Functional Differential Systems Research Articles

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.

Intelligent predictive computing for functional differential system in quantum calculus

Intelligent predictive computing for functional differential system in quantum calculus

Event-triggered delayed impulsive control for functional differential systems on networks

This article investigates the stability of functional differential systems on networks adopting event-triggered multi-delayed impulsive control. We construct a new event-triggered mechanism (ETM) type using the Lyapunov function and the network topology. Combined with the ETM and multi-delayed impulsive control, in which the impulsive jumps are related to the current and past states, we propose a multi-delayed event-triggered impulsive control strategy. This controller will execute the control tasks only when the systems violate the preset ETM. In the view of the Razumikhin method and the graph theory, some criteria are given to avoiding Zeno behavior under this ETM and achieving exponential stability, which is related to the event-triggered parameters, network topology, and impulsive intensity. As an application of our theoretical results, a class of coupled oscillation systems is considered and numerical simulations are presented to verify the practicability and effectiveness.

Morlet Wavelet Neural Network Investigations to Present the Numerical Investigations of the Prediction Differential Model

In this study, a design of Morlet wavelet neural networks (MWNNs) is presented to solve the prediction differential model (PDM) by applying the global approximation capability of a genetic algorithm (GA) and local quick interior-point algorithm scheme (IPAS), i.e., MWNN-GAIPAS. The famous and historical PDM is known as a variant of the functional differential system that works as theopposite of the delay differential models. A fitness function is constructed by using the mean square error and optimized through the GA-IPAS for solving the PDM. Three PDM examples have been presented numerically to check the authenticity of the MWNN-GAIPAS. For the perfection of the designed MWNN-GAIPAS, the comparability of the obtained outputs and exact results is performed. Moreover, the neuron analysis is performed by taking 3, 10, and 20 neurons. The statistical observations have been performed to authenticate the reliability of the MWNN-GAIPAS for solving the PDM.

Stability of multi-link delayed impulsive stochastic complex networks with Markovian switching

This article presents an analysis of p-th moment exponential stability for a class of stochastic functional differential systems with Markovian switching on complex multi-link networks using multi-delayed impulsive control. The multi-delayed impulsive control scheme implies that the impulsive jumps are related to multiple past states of the systems, which increases the complexity of the analysis. Additionally, single-link networks are extended to complex multi-link networks, which are more extensive. By combining graph theory and the Razumikhin method, several criteria for achieving p-th moment exponential stability for systems are derived. The criteria are related to the maximum upper bound of delays, impulsive interval, and network topology. Subsequently, the stability of multi-link stochastic complex networks with time delays and Markovian switching is studied using delayed impulsive control. Finally, the exponential stability of a class of stochastic coupled multi-link oscillators with Markovian switching is analyzed, and several numerical simulations are provided to verify the validity of the theoretical results.

Practical exponential stability of impulsive stochastic functional differential systems with distributed-delay dependent impulses

This paper develops new practical stability criteria for impulsive stochastic functional differential systems with distributed-delay dependent impulses by using the Lyapunov–Razumikhin approach and some inequality techniques. In the given systems, the state variables on the impulses are concerned with a history time period, which is very appropriate for modelling some practical problems. Moreover, different from the existing practical stabilization results for the systems with unstable continuous stochastic dynamics and stabilizing impulsive effects, we take the systems with stable continuous stochastic dynamics and destabilizing impulsive effects into account. It shows that under the impulsive perturbations, the practical exponential stability of the stochastic functional differential systems can remain unchanged when the destabilizing distributed-delay dependent impulses satisfy some conditions on the frequency and amplitude of the impulses. In other words, it reveals that how to control the impulsive perturbations such that the corresponding stochastic functional differential systems still maintain practically exponentially stable. Finally, an example with its numerical simulation is offered to demonstrate the efficiency of the theoretical findings.

