Linear Growth Condition Research Articles

Analysis of Cauchy problem with fractal-fractional differential operators

Abstract Cauchy problems with fractal-fractional differential operators with a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start with deriving some important inequalities, and then by using the linear growth and Lipchitz conditions, we derive the conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to derive a numerical solution to the equation. Some examples of fractal-fractional differential equations were presented, and their exact solutions were obtained and compared with the used numerical scheme. A nonlinear case was considered and solved, and numerical solutions were presented graphically for different values of fractional orders and fractal dimensions.

H∞and Asymptotic Stability via delay feedback for hybrid neutral stochastic delay differential equations with Lévy noise

AbstractThis work addresses existence and stabilization problem for a hybrid neutral stochastic delay differential equations with Lévy noise (HNSDDELN). The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. We first prove the existence and uniqueness of the solution. We then design a delay feedback controller to make an unstable HNSDDELN $H_{\infty }$ and asymptotically stable in ${\mathbb{L}}^{p}$. We end up with a numerical example that corroborates our theoretical findings.

Mean field backward doubly stochastic Volterra integral equations and their applications

This paper deals with a class of mean field backward doubly stochastic Volterra integral equations (MFBDSVIEs, for short). Firstly, we prove the existence and uniqueness results in the sense of M-solutions for MFBDSVIEs as well as a comparison theorem is obtained. Based on comparison theorem, we will study the existence of minimal solution with linear growth condition. Secondly, two duality principles between linear mean field forward doubly stochastic Volterra integral equations (MFFDSVIEs, for short) and MFBDSVIEs are derived. As an application for the duality principles, we show a maximum principle of Pontryagin type for an optimal control problem of MFFDSVIEs.

Global Stabilization for Uncertain Nonlinear Time-Delay Systems With Saturated Input

In this article, global stabilization for a class of nonlinear uncertain time-delay systems with saturated input is considered. The saturated input is handled by a new auxiliary system with dynamic gain and the linear growth condition with unknown growth rates is removed. A novel adaptive observer-controller coupling design framework is put forward, under which the closed-loop system is globally bounded. Finally, a simulation shows that the proposed technique is effective.

Existence and uniqueness for variational data assimilation in continuous time

<p style='text-indent:20px;'>A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.</p>

The generalized Khasminskii-type conditions in establishing existence, uniqueness and moment estimates of solution to neutral stochastic functional differential equations

The main objective of this paper involving neutral stochastic functional differential equations (NSFDEs), with finite or infinite history dependence, is to prove the existence and uniqueness of their global solutions by imposing conditions that hold for highly non-linear NSFDEs? coefficients. For that purpose, Yamada-Watanabe condition or the local Lipschitz condition for the drift and diffusion coefficients are imposed, together with contractivity condition for the neutral term. Also, instead of the linear growth condition for the drift and diffusion coefficients of the equations, generalized Khasminskii-type conditions are applied. The proof of the existence and uniqueness of the solution also leads us to estimates of the moments to the solution. Consequently, we discuss some asymptotic properties of the solution in terms of the generalized Lyapunov exponent. Additionally, we consider a class of neutral stochastic differential equations with state-dependent delay, as a special case of NSFDEs. The theoretical results are illustrated with two examples.

Adaptive Neural Design of Consensus Controllers for Nonlinear Multiagent Systems Under Switching Topologies

Existing adaptive neural control methods for nonlinear multiagent systems (MASs) are only applicable under a fixed topology or are applicable under switching topologies but require some linear growth conditions on the nonlinear functions. Motivated by these limitations, a state-dependent adaptive neural design method is proposed in this article. Technically, our method is developed from a state-dependent Lyapunov function candidate, a switched control law, and a projection-based adaptation mechanism. To overcome the stability analysis difficulty caused by the new design of the Lyapunov function, a nonswitched compensation approach and a modified multiple Lyapunov functions method are proposed to derive a dwell-time condition, under which stability can be preserved. It is proved that in addition to stability, synchronization errors converge to a tunable residual around zero. Besides, the proposed scheme achieves the improvement of transient performance in terms of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> norm and moreover, once there are no more topology switchings, asymptotic convergence of synchronization errors to a prescribed interval recovers automatically.

Strong convergence on a stochastic controlled Lotka-Volterra 3-species model with L´evy jumps

In this paper we study two properties of the numerical solutions of a controlled stochastic Lotka-Volterra one-predator-two-prey model, namely the boundedness in the mean of the numerical solutions and the strong convergence of these solutions. We also establish and solve, by means of the Stochastic Maximum Principle, the corresponding optimal control problem in a population modeled by a Lotka-Volterra system with two types of stochastic environmental fluctuations: white noise and L´evy jumps. Our study shows, assuming standard linear growth and Lipschitz conditions on the drift and diffusion coefficients, that the boundedness of the numerical solutions and the strong convergence of the scheme are preserved in this stochastic model.

