One-sided Lipschitz Condition Research Articles

Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which both the drift and the diffusion coefficients do not satisfy the global Lipschitz and linear growth conditions, especially the diffusion coefficients are highly non-linear growing. It is proved that the split-step theta (SST) method with θ∈[12,1] is strongly convergent to SDEPCAs under the local Lipschitz, monotone and one-sided Lipschitz conditions. It is also obtained that the SST method with θ∈(12,1] preserves the exponential mean square stability of SDEPCAs under the monotone condition and some condition on the step-size. Without any restriction on the step-size, there exists θ∗∈(12,1] such that the SST method with θ∈(θ∗,1] is exponentially stable in mean square. Moreover, for sufficiently small step-size, the rate constant can be reproduced. Some numerical simulations are presented to illustrate the analytical theory.

Strong completeness and semi-flows for stochastic differential equations with monotone drift

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong Δ-completeness which means that there exists a modification of the solution which if restricted to any set A⊂Rd of dimension Δ is almost surely continuous with respect to the initial condition.

Bipartite synchronization in a network of nonlinear systems: A contraction approach

This paper studies the bipartite synchronization in a network of nonlinear systems with collaborative and antagonistic interactions. Under the assumption that the signed graph is structurally balanced and the considered domain does not contain the origin, we use contraction theory to obtain some sufficient conditions such that the network admits a bipartite synchronization solution. These conditions are described by coupling matrices and the contractivity of lower-dimensional dynamic systems. In particular, if the nonlinear system satisfies a one-sided Lipschitz condition and the coupling matrices are identical, we also obtain some sufficient conditions about the second smallest eigenvalue of signed graph for the bipartite synchronization. Some numerical examples are presented to illustrate the effectiveness of the obtained results.

Practical bipartite synchronization via pinning control on a network of nonlinear agents with antagonistic interactions

This paper studies the synchronization phenomena in a network of heterogeneous nonlinear systems over signed graph, which can be considered as the perturbation of a network of homogeneous nonlinear systems. Assume that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz condition, and a leader pins a subset of agents. Under some proper initial conditions of the leader, we derive some conditions under which bipartite synchronization error can be kept arbitrary small by choosing a proper pining scheme. This property is called practical pinning bipartite synchronization as an alternative synchronization notion for network of heterogeneous nonlinear systems over signed graph. Finally, we present a numerical example to illustrate the effectiveness of the obtained results.

Pinning bipartite synchronization for coupled nonlinear systems with antagonistic interactions and switching topologies

This paper studies the bipartite synchronization in a network of nonlinear systems with antagonistic interactions and switching topologies. In order to obtain some conditions such that the network achieves bipartite synchronization for any initial conditions, we design a pinning scheme to pin a part of agents. Under the assumptions that all signed graphs are structurally balanced and the nonlinear system satisfies a one-sided Lipschitz condition, we derive conditions under which the network reaches bipartite synchronization for any initial conditions and arbitrary switching signals. For a general switching signal (especially the periodic switching signal), some conditions related to switching signal are obtained. Finally, we present two numerical examples to illustrate the effectiveness of the obtained results.

Delay-range-dependent synchronization of drive and response systems under input delay and saturation

This paper addresses the synchronization of nonlinear drive and response systems under input saturation and subject to input time-delay. In considering generalized forms of the systems, their dynamics are assumed to satisfy the one-sided Lipschitz condition along with the quadratic inner-boundedness rather than the conventional Lipschitz condition. Further, the time-delays are handled by application of the delay-range-dependent methodology, rather than the delay-dependent one, utilizable for both short and long time-delays. Synchronization controller designs are provided by application of the Lyapunov–Krasovskii functional, local sector condition, generalized Lipschitz continuity, quadratic inner-boundedness criterion and Jensen's inequality. To the best of the authors’ knowledge, a delay-range-dependent synchronization control approach for the one-sided Lipscitz nonlinear systems under input delay and saturation constraints is studied for the first time. A convex-routine-based solution to the controller gain formulation by application of recursive nonlinear optimization using cone complementary linearization is also provided. The proposed methodology is validated for synchronization of modified Chua's circuits under disturbances by considering the input delay and saturation constraints.

