Mittag-Leffler Stabilization Research Articles

Sliding mode control for the stabilization of fractional heat equations subject to boundary uncertainty

By adopting the sliding mode control (SMC) and the generalized Lyapunov method, the boundary feedback stabilization issue is studied for the fractional diffusion system subject to boundary control matched disturbance. The classical sliding surface and the fractional integral sliding function are constructed and sliding mode controllers are designed respectively to realize the Mittag-Leffler (M-L) stabilization of the considered system. The controller based on the newly-introduced fractional integral sliding function not only helps to relax the constraints on the disturbance but also realizes the same stabilization effect as that of the classical one. The well-posedness result of the solution is also obtained for discontinuous fractional heat equations. Besides, a numerical experiment validates the theoretical outcomes.

The stabilization of uncertain dynamic systems involving the generalized Riemann-Liouville fractional derivative via linear state feedback control

The main objective of this paper is to investigate the stabilization problem of uncertain fractional-order dynamic systems (UFDSs) that involve the generalized Riemann-Liouville fractional derivative using a linear feedback controller. We establish several sufficient conditions for achieving Mittag-Leffler stabilization and asymptotic stabilization of UFDSs. The primary methodology employed in our study is Lyapunov's direct method (LDM), along with other inequality techniques, which have demonstrated significant efficacy in studying the stability theory of nonlinear dynamic systems. To demonstrate the effectiveness of the proposed approach, examples and corresponding simulations are provided.

Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control

Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control

Boundary Mittag–Leffler stabilization of a class of time fractional‐order nonlinear reaction–diffusion systems

AbstractBoundary control design of a class of time fractional‐order nonlinear reaction–diffusion systems (FNRDSs) is considered in three cases: domain‐averaged measurement, collocated boundary point measurement, and anti‐collocated boundary point measurement. For domain‐averaged measurement and collocated boundary point measurement, boundary controllers are designed directly based on the measurements, respectively. For the anti‐collocated boundary point measurement, to overcome the difficulty that the measurement cannot be used for the controller design directly, an observer is constructed firstly, and then, an observer‐based boundary controller is designed. Sufficient conditions for Mittag–Leffler (M‐L) stability of the closed‐loop system are all provided in terms of linear matrix inequalities (LMIs). Numerical simulation results are provided to illustrate the feasibility and effectiveness of the proposed methods.

Impulsive control strategies of mRNA and protein dynamics on fractional-order genetic regulatory networks with actuator saturation and its oscillations in repressilator model

In genetic regulatory networks (GRNs), the control strategies of messenger RNA (mRNA) and protein play a key role in regulatory mechanisms of gene expression, especially in translation and transcription. However, the influence of impulsive control strategies on oscillatory gene expression is not well understood. In this article, by considering the impulsive control strategies of mRNA and protein, a novel fractional-order genetic regulatory networks with actuator saturation is proposed. By applying polytopic representation technique, the actuator saturation term is first considered into the design of impulsive controller, and less conservative linear matrix inequalities (LMIs) criteria that guarantee finite-time Mittag-Leffler stabilization problem for fractional-order genetic regulatory networks are given. The derived sufficient conditions can easily be verified by designing impulsive control gains and solving simple LMIs. Finally, to investigate the effectiveness and applicability of the control strategies, an interesting simulation example as a synthetic oscillatory network of transcriptional regulators in Escherichia coli is illustrated.

Synchronization of fractional-order chaotic networks in Presnov form via homogeneous controllers

The Presnov decomposition for continuously differential vector fields consists of a gradient of a scalar field (dissipative term) and a vector field orthogonal to a radial vector of the origin (circulative or non-dissipative term). This paper uses the Presnov decomposition associated with a fractional-order Lorenz family to design algorithms that solve the multi-synchronization problem. Assuming that the whole state is known, two active control solutions are proposed to achieve the desired synchronization. Both control algorithms take advantage of homogeneous functions to have a powerful and simple Lyapunov analysis to show Mittag-Leffler stabilization in the first algorithm. In contrast, the second one achieves synchronization before a predefined time defined by the user. The combination of homogeneous functions and Riemann–Liouville integral in the controllers improve the behavior of the dynamic systems due to the impact on the robustness properties in the control scheme. Numerical simulations are presented to show the reliability of the proposed methods.

Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays

<abstract> <p>Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays are investigated in the article. We derive the relationship between the fractional-order integrals of the state function with and without delays through the division of time interval, using the properties of fractional calculus, and initial conditions. Moreover, by constructing the sequence solution of the system function which converges to a continuous function uniformly with the Arzela-Asoli theorem, a sufficient condition is obtained to ensure the existence of an anti-periodic solution and Mittag-Leffler stabilization of the system. In the final, we verify the correctness of the conclusion by numerical simulation.</p> </abstract>

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives.

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

Mittag-Leffler stabilization of fractional infinite dimensional systems with finite dimensional boundary controller

This paper studies the Mittag-Leffler stabilization for unstable infinite dimensional systems actuated by boundary controllers described by time fractional reaction diffusion equations. Via the Riesz basis method, the stable and the unstable part of the considered system are separated. The Kalman rank criterion, which is a classical linear algebra condition, guarantees the stabilizability of the unstable subsystem. Based on these, a controller governed by a finite dimensional system (so called the finite dimensional controller hereafter) is designed to achieve the Mittag-Leffler stability of the closed loop. From infinite to finite, this methodology is a substantial improvement for the existing control laws.

Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

AbstractThis paper studies the boundary feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems with non‐collocated or collocated outputs in the case that the system state is unmeasurable. By employing the backstepping method, we introduce a target system and analyze its Mittag–Leffler stability. For each pair of sides of the region boundary, assuming that system state is hinged on one side, while the controller is designed on the opposite side. These boundary controllers work together to achieve the state feedback Mittag–Leffler stabilization of the considered system. In addition, for both kinds of outputs, feedback controllers are also established to achieve the asymptotical stability of the corresponding closed‐loop system. Two numerical experiments are carried out to illustrate our results.

Boundary Mittag-Leffler stabilization of coupled time fractional order reaction–advection–diffusion systems with non-constant coefficients

Abstract This paper is concerned with boundary control for a class of coupled time fractional order reaction–advection–diffusion (FRAD) systems with non-constant coefficients (space-dependent coefficients) by state feedback. Partial differential equation (PDE) backstepping makes available to stabilize coupled time FRAD systems modeled by fractional PDEs. With boundary controller design and discussion on well-posedness of control kernel equations, the Mittag-Leffler stability of the closed-loop system is analyzed theoretically by the fractional Lyapunov method. A numerical scheme is constructed for coupled FRAD system to simulate numerical examples when the kernel equations have not the explicit solution. Comments on robustness to perturbation parameters in system coefficients are finally stated.

Boundary Mittag-Leffler Stabilization of a Class of Time Fractional Order Nonlinear Reaction-Diffusion Systems

Boundary Mittag-Leffler Stabilization of a Class of Time Fractional Order Nonlinear Reaction-Diffusion Systems

Boundary Mittag-Leffler stabilization of fractional reaction–diffusion cellular neural networks

Mittag-Leffler stabilization is studied for fractional reaction–diffusion cellular neural networks (FRDCNNs) in this paper. Different from previous literature, the FRDCNNs in this paper are high-dimensional systems, and boundary control and observed-based boundary control are both used to make FRDCNNs achieve Mittag-Leffler stability. First, a state-dependent boundary controller is designed when system states are available. By employing the spatial integral functional method and some inequalities, a criterion ensuring Mittag-Leffler stability of FRDCNNs is presented. Then, when the information of system states is not fully accessible, an observer is presented to estimate the system states based on boundary output and an observer-based boundary controller is provided aiming to stabilize the considered FRDCNNs. Furthermore, a robust observer-based boundary controller is proposed to ensure the Mittag-Leffler stability for FRDCNNs with uncertainties. Examples are given to illustrate the effectiveness of obtained theoretical results.

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

Bilateral Output Feedback Control of Fractional PDEs with Space-Dependent Coefficients

Abstract This paper develops an extension of the bilateral control method for fractional partial differential equations (PDEs) with space-dependent coefficients by output feedback. Using a backstepping transformation, a full state feedback control law is designed. Then the fractional PDE system is folded into two subsystems and Mittag-Leffler convergent state observers of these subsystems are derived. Although the observers are coupled with boundary conditions (BCs), error subsystems are decoupled by assuming some available measurements. Hence, the observer gains are easily obtained. After this, we compose the designed state feedback controller and observers to enable Mittag-Leffler stabilization by output feedback. Finally, a fractional numerical example is provided to support the effectiveness of the proposed synthesis for the case when neither the control kernel nor the estimation kernel has an explicit solution.

Adaptive control of Mittag-Leffler stabilization and synchronization for delayed fractional-order BAM neural networks

This paper is committed to studying adaptive control design of Mittag-Leffler stabilization and synchronization for delayed fractional-order bidirectional associative memory (BAM) neural networks. Considering better dynamic property and steady state performance of the system, we adopt adaptive control approaches to stabilize and synchronize two types of delayed fractional-order BAM neural network. It is a remarkable fact, based on adaptive control scheme, the method of auxiliary functions, Mittag-Leffler stabilization and synchronization theories with respect to fractional-order systems, the adaptive controllers are very well designed in a controlled system and two coupled systems, separately. Two examples are performed to illustrate the advantage of the presented theoretic analysis and results.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay.

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Global Stabilization for a class of nonlinear fractional-order systems

The problem of Mittag–Leffler stabilization (MLS) is studied for a class of nonlinear non-integer order systems. The stabilizer is constructed by using the Lyapunov function and backstepping algorithm. The continuous controller is designed to ensure that the state of the nonlinear fractional-order closed-loop system converges to the equilibrium. Two simulation examples are given to illustrate the effectiveness of the method.

Mittag-Leffler Stabilization of an Unstable Time Fractional Hyperbolic PDE

This paper aims to study the Mittag-Leffler stabilization of an unstable time fractional hyperbolic partial differential equation by boundary control and boundary measurement. The backstepping method, the fractional Lyapunov method, and the semigroup theory are adopted in the investigation. A novel state feedback control via the Dirichlet boundary is designed to stabilize the controlled system. Based on the output signal, we first construct an observer that can recover the state of the original system, and then, we propose an observer-based stabilizing control law, under which the closed-loop system is shown to admit a unique solution and to be Mittag-Leffler stable. Finally, a benchmark example is presented to test the proposed theory.

Observer Design for Semilinear Time Fractional Diffusion Systems with Spatially Varying Parameters

This paper describes how to design observers for the semilinear time fractional diffusion systems with spatially varying parameters. The motivation is that the availability of full-state measurements in some practical cases may be impossible for direct measurement due to the difficulties in measuring. For this, an integral transformation is first proposed to equivalently convert the considered systems into a simpler form with Robin boundary conditions. Besides, we build an extended Luenbergertype observer that uses measurements of the inputs and the outputs of system under consideration to determine its state. MittagLeffler stabilization problem of the corresponding semilinear observer error dynamic systems is then explored via backstepping technique. A numerical illustration is finally included.