Intermittent distributed control for a class of nonlinear reaction-diffusion systems with spatial point measurements

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Abstract

This paper addresses the intermittent distributed stabilization problems of a class of nonlinear reaction-diffusion systems, where the state measurements are available at a finite number of sampling spatial points. By developing piecewise switching-time-dependent Lyapunov function methods combined with the descriptor system approach, L2-norm and H1-norm stability criteria are established, respectively. For the case of Dirichlet boundary condition, it is shown that the H1-norm stability implies the pointwise-in-space stability. The obtained stability conditions establish a quantitative relationship among the upper bound of the spatial sampling intervals, control width, and rest width. In the framework of linear matrix inequalities, sufficient conditions for the existence of intermittent distributed static output-feedback controllers are derived. A comparison between the two stability criteria is given by means of a numerical example.

Introduction

Reaction-diffusion equations are parabolic partial differential equations (PDEs) which usually exist in practice, such as distributed in space, thermodynamics, biology, heat transfer processes, under the influence of local chemical reactions, in which the substances are transformed into each other, and diffusion, which causes the substances to spread out over a surface in space [1], [2]. PDEs have a better approximation of the spatiotemporal actions and interactions of practical systems than ordinary differential equations. In recent years, stability, periodicity, and synchronization of reaction-diffusion PDEs have successfully attracted lots of attention [3], [4], [5], [6], [7], [8], [9], [10]. For examples, the distributed control theory and design algorithms for linear PDE systems were introduced in [3], [4]. The works of [5], [6] studied the stability problem of reaction-diffusion neural networks with distributed delays. In [9], [10], the authors presented impulsive synchronization criteria for two identical reaction-diffusion neural networks with mixed delays.

In recent years, discontinuous control strategies including impulsive control [9], [10], finite-time control [11], sliding-mode control [12], [13] and intermittent control [14], [15], [16] have drawn a lot of attention due to their advantages compared with continuous control strategies, such as the easy implementation, robustness and low maintenance costs. In the framework of intermittent control, two distinct modes operate alternately: the closed-loop mode for the control intervals and the open-loop mode for the rest intervals. It is noteworthy that the intermittent control reduces to the common continuous state-feedback control when the rest interval tends to zero. Moreover, the intermittent control usually includes two types: periodic case [14], [15], [16], [17], [18] and aperiodic case [19], [20], [21], [22], [23]. In fact, the intermittent controlled system provides a general model for some practical situations, such as controller failures and the purposeful suspension of the controller for equipment maintenance or decreased bandwidth usage. The main idea of stability analysis of intermittent control systems is to activate the stable closed-loop mode in control intervals to suppress the unstable open-loop mode in the rest intervals, e.g., see [17], [19], [22], [23]. However, it should be noted that the above-mentioned works related to the reaction-diffusion systems require that the state feedback information are spatially available. This may waste many sensors placed in the space to measure the state of the controlled systems.

Motivated by considerations about the efficient use of the available resources and the development of digital control techniques, there are some recent results on stability and stabilization of reaction-diffusion systems with spatially sampling, see the works [24], [25], [27], [28]. Nevertheless, to the authors best knowledge, there are few results concerning the intermittent distributed control using spatial point measurements. Furthermore, it should be pointed out that the above results related to intermittent distributed control only considered the stability problem in the sense of L2-norm, i.e., z(·,t)L20 as t+, where z(x, t) is the state of the closed-loop system. However, there could appear some “unbounded spikes” for some x in the spatial domain that does not contribute to the L2-norm. Actually, in some practical applications, both the system state and its spatial partial derivative are required to converge to the steady state, i.e., convergence in the sense of H1-norm. In the case of PDE systems with Dirichlet boundary condition, the H1-norm stability implies pointwise-in-space stability, see the work [26]. Therefore, it is of great theoretical significance to study the stability and stabilization in the sense of H1-norm for intermittent control systems with reaction-diffusion term.

This paper investigates intermittent distributed control problem for a class of nonlinear reaction-diffusion systems in which the measured output only can be available in a finite number of fixed sampling spatial points. The notable contributions of this paper are summarized as follows

  • Based on the discrete measurements of system states in space, a static output aperiodically intermittent distributed control strategy, allowing the lengths of the control intervals and rest intervals to be variable, is proposed. Compared with previous works, the proposed intermittent control scheme generalizes the periodic case [14], [15], [16], [17] and the aperiodic case [22], [23].