Existence and uniqueness of solutions for uncertain nonlinear switched systems

In this paper, an uncertain nonlinear switched system is a nonlinear switched system disturbed by subjective uncertainties, which can be formulated by several uncertain differential equations. Few results concerning such type of nonlinear switched systems were published before. To fill this gap, the property of solutions for uncertain nonlinear switched systems with finite-time horizon is discussed in depth. Based on two conditions called linear growth condition and Lipschitz condition, an existence and uniqueness theorem is proposed and demonstrated for the solutions of the uncertain nonlinear switched systems, and the main theoretical bases include uncertainty theory and Banach fixed point theorem. At last, a numerical example and a stock price model are given to illustrate the effectiveness of the results derived.

Multidimensional Markovian BSDEs with Jumps and Continuous Generators

We deal with a multidimensional Markovian backward stochastic differential equation driven by a Poisson random measure and independent Brownian motion (BSDEJ for short). As a first result, we prove, under the Lipschitz condition, that the BSDEJ’s adapted solution can be represented in terms of a given Markov process and some deterministic functions. Then, by means of this representation, we show existence results for such equations assuming that their generators are totally or partially continuous with respect to their variables and satisfy the usual linear growth conditions. The ideas of the proofs are to approximate the generator by a suitable sequence of Lipschitz functions via convolutions with mollifiers and make use of the L2–domination condition, on the law of the underlying Markov process, for which several examples are given.

Stabilization of highly nonlinear hybrid systems driven by Lévy noise and delay feedback control based on discrete-time state observations

There are many hybrid stochastic differential equations (SDEs) in the real-world that don’t satisfy the linear growth condition (namely, SDEs are highly nonlinear), but they have highly nonlinear characteristics. Based on some existing results, the main difficulties here are to deal with those equations if they are driven by Lévy noise and delay terms, then to investigate their stability in this case. The present paper aims to show how to stabilize a given unstable nonlinear hybrid SDEs with Lévy noise by designing delay feedback controls in the both drift and diffusion parts of the given SDEs. The controllers are based on discrete-time state observations which are more realistic and make the cost less in practice. By using the Lyapunov functional method under a set of appropriate assumptions, stability results of the controlled hybrid SDEs are discussed in the sense of pth moment asymptotic stability and exponential stability. As an application, an illustrative example is provided to show the feasibility of our theorem. The results obtained in this paper can be considered as an extension of some conclusions in the stabilization theory.

The Existence, Uniqueness, and Carathéodory’s Successive Approximation of Fractional Neutral Stochastic Differential Equation

The existence, uniqueness, and Carathe´odory’s successive approximation of the fractional neutral stochastic differential equation (FNSDE) in Hilbert space are considered in this paper. First, we give the Carathe´odory’s approximation solution for the FNSDE with variable time delays. We then establish the boundedness and continuity of the mild solution and Carathe´odory’s approximation solution, respectively. We prove that the mean-square error between the exact solution and the approximation solution depends on the supremum of time delay. Next, we give the Carathe´odory’s approximation solution for the general FNSDE without delay. Under uniform Lipschitz condition and linear growth condition, we show that the proof of the convergence of the Carathe´odory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution. Furthermore, under the non-Lipschitz condition, which is weaker than Lipschitz one, we establish the existence and uniqueness theorem of the solution for the FNSDE based on the Carathe´odory’s successive approximation. Finally, a simulation is given to demonstrate the effectiveness of the proposed methods.

Stochastic Wave Equations Defined by Fractal Laplacians on Cantor-Like Sets

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the resolvent density. Afterwards, we establish existence and uniqueness of mild solutions to stochastic wave equations provided some Lipschitz and linear growth conditions. We prove H¨older continuity in space and time and compute the Hölder exponents. Moreover, we are concerned with the phenomenon of weak intermittency.

Predefined-Time Backstepping Stabilization of Autonomous Nonlinear Systems

Dear Editor, In this letter, the backstepping technique is applied to solve the predefined-time stabilization problem for autonomous nonlinear systems, where nonlinear terms are unknown but satisfy a linear growth condition. A linear time-varying control law is designed to drive all states of an autonomous nonlinear system to the origin for an a priori preassigned time. Numerical examples are included to validate the obtained theoretical results and demonstrate effectiveness of the proposed control law. The conducted simulations show that the designed control input provides a lesser magnitude and is easier computable than some existing control laws.

