non-Lipschitz Conditions Research Articles

A new result on semi-synchronous event-triggered backstepping robust control for a class of non-Lipschitzian networked systems

The paper investigates the backstepping robust control problem for a class of nonlinear networked systems. The non-Lipschitz conditions of linear growth condition and Hölder condition are considered in the time-varying disturbance and uncertainty. An important innovation is that the virtual controller is designed to be piecewise constant such that the usage of the command filter is avoided. In the process of designing the robust controller, a semi-synchronous event-triggered scheme is proposed to relieve the communication pressure in high-order systems. Thus, a switched controller without using average dwell time (ADT) is considered. The finite-time practical stability with the form of fast terminal sliding mode (TSM) is also obtained. Then, the Zeno behavior is excluded and a self-triggered scheme is further proposed to avoid the continuous monitoring of the event-triggered conditions. Compared with some existing command-filtered results, the design process is simplified and more suitable for networked systems. The effectiveness of the proposed theory is validated by the numerical simulation of a hypersonic vehicle.

Some existence results for a stochastic differential system with non-Lipschitz conditions

The existence and uniqueness of mild solutions for a fractional stochastic differential system derived by Atangana–Baleanu fractional derivative have been discussed in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard-type iterations, basic concepts of fractional calculus, sectorial operator, and stochastic analysis techniques. At last, an application is discussed to show the applicability of the developed theory.

Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

Exponential Stable Behavior of a Class of Impulsive Partial Stochastic Differential Equations Driven by Lévy Noise

In this paper, using exponential stable property, the stochastic analysis techniques and a fixed-point theorem for condensing maps, we obtain the existence results of piecewise weighted pseudo almost periodic in distribution mild solutions for a class of impulsive partial stochastic differential equations driven by Lévy noise in Hilbert spaces under non-Lipschitz conditions. Furthermore, the exponential stable of mild solution in mean square is investigated. Finally, an example is presented to illustrate the results.

Limit theorems for additive functionals of stochastic functional differential equations with infinite delay

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales.

The non-Lipschitz stochastic Cahn–Hilliard–Navier–Stokes equations in two space dimensions

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.

Approximate controllability of stochastic differential system with non-Lipschitz conditions

This paper includes investigation of approximate controllability results for a multi-term time-fractional stochastic differential system in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard type iterations. The basic tools of fractional calculus, generalization of semigroup, and stochastic analysis techniques have been utilized. Finally, an application is discussed to illustrate the developed theory.

The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

This paper is devoted to presenting an averaging principle for stochastic pantograph equations. Under suitable non-Lipschitz conditions, the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. At last, an example is given to demonstrate the feasibility of obtained results. Moreover, our results have generalized significantly some previous ones.

Pathwise uniqueness of stochastic differential equations driven by Brownian motions and finite variation Lévy processes

We study the pathwise uniqueness of the solutions to one-dimensional stochastic differential equations driven by Brownian motions and Lévy processes with finite variation paths. The driving Lévy processes are not necessarily one-sided jump processes. In this paper, we obtain some non-Lipschitz conditions on the coefficients, under which the pathwise uniqueness of the solution to the equations is established. Some of our results can be applied to the equation with discontinuous coefficients.

Uncertain Fractional Evolution Equations with Non-Lipschitz Conditions Using the Condensing Mapping Approach

This paper studies the solvability of Cauchy problems for fractional evolution equations with uncertainty. By using a new approach based on the concept of non-compactness measure and the principle of condensing mappings in the spaces without linearity, we prove the existence of C0 −solutions without assuming the Lipschitz continuity of the function on the right-hand side. The present results extend previous results when external forces are always required to satisfy some kinds of generalized Lipschitz conditions. Moreover, the principle of condensing mappings for fuzzy-valued functions or set-valued functions found as an application of noncompactness measure is a useful result when studying dynamical systems containing uncertainties.

Anticipated backward stochastic differential equations with non-Lipschitz coefficients

This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies Non-Lipschitz conditions. The existence and uniqueness of solution are obtained.