On observer-based control of one-sided Lipschitz systems

This paper describes the novel observer-based controller design for the one-sided Lipschitz nonlinear systems. An observer-based controller design condition is obtained by application of the Lyapunov theory, by involving the one-sided Lipschitz condition and quadratic inner-boundedness criterion, and by incorporating the matrix inequality procedures. Further, a decoupling condition, necessary and sufficient for the main design approach, is developed to determine the controller and observer gains. Furthermore, two design conditions, a computationally simple sufficient condition and a more generic necessary and sufficient condition, based on the existing and the novel treatments for the one-sided Lipchitz nonlinearity are evaluated to obtain the observer-based control solution. An algorithm for solving the proposed design constraints by combining a nested bilinear-terms-solver approach and a nonlinear-optimization-based cone complementary linearization method is provided. The effectiveness of the proposed observer-based controller design for the one-sided Lipschitz nonlinear systems is demonstrated using a numerical example.

On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

Dissipativity and Stability Analysis for Fractional Functional Differential Equations

Abstract This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions as the classical integer-order functional differential equations (FDEs). The asymptotic stability of F-FDEs are also proved under the one-sided Lipschitz conditions. Those extend the corresponding properties from integer-order FDEs to the Caputo fractional ones. The results can also be directly applied to some special cases of fractional nonlinear equations, such as the fractional delay differential equations (F-DDEs), fractional integro-differential equations (F-IDEs) and fractional delay integro-differential equations (F-DIDEs). The fractional Adams-Bashforth-Moulton algorithm is employed to simulate the F-FDEs, and several numerical examples are given to illustrate the theoretical results.

Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients

We consider the Euler-Maruyama approximation for multi-dimensional stochastic differential equations with irregular coefficients. We provide the rate of strong convergence where the possibly discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is Holder continuous.

Modulus synchronization in a network of nonlinear systems with antagonistic interactions and switching topologies

This paper studies the collective behavior in a network of nonlinear systems with antagonistic interactions and switching topologies. The concept of modulus synchronization is introduced to characterize the case that the moduli of corresponding components of the agent (node) states reach a synchronization. The network topologies are modeled by a set of directed signed graphs. When all directed signed graphs are structurally balanced and the nonlinear system satisfies a one-sided Lipschitz condition, by using matrix measure and contraction theory, we show that modulus synchronization can be evaluated by the time average of some matrix measures. These matrices are about the second smallest eigenvalue of undirected graphs corresponding to directed signed graphs. Finally, we present two numerical examples to illustrate the effectiveness of the obtained results.

Observer design for one-sided Lipschitz descriptor systems

This paper describes the design of a nonlinear observer for a class of nonlinear descriptor systems with one-sided Lipschitz nonlinearities. To provide a general framework applicable to a large class of systems, nonlinearities and disturbances are considered at the state as well as at the output equations. Non-strict and strict bilinear matrix inequality (BMI)-based robust observer synthesis schemes are provided by utilizing the one-sided Lipschitz condition, the concept of quadratic inner boundedness, the generalized Lyapunov theory for singular systems, and the L2 gain reduction. The BMI-based condition is converted into the linear matrix inequality (LMI)-based condition by utilizing change of variables for straightforward computation of the observer gain matrix. In contrast to the traditional observer schemes for one-sided Lipschitz nonlinear systems, the resultant scheme is applicable to singular systems, capable of dealing with nonlinearities at the output equation and appropriate for treating with disturbances. Two simulation examples for the one-sided Lipschitz and the singular one-sided Lipschitz nonlinear systems are provided to validate the proposed observer synthesis methodologies.

Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations

This paper is concerned with pth moment and almost sure exponential stability of the exact and numerical solutions of neutral stochastic delay differential equations (NSDDEs). Moment exponential stability criteria of the continuous and discrete solutions are established by virtue of the Lyapunov method. Then the almost sure exponential stability criterion is derived by the Chebyshev inequality and the Borel–Cantelli lemma. We also examine conditions under which the numerical solution can reproduce the exponential stability of exact solution. It is shown that the linear growth condition is necessary for Euler–Maruyama (EM) method to maintain the moment exponential stability of the exact solution. If the drift coefficient of NSDDE satisfies the one-sided Lipschitz condition, EM method may break down, but we show that the backward EM (BEM) method can share the mean square exponential stability of the exact solution.

On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations

The asymptotic mean-square stability of two-step Maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay (NSDDEs). Under the one-sided Lipschitz condition and the linear growth condition, it is proved that a family of implicit two-step Maruyama methods can preserve the stability of the analytic solution in mean-square sense. Numerical results for both a nonlinear NSDDE and a system show that the family of two-step Maruyama methods have better stability than previous two-step Maruyama methods.