Practical exponential stability of hybrid impulsive stochastic functional differential systems with delayed impulses

SummaryThis paper is concerned with the problem of practical exponential stability for hybrid impulsive stochastic functional differential systems with delayed impulses, which comprise three classes of systems: the systems with unstable continuous stochastic dynamics and stable discrete dynamics, the systems with stable continuous stochastic dynamics and unstable discrete dynamics, and the systems where both the continuous stochastic dynamics and the discrete dynamics are stable. By using the Lyapunov‐Razumikhin approach, several new sufficient conditions for the practical exponential stability are established for each class of systems. It shows that the stabilizing and destabilizing delayed impulses that satisfy some conditions on their frequency and amplitude can stabilize the systems with unstable continuous stochastic dynamics in the practical exponential stability sense and ensure the practical exponential stability of the systems with stable continuous stochastic dynamics, respectively. Conversely, if the continuous stochastic dynamics are practically exponentially stable and the delayed impulses are stabilizing, then the systems can be practically exponentially stable regardless of the restrictions on the delayed impulses frequency and amplitude. Finally, two numerical examples are presented to illustrate the efficiency of the results.

Practical stability analysis of stochastic functional differential systems with G-Brownian motion and impulsive effects

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behavior and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behavior of the addressed system. Two numerical examples are given to verify the validity of our developed results.

Event-triggered impulsive control of stochastic functional differential systems

This article discusses the stabilization problem of stochastic functional differential systems (SFDSs) with event-triggered impulsive control (ETIC). We propose two types of event-triggered mechanism (ETM), namely, continuous ETM with state-dependent waiting time and periodic ETM with state-dependent sampling period. Based on the Lyapunov functionals and mathematical induction, we develop sufficient conditions to achieve the stability of SFDS. Two illustrative examples are given to demonstrate the validity of the theoretical results.

Event-Triggered Control of Stochastic Functional Differential Systems via Degenerate Lyapunov Functionals

This article discusses the problem of event-triggered controls (ETCs) for stochastic functional differential systems (SFDSs). For the stabilization of SFDSs, we design a proper degenerate Lyapunov functional and propose a novel ETC. The degenerate Lyapunov functional is more general than the traditional positive definitive Lyapunov functionals and ETC has a prominent advantage of reducing communication resources compared with periodic sampled-data feedback controls because time updating is based on a certain event. Also, a fixed positive lower bound on the stochastic interexecution times is guaranteed by this novel event-triggering mechanism. Together with ETC and the degenerate Lyapunov functional, the boundedness, mean square exponential stabilization, and globally asymptotical stabilization of SFDSs can be established for different choices of parameters. Finally, the obtained results are validated through numerical simulations.

Stochastic mixed impulsive control and stability for stochastic functional differential systems with semi-Markov jump

This paper focuses on the problem of stochastic mixed impulsive control and stability for stochastic functional differential systems with semi-Markov jump. A new definition of average stochastic impulsive gain is formulated to estimate the stochastic mixed impulsive intensity. Functional systems states are referred to the historical system states in the past time interval rather than in the past time instant of the current time instant. The stability analysis is carried out based on the constructed appropriate Lyapunov functional, Dupire functional Itô’s formula, stochastic analysis theory and graph theory techniques. Moreover, we provide some exponential stability criteria for the investigated systems, which have close relation to not only the topological structures and semi-Markov jump of the functional systems but also average stochastic impulsive gain and stochastic disturbance intensity. Further we put it into practice of a new class of stochastic oscillators systems with semi-Markov jump and the corresponding numerical simulations results are on show to demonstrate the validity of the derived theoretical results.

Mean-Square Stability and Instability Criteria for the Gikhman–Ito Stochastic Diffusion Functional Differential Systems Subject to External Disturbances of the Type of Random Variables

Mean-Square Stability and Instability Criteria for the Gikhman–Ito Stochastic Diffusion Functional Differential Systems Subject to External Disturbances of the Type of Random Variables

Intelligent computing with the knack of Bayesian neural networks for functional differential systems in Quantum calculus model

The purpose behind this research is to utilize the knack of Bayesian solver to determine numerical solution of functional differential equations arising in the quantum calculus models. Functional differential equations having discrete versions are very difficult to solve due to the presence of delay term, here with the implementation of Bayesian solver with means of neural networks, an efficient technique has been developed to overcome the complication in the model. First, the functional differential systems are converted into recurrence relations, then datasets are generated for converted recurrence relations to construct continuous mapping for neural networks. Second, the approximate solutions are determined through employing training and testing steps on generated datasets to learn the neural networks. Furthermore, comprehensive statistical analysis are presented by applying various statistical operators such as, mean squared error (MSE), regression analysis to confirm both accuracy as well as stability of the proposed technique. Moreover, its rapid convergence and reliability is also endorsed by the histogram, training state and correlation plots. Expected level for accuracy of suggested technique is further endorsed with the comparison of attained results with the reference solution. Additionally, accuracy and reliability is also confirmed by absolute error analysis.

