This paper investigates the sliding mode boundary stabilization for uncertain delay Markovian reaction-diffusion neural networks with partially unknown transition rates. First, a mode-dependent sliding mode surface (SMS) and a mode-dependent sliding mode boundary controller are designed, offering enhanced adaptability to different modes. Next, the almost sure finite-time reachability of the proposed SMS is derived, despite the challenges posed by Markovian switching with partially unknown transition rates. Then, by employing the Lyapunov stability method, inequality analysis techniques, and the free-weighting matrices approach, a criterion is established to guarantee the mean-square robust exponential stability of the closed-loop system. The proposed approach effectively overcomes the challenges arising from the coexistence of partially unknown transition rates, diffusion dynamics, and boundary disturbances under Markovian switching. Finally, the proposed theoretical results are validated via application to temperature control in lithium-ion battery packs for electric vehicles, demonstrating their effectiveness.

The control problems are investigated for a stochastic reaction–diffusion system (SRDS) cascaded with an ordinary differential system (ODS). We propose a novel hybrid control strategy that combines boundary control for the SRDS with state feedback control for the ODS. The primary objective is to ensure the cascaded system achieves mean-square exponential stability (MSES). Our research offers a flexible framework for stability analysis and control synthesis in cascaded systems, applicable regardless of the ODS’s inherent stability. We further explore the system’s robust stabilization with uncertainties and its$H_{\infty } $performance to quantify disturbance-rejection capabilities. The key innovations include the development of a unified control framework that accommodates both stable and unstable ODE subsystems, the design of boundary control inputs that ensure MSES even in the presence of parameter uncertainties, and the application of matrix inequality techniques to simplify controller design. The effectiveness of the proposed control strategies is demonstrated by three numerical examples.

This paper considers the interval estimation for delayed linear reaction-diffusion systems. To obtain the estimates of the states for delayed reaction-diffusion systems, a novel interval estimation scheme is proposed based on a spatial finite difference method and decoupling technology. First, the delayed reaction-diffusion system is discretized by the finite difference method, resulting in approximated delay differential equations. The bounds of the discretization errors are presented via a rigorous analysis. Then, we involve decoupling techniques into interval observer design for approximated delay differential equations, and its estimates are obtained. Moreover, combining the observations with the discretization errors, we provide the interval estimation of the delayed reaction-diffusion systems. Furthermore, a controller is designed using the observations to achieve the$H_{\infty}$performance. Finally, the proposed methods are validated by two numerical simulations.Note to Practitioners—This work is motivated by industrial problems instance of the thermal diffusion in CPU chips. Due to the significant cost of real-time measurement of chip’s temperature and the influence of environment, it is more feasible to obtain the thresholds of operating temperature based on known information and available measurements. To obtain the state thresholds, this paper focuses on designing interval observer and analysis of discretization error to guarantee the accuracy while reducing the impact of uncertainties. The methods in this paper is simple and fast to execute in practical engineering, and the discretization error does not need to be obtained through a priori numerical experiment. More importantly, the estimates include the worst-case during system operation, which is relatively practical in real applications. The proposed results aim to provide a helpful reference for controller synthesis and fault detection of spatio-temporal system with time delays, thereby promoting corresponding application research.

This paper mainly analyzes the finite-time stabilization of semi-Markov reaction-diffusion memristive neural networks (R-DMNNs) with unbounded time-varying delays. Firstly, the reaction-diffusion term and semi-Markov jumping are introduced into memristive neural networks, which relaxes the limitation of Markov switching on sojourn time and makes the model more applicable. Secondly, by constructing a suitable comparison function, the states of R-DMNNs converges to 0 directly, which can clearly estimate the upper limit of the settling time and simplify the complexity of the theoretical derivation. Furthermore, this paper removes the requirement of bounded and differentiable time delay, which provides a new perspective for understanding the finite-time stabilization of the neural networks with reaction-diffusion terms. Finally, one example illustrates the usefulness of the analysis results in this research.

This study investigates the stability of reaction-diffusion systems (RDSs) under impulsive disturbances using an event-triggered impulsive control method. Our key contribution is providing Zeno-free conditions for the event-triggered mechanism (ETM), which is crucial due to the potential for impulsive disturbances to trigger the sampling threshold earlier than expected, leading to Zeno behavior. We address this challenge by deriving conditions that ensure the ETM operates without Zeno behavior, essential for practical control implementation. We further derive several sufficient conditions for the asymptotic stability of RDSs based on impulsive control theory. Special cases with specific disturbance rules are also discussed, offering a broader understanding of system stability under various conditions. To demonstrate the applicability of our theoretical findings, we apply our control strategies to an atmospheric pollution model. A numerical example is provided to validate the effectiveness of our approach and to offer theoretical guidance for real-world environmental pollution control.