  • Novel L2-norm and H1-norm stability criteria are separately established, which provide a quantitative relationship between the upper bound of the spatial sampling interval, control width, and rest width. It is worth mentioning that unlike the Lyapunov direct methods for the L2-norm stability [14], [15], [16], [17], [22], [23], [28], a novel piecewise diffusion-dependent Lyapunov function method combined with descriptor system approach is developed for analyzing the H1-norm stability. Additionally, it is shown with a numerical example that the H1-norm stability criterion is less conservative than the L2-norm stability criterion.

  • Both the Neumann and the Dirichlet boundary conditions are studied. Moreover, it is proved that the obtained stability criteria for the Dirichlet boundary condition are less conservative than the ones for the Neumann boundary condition.

The rest of this paper is structured as follows. Section 2 presents system description and preliminaries. Section 3 establishes two stability criteria, namely, L2-norm stability and H1-norm stability, for reaction-diffusion systems via aperiodic intermittent distributed control strategy. Section 4 provides two sufficient conditions to design intermittent controllers. In Section 5, a numerical example is presented to make a comparison between the two kinds of stability.

Section snippets

Problem formulation

Notation. Let R and R+ denote the set of real numbers and the set of nonnegative real numbers, respectively. N represents the set of positive integers, and let N0=N{0}. For a real symmetric matrix A, the notation A > 0 (A < 0) means that the matrix A is positive (negative) definite. The symmetric elements of a symmetric matrix are denoted by *. For two integers n1 and n2 with n1 ≤ n2, the notation n1,n2¯ represents the set of {n1,n1+1,,n2}. C and C1 stand for the sets of functions which are

Main results

In this section, by introducing two switching-time-dependent Lyapunov functions, two aperiodically intermittent stabilization criteria will be established for system (1). For this purpose, for kN0, define some piecewise linear functions as follows:ρ11k(t)=ttksktk,ρ12k(t)=sktsktk,ρ10(t)=1sktk,tJ1,k,ρ21k(t)=tsktk+1sk,ρ22k(t)=tk+1ttk+1sk,ρ20(t)=1tk+1sk,tJ2,k.It is easy to see that ρijk(t) ∈ [0, 1], tR,kN0, andρ11k(tk)=ρ22k(tk+1)=ρ21k(sk)=ρ12k(sk)=0,ρ12k(tk)=ρ21k(tk+1)=ρ22k(sk)=ρ

Controllers design

If the intermittent gain matrix is unknown, then the matrix inequalities of Theorem 1 or Theorem 2 are nonlinear. In order to solve these matrix inequalities, there are some algorithm can be adopted, such as [8]. In this section, based on Theorem 1 and Theorem 2, simple methods are given to linearize these inequalities for designing the intermittent gain matrix Kj.

Theorem 3

For given a class of switching signals Sσ(δ11,δ12;δ21,δ22), and the step-size of spatial sampling Δj,j1,N¯. If for prescribed

An illustrative example

In this section, a numerical example is presented to demonstrate the efficiency of the obtained theoretical results.

Consider the feedback control problem for a heat equation with the following form:zt(x,t)=zxx(x,t)+1.8z(x,t)+j=1Nbj(x)uj(t),t>0,x[0,π],where the intermittent controller (7) is chosen with Kj=3,j1,N¯, and N to be determined later. To show the differences between the L2-norm stability and H1-norm stability, the following two cases are considered.

Case 1: Calculating the maximum

Conclusion

The intermittent distributed control problem for a class of nonlinear reaction-diffusion systems with Neumann boundary condition or Dirichlet boundary condition has been addressed. Different from the previous works concerning intermittent stabilization of these systems, it is assumed that a finite number of point spatial state measurements are available, and both the control width and the rest width can be variable. By using time-dependent Lyapunov function approaches and Wirtinger’s inequality

Acknowledgment

The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. This work is supported in part by National Natural Science Foundation of China [grant number 61873061].

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