Linear Growth Condition Research Articles

The Improved Stability Analysis of Numerical Method for Stochastic Delay Differential Equations

In this paper, the improved split-step θ method, named the split-step composite θ method, is proposed to study the mean-square stability for stochastic differential equations with a fixed time delay. Under the global Lipschitz and linear growth conditions, it is proved that the split-step composite θ method with θ≥0.5 shows mean-square stability. An approach to improving numerical stability is illustrated by choices of parameters of this method. Some numerical examples are presented to show the accordance between the theoretical and numerical results.

Well-posedness and regularity of some stochastic time-fractional integral equations in Hilbert space

In the current work, we deal with a class of stochastic time-fractional integral equations in Hilbert space by studying their well-posedness and regularity. Precisely, we use the celebrity fixed point theorem to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Further, we prove the spatial and temporal regularity by imposing only a regularity condition on the initial value. An important example is considered in order to confirm and support the validity of our theoretical results.

Modern and generalized analysis of exogenous growth models

The work presents a critical analysis of the model that could replicate the dynamic of long-run economic growth by regarding the capital accumulation, the labor or the population growth and increases in productivity that depends on technological progress. In this case, we calculated the equilibrium point and gave the model’s existence and uniqueness conditions. We introduced an analysis of the Lyapunov function for the first and second derivatives. Infections point analysis was presented in detail, and a numerical scheme based on the Lagrange polynomial interpolation was used to derive the numerical solution of the classical model. To introduce into mathematical formulation nonlocal behaviors, four cases were considered, including the effect of the power-law, the impact of fading memory, the development of crossover from stretched exponential to power-law and crossover behaviors from power law to stochastic. For each case, Linear growth and Lipschitz conditions were used to establish the uniqueness of the exact solution. Further, different numerical techniques were used to prove convergence of the used numerical scheme. Finally, numerical simulations were performed for other cases.

Stabilisation of stochastic hybrid high-order nonlinear coupled systems by intermittent control

This paper considers the stabilisation of high-order nonlinear stochastic delayed coupled systems with Markovian switching (HNSMS) by periodically intermittent control (PIC). Notably, the coefficients in HNSMS no longer satisfy the linear growth condition. In the mean time, PIC is first introduced to ensure the stabilisation of HNSMS. Since current Halanay-type differential inequalities (H-type inequalities) are not applicable to high-order nonlinear systems with PIC, a generalised H-type inequality with high-order terms for such systems is established, in which the order of high-order terms can be different. Together with graph theory and Lyapunov method, some sufficient conditions reaching the stability are presented. Then, our results are used to study the stability of modified Van der Pol–Duffing oscillators with Markovian switching. Ultimately, the effectiveness of theoretical results is checked by a numerical test.

Neural network-based smooth fixed-time cooperative control of high-Order multi-agent systems with time-varying failures

This article tries to achieve smooth fixed-time cooperative control for high-order multi-agent systems in the presence of nonlinear uncertainties, time-varying failures and single-way communication. Due to the lemma of fixed-time stabilization involving fractional and high-power terms, it is nontrivial to design C1 smooth controllers for such systems. To eliminate linear growth conditions of uncertainties, neural networks are introduced; To ensure all fixed-time controllers are C1 continuous, a novel smooth fixed-time cooperative control framework is provided by designing C1 smooth switching and implementing dynamic surface control such that singularity and chattering problem are eliminated completely; To achieve smooth fixed-time fault-tolerant control, based on bound estimation method, the fixed-time compensation of time-varying failures is made by designing two continuous adaptive laws. Then, a novel smooth fixed-time fault-tolerant cooperative control scheme is proposed to guarantee that all control signals are continuous, smooth and bounded. Meanwhile, all internal errors will converge to arbitrarily small zones of zero within fixed time. To confirm the effectiveness of the proposed smooth fixed-time fault-tolerant cooperative control schemes, simulations based on a numerical example and a practical example are made.

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.

Khasminskii-type theorem for a class of stochastic functional differential equations

Abstract This paper is concerned with the existence and uniqueness theorems for stochastic functional differential equations with Markovian switching and jump, where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Lyapunov-type functions.

Stability of highly nonlinear neutral stochastic delay systems with non-random switching signals

For the last few decades, many works on stability criteria have been established for neutral stochastic delay systems, which are modeled by stochastic differential delay equations. Some of these works do not involve switching signals, and others contain Markovian switching signals. This paper studies highly nonlinear neutral stochastic delay systems with non-random switching signals. Firstly, the rule of non-random switching signals is imposed. Then, by using the Lyapunov function method and the mode-dependent average dwell time approach, the existence and boundedness of unique global solution is discussed for highly nonlinear neutral stochastic delay systems with non-random switching signals. In addition, without the linear growth condition, asymptotic stability and exponential stability are obtained. In the last part of the article, an illustrative example is provided to show the effectiveness of the obtained results.

Exponential Stabilization for a Class of Strict-Feedback Nonlinear Time Delay Systems via State Feedback Control Scheme

This paper considers the exponential stabilization problem for a class of strict-feedback nonlinear systems with multiple time-varying delays, whose nonlinear terms satisfy the linear growth condition. The state feedback controller that relies on a positive parameter to be determined is constructed to deal with nonlinear terms. By tactfully introducing the Lyapunov–Krasovskii functional with an exponential function and selecting the applicable parameter to be determined, the implementable state feedback controller can be obtained to guarantee that the closed-loop system is exponentially stable. The proposed state feedback control scheme is finally applied to the control design of two-stage chemical reactor system, which illustrates the effectiveness of the control method.

