Stochastic Differential Systems Research Articles

Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

This study focuses on optimal feedback control for stochastic Caputo fractional evolution systems in separable Hilbert spaces. Initially, we deal with the existence of a mild solution, establishing it through the application of the generalised Clarke's subdifferential and a fixed point theorem for multivalued maps. Then, we utilise the Cesari property and the Filippov theorem to demonstrate the existence of feasible pairs. Additionally, we provide pairs of suitable feedback controls under specific conditions. Finally, we present a simple illustration to support our theoretical findings.

Stability analysis of stochastic differential systems with state‐dependent delay subject to average‐delay impulses

In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.

Approximate controllability for Hilfer fractional stochastic differential systems of order 1

In this paper, we analyse the approximate controllability of Hilfer fractional stochastic differential systems of order 1 < μ < 2 in Hilbert spaces. The primary findings are carried out by using fractional calculus, stochastic analysis theory, cosine family, and Schauder's fixed point theorem. In particular, we draw a new set of sufficient conditions for the approximate controllability of Hilfer fractional stochastic differential equations of order 1 < μ < 2 under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the derived theory.

Existence Result for a Stochastic Functional Differential System Driven by G-Brownian Motion with Infinite Delay

Abstract In this article, we are interested in the study of a class of stochastic functional differential systems driven by G-Brownian motion with infinite delay. We prove the existence and uniqueness of the solutions when two basic conditions are met: the linear growth condition and the Lipschitz condition.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Modeling the dengue control dynamics based on a delay stochastic differential system

Modeling the dengue control dynamics based on a delay stochastic differential system

On the Asymptotic Stability of Hilfer Fractional Neutral Stochastic Differential Systems with Infinite Delay

On the Asymptotic Stability of Hilfer Fractional Neutral Stochastic Differential Systems with Infinite Delay

Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Lévy jumps under different structures

This paper investigates the boundedness and stability of a class of nonlinear hybrid neutral stochastic differential delay systems with Lévy jumps and different structures. The coefficients in this system satisfy the local Lipschitz condition and a suitable Khasminskii-type condition, and the state space of the system is separated into two subsets, the existence uniqueness, asymptotic boundedness, and exponential stability of the system are obtained by designing a new Lyapunov function and applying the M-matrix technique as well as dealing with the non-differentiable delay function. Different with the existing work, we not only consider the neutral term, but also the case of the delay function being bounded and non-differentiable. At last, numerical examples are performed to manifest the obtained results.

Application of the Mittag-Leffler kernel in stochastic differential systems for approximating the controllability of nonlocal fractional derivatives

This research focuses on utilizing the Mittag-Leffler kernel within stochastic differential systems to estimate the controllability of nonlocal Atangana–Baleanu fractional derivatives. By assuming the automatic control of the corresponding linear system, a novel set of necessary and sufficient conditions for the approximate controllability of the fractional stochastic differential inclusions of Atangana–Baleanu is established. Furthermore, the study explores the approximate controllability of the proposed system with infinite delay. The investigation relies on the fixed-point theorem for multivalued operators and fractional calculus to derive these outcomes. Lastly, an illustrative example is provided to highlight the practical implications of the research findings.

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

New Criteria on Asymptotic Stability of Stochastic Differential Systems via Impulsive Control

This brief is concerned with the stability issue for stochastic differential systems (SDSs). Some new criteria on asymptotic stability are established for SDSs with impulsive control based on Lyapunov stability theory, bounded difference sequences and martingale convergence theorems. The efficiency of the theory is demonstrated by two examples, i.e. a Chua’s circuit system and a numerical example, which show that the designed impulse controller can stabilize the SDSs.

Existence theory and stability analysis of neutral $ \psi $–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses

&lt;abstract&gt;&lt;p&gt;In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear $ \psi $-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index $ H \in (\frac{1}{2}, 1) $. At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps

&lt;abstract&gt;&lt;p&gt;This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.&lt;/p&gt;&lt;/abstract&gt;

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses

Approximate controllability of Atangana- Baleanu fractional stochastic differential systems with non-Gaussian process and impulses

Novel intelligent predictive networks for analysis of chaos in stochastic differential SIS epidemic model with vaccination impact

In this paper, stochastic predictive computing networks are exploited to investigate the dynamics of the SIS with vaccination impact based epidemic model (SISV-EM) represented by nonlinear systems of stochastic differential equations (SDEs) by exploitation of artificial neural networks (ANNs) with the backpropagated Levenberg-Marquardt technique (BLMT) i.e., (ANNs-BLMT) to approximate the solution behavior. The stochastic nonlinear SISV-EM is governed with three classes: susceptible, infectious, and vaccinated populations. The referenced or target datasets for ANNs-BLMT are constructed by employing Euler-Maruyama (EM) scheme for solving stochastic differential systems in case of sufficiently various nonlinear SISV-EM scenarios by varying the percentage of vaccination for newly born, the coefficient of transmission, the natural mortality rates, the infectious rates of recovery, the rate at which vaccinated people lose their immunity, the rate of death caused by disease, the proportion of vaccinated against susceptible and the white noise in the environment. Based on arbitrary training, testing, and validation samples from the referenced dataset, the ANNs-BLMT provides an approximate solution for the stochastic nonlinear SISV-EM, with significant correlations to the referenced results. Exhaustive simulation-based results using error histograms, mean square errors, and regression analyses further demonstrate that the proposed ANNs-BLMT is efficient, consistent, and accurate for solving SISV-EM.

Boundedness analysis of neutral stochastic systems driven by [formula omitted]-Brownian motion

This paper is concerned with exponential ultimate boundedness problem for a class of neutral stochastic differential systems driven by G-Brownian motion. By developing a method that combines the comparison principle and the reduction to absurdity, we provide sufficient conditions for exponential ultimate boundedness in the mean square of neutral stochastic differential systems driven by G-Brownian motion. The obtained results improve greatly some previous works given in the literature. Finally, some comparisons with existing results and examples are provided to show the effectiveness and superiority of the theoretical results.

Almost sure polynomial stability and stabilization of stochastic differential systems with impulsive effects

In this paper, we first consider the almost surely and pth moment polynomial stability of stochastic differential equations with impulsive effects. Sufficient conditions are given to obtain the polynomial stability. Then the stabilization of stochastic differential systems is studied, which shows that impulse can make stochastic differential equations stable.

Input-to-state stability of stochastic differential systems with hybrid delay-dependent impulses

This paper discusses the pth moment input-to-state stability(p-ISS) and pth moment integral input-to-state stability(p-iISS) for impulsive stochastic systems with time-varying delays and hybrid delayed impulses. With the Lyapunov method and some techniques, the criteria for the impulsive stochastic system p-ISS and p-iISS are established under two common hypotheses, while our criterion allows the time delay to be less than, equal to, or greater than the length of the impulsive interval. The results show that time delay, intensity, and density of hybrid delayed impulses are the main factors affecting the p-ISS and p-iISS of the system; In other words, changing one or more of these factors can stabilize the unstable system. In addition, if both delayed impulses and continuous dynamics are stable, then time delay does not affect the stability of impulsive stochastic systems. The impulsive systems may be p-ISS and p-iISS when unstable delayed impulses perturb the stable continuous dynamics. Finally, two examples are presented to show the validity and correctness of results.