Robust finite-time [formula omitted] control for one-sided Lipschitz nonlinear systems via state feedback and output feedback

Robust finite-time H∞ control of a class of continuous-time nonlinear system with parameter uncertainties and disturbance input is discussed in this note. The nonlinear function is considered to satisfy the one-sided Lipschitz condition, which has less conservative than the well-known global Lipschitz nonlinear condition. By means of linear matrix inequality (LMI) techniques, both design algorithms of state-feedback controller and static output-feedback controller are developed. The designed controllers are proved to guarantee the corresponding closed-loop systems that are finite-time boundedness (FTB) with a desired H∞ performance index. And the best scalars selection criterion is used to determine the finite-time scalars such that the LMI-based conditions with the best feasibility in the global field. Finally, a numerical example is included to verify the efficiency of the proposed methods.

New Stability Criteria for Uncertain Nonlinear Stochastic Time-Delay Systems

This paper deals with the problem of robust stabilization and non-fragile robust control for a class of uncertain stochastic nonlinear time-delay systems that satisfy a one-sided Lipschitz condition. The parametric uncertainties are assumed to be real time-varying and norm bounded. Based on the one-sided Lipschitz condition including useful information of the nonlinear part, a new stability criterion for this class of nonlinear systems is provided. A memoryless non-fragile state-feedback controller is designed to guarantee robust stochastic stability of closed-loop systems. The approach of linear matrix inequalities is proposed to solve the robust stability for stochastic nonlinear systems with time-varying delay, and to obtain new delay-dependent sufficient conditions. Numerical examples are given to illustrate the validity and advantages of the proposed theoretical results.

Observer Design for One-Sided Lipschitz Nonlinear Systems Subject to Measurement Delays

This paper presents a novel nonlinear observer-design approach to one-sided Lipschitz nonlinear systems in the presence of output delays. The crux of the approach is to overcome the practical consequences of time delays, encountered due to distant sensor position and time lag in measurement, for estimation of physical and engineering nonlinear system states. A Lyapunov-Krasovskii functional is employed, the time derivative of which is solved using Jensen’s inequality, one-sided Lipschitz condition, and quadratic inner-boundedness, and, accordingly, design conditions for delay-range-dependent nonlinear observer for delayed one-sided Lipschitz systems are derived. Further, novel solutions to the problems of delay-dependent observer synthesis of one-sided Lipschitz models and delay-range-dependent state estimation of linear and Lipschitz nonlinear systems are deduced from the present delay-range-dependent technique. An observer formulation methodology for retrieval of one-sided Lipschitz nonlinear-system states, which is robust againstL2norm-bounded perturbations, is devised. The resultant design conditions, in contrast to the conventional procedures, can be solved via less conservative linear matrix inequality- (LMI-) based routines that succeed by virtue of additional LMI variables, meaningful transformations, and cone complementary linearization algorithm. Numerical examples are worked out to illustrate the effectiveness of the proposed observer-synthesis approach for delayed one-sided Lipschitz systems.

Dissipativity and contractivity for fractional-order systems

This paper concerns the dissipativity and contractivity of the Caputo fractional initial value problems. We prove that the systems have an absorbing set under the same assumptions as the classic integer-order systems. This directly extends the dissipativity from integer-order systems to the Caputo fractional-order ones. The fractional dissipativity conditions can be satisfied by many fractional chaotic systems and the systems from the spatial discretization of some time-fractional partial differential equations. The fractional-order systems satisfying the so-called one-sided Lipschitz condition are also considered in a similar way, and the contractivity property of their solutions is proved. Two numerical examples are provided to illustrate the theoretical results.

Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented.

Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay

This paper can be regarded as the continuation of the work contained in papers Milošević (2011, 2013). At the same time, it represents the extension of the paper Wu et al. (2010). In this paper, the one-sided Lipschitz condition is employed in the context of the backward Euler method, for a class of neutral stochastic differential equations with constant delay. Sufficient conditions for this method to be well defined are revealed. Under certain nonlinear growth conditions, the convergence in probability is established for the continuous forward–backward Euler method, as well as for the discrete backward Euler method. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without requiring for the drift coefficient to satisfy the linear growth condition.