Polynomial Growth Condition Research Articles

Optimal decay rates for nonlinear Moore–Gibson–Thompson equation with memory and Neumann boundary

We consider the initial value problem of Moore–Gibson–Thompson equation with memory effect, being subject to the Neumann boundary condition. Here, the nonlinearity f is conservative and satisfies some polynomial growth conditions. Under the condition that g ( t ) decays exponentially, we obtain the uniform polynomial decaying rate of energy and the solution. Moreover, we show this decay rate is optimal in the sense that there exists a class of solutions decaying exactly at this rate.

Practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise based on aperiodically intermittent discrete-time observation control

This paper investigates practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise (MHFNLs) via aperiodically intermittent discrete-time observation control (AIDOC). It is worth pointing out that AIDOC is considered into MHFNLs for the first time. Due to the difficulty of satisfying linear conditions in real life, polynomial growth conditions are used instead of linear growth conditions. Meanwhile, a global Lyapunov function is constructed and sufficient conditions which can remain MHFNLs practically stable are gained. Finally, an application and its numerical simulations are presented to verify the effectiveness and availability of the theoretical results.

Discrete-time feedback control of highly nonlinear hybrid stochastic neural networks with non-differentiable delays

This paper investigates the stability of a class of highly nonlinear hybrid stochastic neural networks (HSNNs). Distinct from existing results, the system coefficients satisfy polynomial growth conditions instead of the usual linear growth conditions; the delays in the considered systems are time-varying delays with non-differentiable. In this paper, feedback control based on discrete-time state and mode observations is used to make the system stable. By applying the Lyapunov functional method, four results on the stabilisation of the controlled systems are established. Moreover, the upper bound on the duration τ between two consecutive state and mode observations is gained. At last, two examples are given to illustrate the validity of the theoretical results.

Convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching

In this paper, we concern the strong convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching (SPDEs-MS). We present the balanced Euler method of SPDEs-MS and consider its moment boundedness under polynomial growth condition plus Khasminskii-type condition. We also study its strong convergence order and its mean-square stability. Two numerical examples are given to illustrate the theoretical results.

Finite-time semi-global stabilisation of polynomial growth nonlinear systems by linear time-varying feedback

The paper studied the finite-time semi-global stabilisation problems for a class of nonlinear systems with a novel polynomial growth condition. By using some key properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, two time-varying high-gain controllers, where scalar time-varying functions are generated by differential equations, are designed. Both state feedback and observer-based output feedback are considered. It is shown that the proposed methods can achieve finite-time semi-global stabilisation for closed-loop systems and the controls are bound. Two numerical examples verify the effectiveness of the established results.

Periodic self-triggered strategy for highly non-linear hybrid stochastic delayed networks based on impulsive control

This article investigates the stabilization issue of highly non-linear hybrid stochastic delayed networks (HSDNs) via periodic self-triggered control under impulse (PS-TCI). Firstly, the existence of a unique global solution for highly non-linear HSDNs under PS-TCI is studied. Then, a stabilization criterion for highly non-linear HSDNs is established, by combining a graph-theoretic approach with a novel Lyapunov-based analysis, based on a ‘genuine’ Lyapunov function defined by introducing an auxiliary timer. Therein, the less conservative polynomial growth condition and local Lipschitz condition for the drift and diffusion coefficients are used than the linear growth condition and global Lipschitz condition. Meanwhile, the design idea of PS-TCI is based on the evolution of an upper bound of the mathematical expectation for Lyapunov function (not directly Lyapunov function or system state), which implies that the triggered instant of PS-TCI is not a random variable. Finally the theoretical results are employed to study the stability of a class of FitzHugh–Nagumo circuits networks and the central pattern generators networks of a hexapod robot, and correlative numerical simulations are provided for demonstration.

Approximation of deterministic mean field games under polynomial growth conditions on the data

Approximation of deterministic mean field games under polynomial growth conditions on the data

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.

Dynamic of the nonclassical diffusion equation with memory

In this paper, we consider the nonclassical diffusion equation with memory and the nonlinearity of the polynomial growth condition of arbitrary order in the time-dependent space. First, the well-posedness of the solution for the equation is obtained in the time-dependent space mathscr{U}_{t}. Then, we establish the existence and regularity of the time-dependent global attractor. Finally, we also conclude that the fractal dimension of the time-dependent attractor is finite.

