Discrete-time State Observations Research Articles

Some Results of Stochastic Differential Equations

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory.

Stabilization of impulsive hybrid stochastic differential equations with Lévy noise by feedback control based on discrete-time state observations

In this paper, we investigate the problem of the mean-square exponential stabilization for a class of unstable impulsive hybrid stochastic differential equations with Lévy noise (IHSDEs-LN) via feedback control based on discrete-time state observations. Our results show that if feedback control of continuous-time observations can stabilize the controlled system in the sense of mean-square exponential stability, then feedback control of discrete-time state observations can also stabilize the controlled system. And there exists an upper bound on the interval between adjacent observation moments that can be accurately computed. Based on this new theory, we first design continuous-time feedback control for unstable IHSDEs-LN to achieve the stabilization problem, and then improve the continuous-time feedback control to discrete-time state observations feedback control.

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.

Semi-global exponential stabilization of stochastic nonlinear systems with distributed delay under discrete-time state observations and application to circuit systems

This paper is aimed at studying the semi-global exponential stabilization problem for stochastic nonlinear systems with distributed delay under discrete-time state observations. An extended definition of horizontal translation of the segment function along the investigated systems is given, related to the stochastic disturbance and Brownian motion. Basically, the right and upper Dini’s derivative of the constructed functional is presented. Combining this with the global exponential stability of the stochastic nonlinear systems with distributed delay under continuous-time feedback control, the Lyapunov method and Halanay inequality, the maximum observation interval and some stabilization criteria to make the investigated systems under discrete-time state observations reach semi-global exponential stabilization are provided. The results indicate that the semi-global exponential stabilization is dependent upon stochastic disturbance, initial condition range, and maximum observation interval. Finally, an application to the circuit systems is discussed and the involved sufficient conditions for the circuit systems to keep semi-global stabilization are also proposed. Besides, the numerical simulation results confirm the effectiveness of the theoretical results.

Almost sure exponential stabilization of impulsive Markov switching systems via discrete-time stochastic feedback control

It is well known that noise can stabilize an unstable Markov switching system (MSS). However, given an unstable impulsive Markov switching system (IMSS), can it be stabilized by noise? To the best of the author’s knowledge, this question has been rarely explored. If the stochastic feedback controller is observed in discrete time, the research is more difficult. In this paper, a comparative method is used to establish a noise stabilization strategy based on discrete-time state and mode observations for IMSSs. In addition, we use the stationary distribution of the ergodic Markov chain instead of the M-matrix technique to obtain the pth moment estimation on the gap between the discrete-time observed controlled system’s states and the auxiliary system’s states, so that the control strategy can be applied to a wider class of IMSSs. Finally, we applied the proposed control scheme to impulsive neural networks with Markovian switching (INNs-MS) and impulsive boost converter systems (IBCSs). Through an example and several simulations, we validated the effectiveness and feasibility of this control scheme.

Delay Feedback Control Based on Discrete-time State Observations for Highly Nonlinear Hybrid Stochastic Functional Differential Equations With Infinite Delay

Delay Feedback Control Based on Discrete-time State Observations for Highly Nonlinear Hybrid Stochastic Functional Differential Equations With Infinite Delay

Stabilization of nonlinear hybrid stochastic time-delay neural networks with Lévy noise using discrete-time feedback control

<p>This paper aims to formulate a class of nonlinear hybrid stochastic time-delay neural networks (STDNNs) with Lévy noise. Specifically, the coefficients of networks grow polynomially instead of linearly, and the time delay of given neural networks is non-differentiable. In many practical situations, nonlinear hybrid STDNNs with Lévy noise are unstable. Hence, this paper uses feedback control based on discrete-time state and mode observations to stabilize the considered nonlinear hybrid STDNNs with Lévy noise. Then, we establish stabilization criteria of $ H_{\infty} $ stability, asymptotic stability, and exponential stability for the controlled nonlinear hybrid STDNNs with Lévy noise. Finally, a numerical example illustrating the usefulness of theoretical results is provided.</p>

Stabilization of highly nonlinear stochastic coupled systems with Markovian switching under discrete-time state observations control

This paper investigates the stabilization of highly nonlinear stochastic coupled systems (HNSCSs) with Markovian switching under discrete-time state observations control. Different from most existing literature, the condition in which the diffusion coefficient and drift coefficient fulfill the linear growth condition is removed. In other words, the coupled systems considered are highly nonlinear. Combining the graph theory and Lyapunov method, we establish the stability criteria for HNSCSs with Markovian switching. Meanwhile, the upper bounded for duration between two successive state observations is proposed. Furthermore, the theoretical results are applied to study the stability of nonlinear electric RLC circuits. Lastly, a numerical example with simulations is provided to show the viability of obtained results.

Synchronization for stochastic large-scale systems via intermittent delay discrete observation control

This paper proposes a novel intermittent delay discrete observation control for synchronization of stochastic large-scale systems (LSs) with semi-Markov jump. Different from the existing intermittent delay control, our proposed control strategy is based on discrete-time state observations rather than continuous-time state observations during the work time. And by applying the method of average control rate, the control strategy becomes more practical. Then we establish a lower conservative differential inequality, which is significantly conducive to analyze the synchronization of LSs with multiple delays. Based on the generalized Halanay-type inequality, Lyapunov method and stochastic analysis techniques, several sufficient conditions are derived. Finally, we apply the theoretical results to Chua’s circuit with numerical examples being given.