Bistable traveling waves in a delayed competitive system

<p style='text-indent:20px;'>This paper is concerned with the existence, uniqueness and asymptotic stability of traveling wave solutions in a delayed competitive system with spatial diffusion. When the interspecific competition is strong, the corresponding functional differential system has two stable and two unstable steady states. Firstly, we establish the existence of bistable traveling wave solutions by the theory of monotone semiflows. With the help of squeezing technique based on comparison principle, the asymptotic stability of traveling wave solutions is proved, which further implies the uniqueness of wave speed and wave profile in the sense of phase shift.</p>

Intelligent computing networks for nonlinear influenza-A epidemic model

The differential equations having delays take paramount interest in the research community due to their fundamental role to interpret and analyze the mathematical models arising in biological studies. This study deals with the exploitation of knack of artificial intelligence-based computing paradigm for numerical treatment of the functional delay differential systems that portray the dynamics of the nonlinear influenza-A epidemic model (IA-EM) by implementation of neural network backpropagation with Levenberg–Marquardt scheme (NNBLMS). The nonlinear IA-EM represented four classes of the population dynamics including susceptible, exposed, infectious and recovered individuals. The referenced datasets for NNBLMS are assembled by employing the Adams method for sufficient large number of scenarios of nonlinear IA-EM through the variation in the infection, turnover, disease associated death and recovery rates. The arbitrary selection of training, testing as well as validation samples of dataset are utilizing by designed NNBLMS to calculate the approximate numerical solutions of the nonlinear IA-EM develop a good agreement with the reference results. The proficiency, reliability and accuracy of the designed NNBLMS are further substantiated via exhaustive simulations-based outcomes in terms of mean square error, regression index and error histogram studies.

Pth moment input-to-state stability of impulsive stochastic functional differential systems with Markovian switching

The paper introduces a unified criterion for pth moment input-to-state stability (p-ISS), pth moment integral input-to-state stability (p-iISS), and eσt-weighted pth moment integral input-to-state stability (eσt-p-ISS) of impulsive stochastic function differential system with Markovian switching under the perturbation of the stabilizing impulse and destabilizing impulse. The Lyapunov function approach, comparison principle, and impulsive average dwell-time method are applied in this paper. The linear coefficients of the upper bound of Lyapunov functional differential operators are time-varying functions, including the case of constants, which advances and improves the existing results. In addition, the same results were obtained by applying impulse differential inequality. At the end of this paper, we use a numerical example to verify the validity of the results.

Exponential stability analysis for a class of switched nonlinear time-varying functional differential systems

This paper proposes a unified approach for studying global exponential stability of a general class of switched systems described by time-varying nonlinear functional differential equations. Some new delay-independent criteria of global exponential stability are established for this class of systems under arbitrary switching which satisfies some assumptions on the average dwell time. The obtained criteria are shown to cover and improve many previously known results, including, in particular, sufficient conditions for absolute exponential stability of switched time-delay systems with sector nonlinearities. Some simple examples are given to illustrate the proposed method.

ON THE CONTROLLABILITY OF LINEAR EVOLUTIONARY FUNCTIONAL-DIFFERENTIAL SYSTEMS

New theorem on the controllability of linear functional-differential system of evolutionary type and algorithm for practical verification of controllability are obtained, which can be applied even in the case when the coefficients of system are not continuously differentiable over the time interval under consideration. Special cases of this system are nonstationary systems with distributed and concentrated delay, integro-differential systems with a Voltaire integral and ordinary differential systems. The main results obtained are formulated in the form of 12 theorems and 3 corollaries. On their basis an algorithm for practical verification of the controllability of the system under consideration using a computer is built. Examples illustrating the operability of the obtained theorems and the algorithm are given. The algorithm is implemented in the Maple 17 package for examples of second-order differential systems with delay.

Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h(⋅)↦∫0⋅U(⋅,s)h(s)ds:L1([0,T];Y)→C([0,T];Y), where Y is a Banach space and U(⋅,⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.

A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.