Stabilization of Stochastic Highly Nonlinear Delay Systems With Neutral Term

In this article, stochastic highly nonlinear delay systems with neutral term are studied, which does not satisfy the linear growth condition. Under the local Lipschtiz condition and the polynomial growth condition, we study the existence and uniqueness, as well as boundedness, of global solution for stochastic highly nonlinear delay systems with neutral term. By designing a discrete-time feedback control function, the stabilization of stochastic highly nonlinear delay systems with neutral term is presented. Different from the previous literature, our discrete-time feedback control function depends on Markov switching signals with a delay. An illustrative example is given to show the effectiveness of the obtained results.

Stability analysis for a class of stochastic delay nonlinear systems driven by G-Lévy Process

The regularity and stability of the solution to a class of stochastic delay differential equation driven by G-Lévy processes are studied in this paper. Firstly, we introduce a new Burkholder–Davis–Gundy (BDG) inequality involving the jump measure. Secondly, we use the BDG inequality to establish the existence and uniqueness of the solution under non-Lipschitz condition. Thirdly, we establish the existence, uniqueness, quasi-sure exponential stability and pth moment exponential stability of the solution under local Lipschitz condition and one-sided polynomial growth condition.

Asymptotic stability of stochastic differential equations driven by G-Lévy process with delay feedback control

Given an unstable stochastic differential equations, the stabilisation by delay feedback controls for such equations under Lipschitz conditions or highly nonlinear conditions have been discussed by several authors. However, there is few works on the stabilisation by delay feedback controls under the sub-linear expectation associated with a G-L?vy process. The aim of this paper is to design delay feedback controls in the drift part and obtain the asymptotical stability in mean square and quasi-surely asymptotical stability for the stochastic differential equations driven by G-L?vy process with the polynomial growth condition. Lastly, we give an example to verify the obtained theory.

Multiple Nonnegative Solutions for a Class of Fourth-Order BVPs Via a New Topological Approach

In this paper, we study a class of fourth-order boundary value problems with integral boundary conditions. The nonlinearity may have time-singularity and change sign. Moreover, it satisfies general polynomial growth conditions. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. An example of application illustrates the existence result.

WHEN IS A COMPLETION OF THE UNIVERSAL ENVELOPING ALGEBRA A BANACH PI-ALGEBRA?

AbstractWe prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

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.

General decay stability of backward Euler–Maruyama method for nonlinear stochastic integro-differential equations

The general decay stability of backward Euler–Maruyama(EM) method is discussed for stochastic integro-differential equations(SIDEs) in this paper. Under polynomial growth condition, both continuous and discrete solutions admit high nonlinearity of the underlying equations. Besides, the considered stability namely the general decay stability is a generalized stability including almost sure polynomially stability and almost sure exponentially stability. Making using of the nonnegative semi-martingale convergence theorem, for SIDEs we propose sufficient conditions to obtain the general decay stability of backward EM method.

Cooperative output feedback consensus control of second-order nonlinear multi-agent systems under Markovian switching topologies

This paper deals with the leader-following consensus problem for a class of second-order nonlinear multi-agent systems by output feedback. The communication topology is characterized by a Markovian switching graph. Firstly, an input-driven observer is introduced to estimate the consensus error of each follower agent. Then, a cooperative nonlinear control law is constructed using the relative output information between neighboring agents by employing the backstepping methodology, which achieves leader-following consensus in mean square sense. Compared with the existing results, the nonlinear functions are required to satisfy polynomial growth condition rather than globally Lipschitz growth or Lipschitz-like growth condition. A numerical example is given to illustrate the theoretical results.

A sharp Bernstein\u2013type inequality and application to the Carleson embedding theorem with matrix weights

We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.

Strong convergence of averaging principle for the non‐autonomous slow‐fast systems of SPDEs with polynomial growth

In this work, we study a class of non‐autonomous two‐time‐scale stochastic reaction‐diffusion equations driven by Poisson random measures, in which the coefficients satisfy the polynomial growth condition and local Lipschitz condition. First, the existence and uniqueness of the mild solution are proved by constructing auxiliary equations and using the technique of stopping time. Then, consider that the time dependent of the coefficients, the averaged equation is redefined by studying the existence of time‐dependent evolution family of measures associated with the frozen fast equation. Further, the slow component that strongly converges to the solution of the corresponding averaged equation is verified by using the classical Khasminskii method.

Stabilization by variable-delay feedback control for highly nonlinear hybrid stochastic differential delay equations

We consider a hybrid SDE (stochastic differential equation) with variable delays, which is unstable and its coefficients satisfy some polynomial growth conditions. We want to find a variable-delays feedback control such that the controlled hybrid SDE with variable delay becomes stable. We establish some novel stability criteria of the controlled SDE with variable delays. Our results improve the existing results from hybrid SDEs with constant-delays to hybrid SDEs with variable-delays. Finally, an example is given to confirm our results.