Structured stabilisation of superlinear delay systems by bounded discrete-time state feedback control

Taking different system structures in different Markovian modes into consideration, this paper studies the structured stabilisation of a class of superlinear hybrid stochastic delay systems by feedback control based on discrete-time state observations. The controller is designed in a bounded state area, rather than every observable state, in order to reduce control cost. The time delay is more general in terms of the classical differentiability assumption being relaxed. Compared with the existing papers on discrete-state-feedback stabilisation problem, a new method to estimate the difference between current-time state and discrete-time state is presented, as a result of which the conditions imposed on the underlying system and the control function are less restrictive. Meanwhile, the Lyapunov functional used in this paper is modified to adapt to this change. Finally, an application to stochastic structured neural networks is given to demonstrate the practicability of the developed theory.

Exponential bipartite synchronisation of stochastic signed networks via intermittent pinning discrete-time state observations control

This paper studies the exponential bipartite synchronisation in mean square (EBS-MS) of stochastic signed networks (SSNs) via periodically intermittent pinning control. It is important to note that intermittent control is an extension of existing literature, which depends on discrete observations instead of continuous observations during the work period. Additionally, only a small percentage of the network nodes are controlled by intermittent control. Several sufficient conditions are established by the Lyapunov method and graph theory. Especially, when the duration τ between two consecutive observations tends to zero, our control will degenerate into common periodically intermittent pinning control. Meanwhile, a corollary about the EBS-MS of SSNs is presented. Besides, theoretical results are applied to a single-link robot arm system and an oscillator system. Finally, numerical simulations are presented to show the feasibility of the theoretical results.

Periodically intermittent discrete observations control for stabilization of highly nonlinear hybrid stochastic differential equations

In this paper, the stabilization of a class of hybrid stochastic systems whose drift coefficients satisfy high nonlinear growth conditions is studied. A periodically intermittent controller based on discrete time state observations is designed. The core idea of this paper is to derive the stability of the controlled system by constructing Lyapunov function. An example is given to prove the validity of the conclusion.

Discrete-time state observations control to synchronization of hybrid-impulses complex-valued multi-links coupled systems

ABSTRACT The synchronization of stochastic hybrid-impulses complex-valued multi-links coupled systems (SHCMCS) based on discrete-time state observations control is studied. Therein, hybrid impulses, multiple links and complex-valued factors are considered simultaneously, which make the model more common. By using the Lyapunov method and Kirchhoff's Matrix Tree Theorem, average impulsive interval and average impulsive gain, the synchronization is achieved. Moreover, synchronization criteria are obtained, whose sufficient conditions reflect that the realization of synchronization depends on coupled strength and stochastic disturbance strength. In addition, stochastic complex-valued multi-links coupled oscillators with hybrid impulses are studied. Finally, a numerical test is presented to illustrate the validity of results.

Stochastic stabilization of hybrid neural networks by periodically intermittent control based on discrete-time state observations

This paper is concerned with stabilization of hybrid neural networks by intermittent control based on continuous or discrete-time state observations. By means of exponential martingale inequality and the ergodic property of the Markov chain, we establish a sufficient stability criterion on hybrid neural networks by intermittent control based on continuous-time state observations. Meantime, by M-matrix theory and comparison method, we show that hybrid neural networks can be stabilized by intermittent control based on discrete-time state observations. Finally, two examples are presented to illustrate our theory.

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.

Quasi-Synchronization of Fuzzy Heterogeneous Complex Networks via Intermittent Discrete-Time State Observations Control

This article focuses on quasi-synchronization of fuzzy heterogeneous complex networks. An aperiodically intermittent control strategy based on discrete-time state observations (aperiodically intermittent discrete-time state observations control for short) is designed. Different from intermittent control strategies referred in existing literatures, the control duration of the control strategy used in this article is based on discrete-time state observations, which makes the control used in this article somewhat less demanding and more effective. Under the aperiodically intermittent discrete-time state observations control strategy, a valid approach combining Lyapunov method with graph theory is proposed in this article. Throughout this article, a theorem and a corollary to the quasi-synchronization criterion of fuzzy heterogeneous complex networks are established. The results show that when the control gain is larger, the convergence domain is smaller. Finally, an illustrative example is presented and the simulation of this example shows the feasibility and validness of the obtained results.

Synchronization of complex-valued stochastic coupled systems with hybrid impulses via discrete-time state observations control

Synchronization of complex-valued stochastic coupled systems with hybrid impulses via discrete-time state observations control

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.

Stabilisation in distribution of hybrid stochastic differential equations by feedback control based on discrete-time state observations

A new concept of stabilisation of hybrid stochastic systems in distribution by feedback controls based on discrete-time state observations is initialised. This is to design a controller to stabilise the unstable system such that the distribution of the solution process tends to a probability distribution. In addition, the discrete-time state observations are also taken into consideration to make the design of the controller more practical. Theorems on the stabilisation of hybrid stochastic systems in distribution are proved. The lower bound of the duration between two consecutive state observations is obtained. The implementation of theorems are demonstrated by designing the feedback controls in the structure cases and easy-to-rules are provided for the user. Numerical examples are discussed to illustrate the theoretical results.

Impulsive Exponential Synchronization of Fractional-Order Complex Dynamical Networks with Derivative Couplings via Feedback Control Based on Discrete Time State Observations

Impulsive Exponential Synchronization of Fractional-Order Complex Dynamical Networks with Derivative Couplings via Feedback Control Based on Discrete Time State Observations