Adaptive Control of Uncertain Nonlinear Systems With Discontinuous Input and Time-Varying Input Delay

In this note, we investigate the adaptive quantized or event-triggered control designs for strict-feedback nonlinear systems with time-varying input delay. Because the control signal is discontinuous in the quantized control and event-triggered control, many existing methods to deal with the input delay are no longer applicable. Through constructing an auxiliary tracking error and an auxiliary system, the input-quantized control is implemented for nonlinear systems with unknown control gain and unknown input delay. With the well-designed Lyapunov–Krasovskii functionals and the linear growth condition of input delay, the stability analysis is achieved. The studied method is also applicable to the event-triggered control for systems possessing unknown input delay. The stability analysis shows that all the signals are semiglobally uniformly ultimately bounded (SGUUB). The use of dynamic surface control (DSC) effectively simplifies the controller structure. The quantized and event-triggered control simulations illustrate that the proposed schemes are effective.

Boundedness and stability of nonlinear hybrid stochastic differential delay equation disturbed by Lévy noise

This study focuses on the boundedness and stability of the nonlinear hybrid stochastic differential delay equation (NHSDDE) disturbed by Lévy noise (LN). Different from the previous systems with Markovian switching, not only different parameters under different modes, but also different structures under different modes are considered, where a novel Lyapunov function is constructed to handle the impacts of different structures under different modes. Without the linear growth condition, the existence of the global solution is proved for the NHSDDE with LN. Then, the M-matrix technique is applied to build some reasonable theorems upon the vth moment asymptotic boundedness, the vth moment exponential stability and the almost surely exponential stability for the NHSDDE driven by LN. An approach to solving the admissible upper bound of the time-delay derivative is given for guaranteeing the asymptotic boundedness and the exponential stability of the global solution. Finally, two numerical examples and homologous simulated outcomes are provided to indicate the usefulness of the presented theoretical findings.

Dynamic event‐triggered control for a class of p‐normal nonlinear time‐delay systems via output feedback

ABSTRACTIn this article, the event‐triggered (ET) output feedback control problem is discussed for a class of ‐normal nonlinear time‐delay systems. Different from the related existing literature, the linear growth condition is relaxed to the homogeneous one and the growth rate is unknown. In the case of unknown time‐varying delays and control coefficients, nonsmooth control law and some extra redundant terms will be encountered in the design of ET controller, which would bring substantial challenges to the achievement of global convergence. To cope with unknown growth rate and time‐varying delays, two dynamic gains and an appropriate Lyapunov–Krasovskii functional (LKF) are introduced. Then, a new dynamic ET mechanism is given, in which the triggering threshold can be tuned dynamically. It is proved that all the signals of the closed‐loop system (CLS) are bounded. Corresponding examples are given to indicate the validity of the developed theoretical results.

Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks

This paper deals with the balanced numerical schemes of the stochastic delay Hopfield neural networks. The balanced methods have a strong convergence rate of at least 12 and the balanced schemes have almost sure exponential stability under certain conditions. Under the Lipchitz and linear growth conditions, the balanced Euler methods are proved to have a strong convergence of order 12 in mean-square sense. Using the Lipchitz conditions on the various parameters of the model, based on the semimartingale convergence theorem and some reasonable assumptions, the balanced Euler methods of the stochastic delay Hopfield neural networks are proved to be almost sure exponentially stable. Numerical experiments are provided to illustrate the theoretical results which are derived in this paper. The computational efficiency of the balanced methods is demonstrated by numerical tests and compared to the Euler–Maruyama approximation scheme of the stochastic delay Hopfield neural networks. Furthermore, the obtained numerical results show that the balanced numerical methods of stochastic delay Hopfield neural networks are very efficient with the least error and have the best step size region for almost sure mean square stable.

Almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion

G-Brownian motion has potential applications in uncertainty problems and risk measures, which has attracted the attention of many scholars. This study investigates the almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion. Due to the non-linearity of G-expectation and distribution uncertainty of G-Brownian motion, it is difficult to study this issue. Firstly, the existence of the global unique solution is derived under the linear growth condition and local Lipschitz condition. Secondly, the almost sure exponential stability of the system is analyzed by applying the G-Lyapunov function and G-Itô formula. Finally, an example is introduced to illustrate the stability. The conclusions of this paper can be applied to the stability and risk management of uncertain financial markets.

Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ N

ABSTRACT In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems where , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for and b large enough.