This article tackles asynchronous control issue for a class of stochastic Markovian reaction-diffusion neural networks with mode-dependent delays (MDDs). Taking into account the spatio-temporal distribution of such networks, we propose a boundary control (BC) scheme combined with asynchronous control to reduce control implementation cost and overcome environmental constraint. By incorporating a hidden Markov model to manage the mode asynchrony, we develop an integral asynchronous boundary controller for Neumann boundary conditions, as well as an innovative one for Dirichlet boundary conditions. We then derive an exponential stability criterion specific to MDDs and introduce a novel asynchronous BC synthesis approach. Additionally, we extend our findings to the leader-follower synchronization of these neural networks. The validity, superiority, and practicality of the proposed control design approach are demonstrated via three numerical examples, respectively.

This work investigates the observer-based asynchronous boundary stabilization for a kind of stochastic Markovian reaction-diffusion neural networks with exogenous disturbances. Specifically, parameter uncertainties are considered in the drift item. First, a hidden Markov model is introduced that guarantees the observer modes run asynchronously with the system modes. It should be noted that the asynchronous observer constructed in this work only uses the boundary measurement information. Then a nonfragile asynchronous observer-based boundary controller is designed. Taking advantage of inequality techniques and stochastic analysis method, sufficient criterion is provided to satisfy input-to-state exponentially mean-square stability, and the asynchronous boundary observer/controller gains are further derived. As a special case, the synchronous observer-based boundary stabilization is also obtained. Finally, a numerical example is exploited to manifest the validity of the established results.

This article concerns with the asynchronous boundary control for a class of Markov jump reaction-diffusion neural networks (MJRDNNs). In consideration of nonsynchronous behavior between the system modes and controller modes, a novel asynchronous boundary control design is proposed for MJRDNNs. Based on the designed asynchronous boundary controller, a sufficient criterion is established to ensure the stochastic finite-time boundedness for the considered MJRDNNs by constructing a Lyapunov–Krasovskii functional and utilizing Wirtinger-type inequality. Then, a sufficient condition is acquired to guarantee that MJRDNNs are stochastic finite-time bounded with$H_{\infty }$performance. Finally, a numerical example is provided to illustrate the effectiveness of the proposed design method.

This article studies the synchronization of new coupled fractional delayed reaction–diffusion neural networks with reaction terms satisfying the global Lipschitz condition via time-continuous and time-discontinuous boundary controllers. The realization of neural networks inevitably involves diffusion phenomena and time delays, and all the neurons of neural networks are interrelated. Considering these aspects, this study focuses on coupled fractional neural networks with time-delay and diffusion terms. A state-dependent boundary control (BC) is designed for when the state information is available, and a criterion is presented to ensure the synchronization of the considered systems. Considering the advantages of a time-discontinuous controller, an intermittent BC and a criterion of synchronization are given. When the state information cannot be fully obtained, a boundary-output-based observer is provided for estimating the states. Then, an observer-based intermittent boundary controller is given to ensure the synchronization. From the given criteria, the effects of time delay and the control time length on synchronization are analyzed. This research involves two key challenges: 1) consideration of the BC and intermittent control parameters in the system performance analysis and 2) clarification of the influence of system parameters on synchronization. These challenges are addressed using Poincaré’s inequality, the fractional Razumikhin-type theorem, and several properties of the Mittag-Leffler function are used to deal with the above difficulties. Examples show that our results are valid.

This study considers the boundary stabilization for stochastic delayed Cohen-Grossberg neural networks (SDCGNNs) with diffusion terms by the Lyapunov functional method. In the realization of NNs, sometimes time delays and diffusion phenomenon cannot be ignored, so Cohen-Grossberg NNs with time delays and diffusion terms are studied in this article. Moreover, different from the previously distributed control, the boundary control is used to stabilize the system, which can reduce the spatial cost of the controller and is easy to implement. Boundary controllers are presented for system with Neumann boundary and mixed boundary conditions, and criteria are derived such that the controlled system achieves mean-square exponential stabilization. Based on the criterion, the effects of diffusion matrix, coupling strength, coupling matrix, and time delays on exponentially stability are analyzed. In the process of analysis, two difficulties need to be addressed: 1) how to introduce boundary control into system analysis? and 2) how to analyze the influence of system parameters on stability? We deal with these problems by using Poincaré's inequality and Schur's complement lemma. Moreover, mean-square exponential synchronization of stochastic delayed Hopfield NNs with diffusion terms, as an application of the theoretical result, is considered under the boundary control. Examples are given to illustrate the effectiveness of the theoretical results.

Dissipativity-based asynchronous boundary stabilization problem is addressed for stochastic Markov jump reaction-diffusion systems (SMJRDSs). In practical engineering, nonsynchronous behavior between system modes and controller modes is inevitable, and the incomplete matrix information makes the problem analysis difficult, so this work considers the asynchronous stabilization. Different from the distributed control, we apply a simple boundary control strategy, which greatly reduces the cost of the control design. Note that three issues need to be addressed: 1) how to model the asynchronous behavior? 2) how to design the asynchronous boundary controller? and 3) how to process the incomplete matrix information? We deal with these problems one by one. Based on a general hidden Markov model (HMM), an asynchronous boundary feedback controller is considered. Via the Wirtinger-type inequality, Schur complement technique, and transition matrix properties, sufficient conditions ensuring exponentially mean square stability and strictly$(W, P, R)-\alpha $dissipativity are established, which covers several special cases. Finally, a numerical example is presented to illustrate the proposed control strategies.