensional stochastic differential equations involving the local time (SDELT) at point zero. The existence and pathwise uniqueness theorem for class of SDELT is established under the drift coefficient satisfies a one-sided Lipschitz condition plus the superlinear condition.

State and disturbance simultaneous estimation for a class of nonlinear time-delay fractional-order systems

This article addresses the problem of estimating simultaneously the state and unknown disturbance of one-sided Lipschitz fractional-order systems with time-delay. The nominal models of nonlinearities are assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition. Different from the state observer reported in the literature, which only dealt with one-sided Lipschitz integer-order time-delay systems or nonlinear fractional-order time-delay systems where the nonlinear function satisfying Lipschitz condition, the state observers in this article can be applied to a wide class of nonlinear time-delay systems (one-sided Lipschitz fractional-order time-delay systems and one-sided Lipschitz integer-order time-delay systems). We employ the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function to derive a sufficient condition for the asymptotic stability of the observer error dynamic system. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved using the MATLAB LMI Control Toolbox. Two examples are provided to show the effectiveness of the proposed design approach.

Optimal Control of Mean Field Equations with Monotone Coefficients and Applications in Neuroscience

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X=X^alpha of the stochastic mean-field type evolution equation in {mathbb {R}}^ddXt=b(t,Xt,L(Xt),αt)dt+σ(t,Xt,L(Xt),αt)dWt,X0∼μ(μgiven),(1)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} dX_t=b(t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dt+\\sigma (t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dW_t\\,, \\quad X_0\\sim \\mu (\\mu \\text { given),}\\qquad (1) \\end{aligned}$$\\end{document}under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control alpha _t is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipschitz condition, and that the dynamics (2) satisfies an almost sure boundedness property of the form pi (X_t)le 0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipschitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control.

Stability analysis of quasi one-sided Lipschitz non-linear multi-agent system via sampled data control subject to directed switching topology

Abstract This paper is concerned with the problem of stability and consensus of non-linear multi-agent system by utilizing the sampled-data control. The innovative part of this paper is that the nonlinearity of this class of nonlinear systems is considered to satisfy a quasi one-sided Lipschitz condition. Communication among agents are assumed to be a switching directed graph. The principle target of this paper is to design a sampled data controller such that for all permissible uncertainties, the resulting closed-loop system is stable in the sense of mean square. For this reason, through the development of an appropriate Lyapunov–Krasovskii functional with dual integral terms and usage of Kronecker product properties alongside the matrix inequality techniques, a new set of stability and consensus conditions for the prescribed system is obtained in the form of a linear matrix inequality, which can be easily solved by the well-known effective numerical programming. Finally numerical examples are given to show the validity of the proposed hypothetical results.

A novel approach for event-triggered state-estimation of one-sided Lipschitz systems for efficient bandwidth utilization

This paper proposes a robust event-triggered observer design scheme for the systems fulfilling the one-sided Lipschitz and quadratic inner-boundedness conditions. The suggested observer scheme differs from the traditional methods because it deals with a less conservative and a more generalized class of nonlinear systems while utilizing fewer resources and meeting performance requirements. In contrast to the conventional Lipschitz methods, the proposed method considers a broader class of OSL systems, and it can avoid a high-gain observer, which can be sensitive to the measurement noise. The proposed linear matrix inequalities-based observer design guarantees the uniformly ultimately bounded convergence of the observation error with an exponential decay under disturbances. Also, several scenarios selecting a threshold for event-triggering schemes from an observer design perspective are addressed along with their advantages and issues. Simulation results are provided to validate the established results.