ON THE WELL-POSEDNESS OF THE STOCHASTIC 2D PRIMITIVE EQUATIONS

<abstract> Under non-Lipschitz conditions for the external force term and noise term, the two-dimensional stochastic primitive equations are studied in this paper. Based on Galerkin method, iterative method and the moment estimations, we prove the existence and uniqueness of the solutions in a fixed probability space. </abstract>

Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs

In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean–Vlasov, driven by a spectrally-positive Lévy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type without non-degeneracy assumption following the approach developed by Li and Mytnik (2011). The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler–Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler–Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the Lévy measure around zero.

Existence and uniqueness results for a class of fractional stochastic neutral differential equations

In this paper, we investigate new results on the existence and uniqueness of mild solutions to stochastic neutral differential equations involving Caputo fractional time derivative operator with Lipschitz coefficients and under some Carathéodory-type conditions on the coefficients through the Picard approximation technique. To do so, we derive a stochastic version of variation of constants formula for Caputo fractional differential systems whose coefficients satisfy standard Lipschitz and non-Lipschitz conditions. The main points are to prove a coincidence between the integral equation and the mild solution by applying Itô’s isometry, martingale representation theorem, and the weighted maximum norm for a class of fractional stochastic neutral differential equations. Finally, examples are provided to support the efficiency of the main theory.

Non-Lipschitz stochastic functional differential equations on Riemanian manifolds

In this paper, we investigate the existence and uniqueness of solution for a class of stochastic functional differential equations with delay on compact manifolds. The results are established under non-Lipschitz conditions on the coefficients. As an application, a geometrical example is provided to illustrate the results obtained.

Controllability and Optimal Control for a Class of Time-Delayed Fractional Stochastic Integro-Differential Systems

In this paper, we study the controllability and optimal control for a class of time-delayed fractional stochastic integro-differential system with Poisson jumps. A set of sufficient conditions is established for complete and approximate controllability by assuming non-Lipschitz conditions and pth mean square norm. We also give an existence of optimal control for Bolza problem. Our result is valid for fractional order $$\alpha >\frac{p-1}{p},\ p\ge 2.$$ Finally, an example is provided to illustrate the efficiency of the obtained theoretical results.

Averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noise

Stability of stochastic differential equations driven by Lévy noise with Markovian switching has recently received a lot of attention. Different from the integer-order stochastic differential equations, stochastic fractional differential equations play a circular role in describing many practical processes and systems. In this paper, our aims are to study the averaging principle of the solution of hybrid stochastic fractional differential equations driven by Lévy noise under non-Lipschitz conditions which include classical Lipschitz conditions as special cases and propose several sufficient conditions for asymptotic stability in the pth moment of the solution. Two examples with numerical simulation are given to illustrate the obtained theory.

Existence and stability results for stochastic functional integro-differential equations with poisson jumps under non-Lipschitz conditions

Existence and stability results for stochastic functional integro-differential equations with poisson jumps under non-Lipschitz conditions

Stepanov-like pseudo almost periodicity in distribution and optimal control to impulsive partial stochastic differential equations

ABSTRACT This paper concerns the Stepanov-like pseudo almost periodicity in distribution for a stochastic process and optimal control. Using the analytic semigroup theory, fractional power operators, stochastic analysis theory and fixed point strategy, we obtain the existence of Stepanov-like pseudo almost periodic in distribution mild solutions for impulsive partial stochastic functional differential equations under non-Lipschitz conditions. The existence of optimal pairs of system governed by impulsive partial stochastic control systems is also presented. Finally, an example is given for demonstration.

An Averaging Principle for the Time-Dependent Abstract Stochastic Evolution Equations with Infinite Delay and Wiener Process

In this paper, averaging principle for the time-dependent stochastic evolution equations (TDSEEs) with infinite delay is investigated. In proper non-Lipschitz conditions, we prove that the mild solution of the averaged stochastic evolution equations (ASEEs), governed by time-independent family of linear operators, converges to that of the original one in $$L^{2}$$ sense. At last, an application example is presented to prove the obtained theory.