Backward Stochastic Differential Equations Driven by a Jump Markov Process with Continuous and Non-Necessary Continuous Generators

We deal with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs for short). We start by proving the existence and uniqueness of the solutions for this type of equation and present a comparison of the solutions in the case of Lipschitz conditions in the generator. With these tools in hand, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ.

Convolutional neural network based simulation and analysis for backward stochastic partial differential equations

We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the 2-tuple adapted strong solution to a unified system of backward stochastic partial differential equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system dynamics such as optimal control and differential game problems. The dynamics of the scheme is modeled by a CNN through conditional expectation projection. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions.

Stabilization of stochastic highly non-linear multi-links systems via aperiodically intermittent control

This paper is concerned with the exponential stabilization of stochastic highly non-linear multi-links systems (SHNMSs) by utilizing aperiodically intermittent control (AIC). Wherein the linear growth condition is removed, which weakens previous stability conditions. The factor of multi-links is first introduced into highly non-linear systems, which makes our model more in line with the practical situations. However, when considering highly non-linear systems, the existing results are not suitable for AIC. Therefore, in order to solve this difficulty, a new Halanay-type differential inequality is established in this paper, which not only extends the classic Halanay inequality, but also expands the application scopes of AIC. In addition, utilizing the Lyapunov method and the graph theory, it can be proved that under suitable conditions, SHNMSs exponentially reach stabilization. Furthermore, the theoretical results are applied to modified stochastic multi-links coupled Fitzhugh–Nagumo models. Finally, the availability of the proposed results is demonstrated by two numerical examples.

Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs

In this paper, we investigate the existence and multiplicity of nontrivial solutions for poly-Laplacian system on a finite graph G=(V, E), which is a generalization of the Yamabe equation on a finite graph. When the nonlinear term F satisfies the super-(p, q)-linear growth condition, by using the mountain pass theorem we obtain that the system has at least one nontrivial solution, and by using the symmetric mountain pass theorem, we obtain that the system has at least dim W nontrivial solutions, where W is the working space of the poly-Laplacian system. We also obtain the corresponding result for the poly-Laplacian equation. In some sense, our results improve some results in (Grigor’yan et al. in J. Differ. Equ. 261(9):4924–4943, 2016).

Digital control of a family of time-delay feedforward systems with sparse sampling

For a family of time-delay feedforward systems satisfying a linear growth condition, we study the problem of global asymptotic stabilization by digital state and output feedback, respectively, under arbitrarily sparse sampling. Using the Lyapunov–Krasovskii functional method and the feedback domination design, we construct memoryless sampled-data state and/or output feedback controllers that achieve global asymptotic stability of the hybrid closed-loop systems with large delays. Applications of the sampled-data control schemes are demonstrated through an example with simulations.

A global method for deterministic and stochastic homogenisation in BV

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.

Stabilisation of hybrid system with different structures by feedback control based on discrete-time state observations

This paper considers a class of hybrid stochastic differential equations (SDEs) with different structures in different modes. In some modes, the coefficients of the SDEs satisfy the linear growth condition, while in the other modes, the coefficients are highly nonlinear. These systems are often unstable. This paper aims to design a discrete-time feedback control which is only put in a part of modes where the coefficients are highly nonlinear, such that these systems become stable. The stabilities concerned include H∞-stability and exponential stability in the moment, as well as almost sure stability. Finally, an example is given to illustrate these results.

Finite-time and fixed-time stabilization of multiple memristive neural networks with nonlinear coupling.

This brief presents the finite-time stabilization and fixed-time stabilization of multiple memristor-based neural networks (MMNNs) with nonlinear coupling. Under the retarded memristive theory, the generalized Lyapunov functional method, extended Filippov-framework and Laplacian matrix theory, we can realize both the finite-time stabilization and fixed-time stabilization problem of MMNNs by designing novel state-feedback controller and the corresponding adaptive controller with regulate parameters. Moreover, we assess the bounds of settling time for the both two kinds of stabilization respectively, and we deeply analyze the influence of initial desiring values and the linear growth condition of the controller on the system. Finally, the benefits of the proposed approach and the experimental analysis are demonstrated by numerical examples.

Relaxation for an optimal design problem inBD(Ω)

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair$(\chi,u)$, where$\chi$is the characteristic function of a set of finite perimeter and$u$is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

Adequate Vitamin A Levels with Stunting Adolescents of Minangkabau Ethnicity in Indonesia: A Case-Control Study

Background: Stunting is a type of linear growth condition. The release of growth hormone in long bones can be hampered by a lack of micronutrients such as vitamin A. This study was performed to determine adequate vitamin a levels with stunting adolescents of Minangkabau ethnicity in Indonesia.
 Methods: A case-control study was used in this study. This study was undertaken at several senior high schools in Padang, Indonesia. Stunting adolescents aged 16 to 18 years of Minangkabau ethnicity were included in the study. Adolescents with stunting were in the case group, whereas those who did not have stunting were in the control group. The age and sex of the cases and controls were matched. There were 42 cases and 42 controls in the research. Vitamin A measurement used SQ-FFQ. The Independent sample T test and the Chi-square test were used to analyze the data. Statistical significance was defined as a two-tailed p-value < 0.05. GraphPad Prism 7.00 program was used to gather and analyze data.
 Conclusion: Our research found that stunted adolescent of the Minangkabau ethnic group in Indonesia have low vitamin A levels.

Strong convergence of the Euler-Maruyama approximation for SDEs with unbounded drift

In this work, we prove strong convergence on small time interval of order for arbitrarily small of the Euler-Maruyama approximation for additive Brownian motion with Hölder continuous drift satisfying a linear growth condition. The proof is based on direct estimations of functional of the Euler-Maruyama approximation. The order of convergence does not depend on the Hölder index of the drift, thus generalizing the results obtained in [10] to both Linear growth and to an optimal convergence order.