Stability and 𝑙₁-Gain Analysis for Positive 2D Systems with State Delays in the Roesser Model

Abstract

This paper considers the problem of delay-dependent stability and 𝑙₁-gain analysis for positive 2D systems with state delays described by the Roesser model. Firstly, the copositive-type Lyapunov function method is used to establish the sufficient conditions for the addressed positive 2D system to be asymptotically stable. Then, 𝑙₁-gain performance for the system is also analyzed. All the obtained results are formulated in the form of linear matrix inequalities (LMIs) which are computationally tractable. Finally, an illustrative example is given to verify the effectiveness of the proposed results.

1. Introduction

2D systems exist in many practical applications, such as circuits analysis, digital image processing, signal filtering, and thermal power engineering [1–4]. Thus the analysis and synthesis of 2D systems are interesting and challenging problems, and they have received considerable attention; for example, 2D state-space realization theory was researched in [5], the stability and 2D optimal control theory was studied in [6, 7], and 𝐻_∞ control and filtering problem for 2D systems were addressed in [8–11]. In addition, linear repetitive processes, a distinct class of 2D systems, have also been investigated. For example, the quasi-sliding mode control problem for linear repetitive processes with unknown input disturbance was solved in [12].

The most popular models of two-dimensional (2D) linear systems were introduced by Roesser [13], Fornasini and Marchesini [5, 14], and Kurek [15]. These models have been extended to positive systems in [16–19]. A positive system means that its state and output are nonnegative whenever the initial condition and input are nonnegative [19–21]. Positive 2D systems are needed in many cases such as the wave equation in fluid dynamics and the heat equation which describes the temperature (using thermodynamic temperature scale) in a given region over time and the Poisson’s equation. These facts stimulate the research on 2D positive discrete systems. Reference [22] investigated the choice of the forms of Lyapunov functions for positive 2D Roesser model. The problem of stability analysis for 2D positive systems has been investigated in [17, 23–25]. It should be noted that although positive 2D systems have been discussed in control engineering and mathematics literature recently, there are still many questions which deserve further investigation.

On the other hand, the reaction of real-world systems to exogenous signals is never instantaneous and, always infected by certain time delays. For general systems, even nominal stable systems when were affected by delays may inherit very complex behaviors such as oscillations, instability, and bad performance [26], and delayed systems have attracted many researchers’ attention [27–32]. The reachability, minimum energy control, and realization problem for positive 2D discrete-time systems with delays has been analyzed in [18, 33]. And the stability analysis for 2D positive delayed systems has been investigated in [34–36]. In addition, perturbations and uncertainties widely exist in the practical systems. In some cases, the perturbations and unmodeled errors can be merged into disturbances, which can be supposed to be bounded in the appropriate norms. It is important and necessary to establish a criterion evaluating the disturbance attenuation performance for the positive 2D discrete-time systems. However, to the best of our knowledge, there has been no literature considering the disturbance attenuation performance for positive 2D systems, which motivates the present study.

In this paper, we will study the problem of delay-dependent stability and 𝑙₁-gain analysis for positive 2D linear systems with delays. The main theoretical contributions of this paper are as follows (1) We use 𝑙₁-gain to evaluate the disturbance attenuation performance of positive 2D linear systems. This important performance is firstly considered for positive 2D systems, and a delay-dependent stability criterion of these systems with state delays is developed. (2) Copositive-type Lyapunov function method is firstly used to analyze delay-dependent stability and 𝑙₁-gain performance for positive 2D linear systems. (3) It is significant to characterize conditions under which the positive 2D delayed system is asymptotically stable. All the developed results are expressed in terms of feasibility testing of LMIs which is computationally tractable.

The paper is organized as follows. In Section 2, problem statement and some definitions concerning the positive 2D linear systems with delays are given. In Section 3, some theorems concerning the delay-dependent stability and 𝑙₁-gain analysis of positive 2D linear systems are presented. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed results. Finally, concluding remarks are provided in Section 5.

Notations. In this paper, the superscript “𝑇” denotes the transpose. The notation 𝑋>𝑌 (𝑋≥𝑌) means that matrix 𝑋-𝑌 is positive definite (positive semidefinite, resp.). 𝐴≽0 (⪯0) means that all entries of matrix 𝐴 are nonnegative (nonpositive). 𝐴≻0 (≺0) means that all entries of matrix 𝐴 are positive (negative). 𝑅^𝑛×𝑚 denotes the set of 𝑛×𝑚 real matrices. The set of real 𝑛×𝑚 matrices with nonnegative entries will be denoted by 𝑅₊^𝑛×𝑚, 𝑅^𝑛₊ denotes the set of vectors with nonnegative entries, and the set of nonnegative integers will be denoted by 𝑍₊. The 𝑛×𝑛 identity matrix will be denoted by 𝐼_𝑛. The 𝑙₁ norm of a 2D signal 𝑤(𝑖,𝑗)=[𝑤₁(𝑖,𝑗),𝑤₂(𝑖,𝑗),…,𝑤_𝑚(𝑖,𝑗)]^𝑇 is given by 𝑤𝑖,𝑗₁=^𝑚_𝑘=1𝑤_𝑘𝑖,𝑗. (1) And we say 𝑤(𝑖,𝑗)∈𝑙₁, if ‖𝑤(𝑖,𝑗)‖₁<∞.

2. Problem Formulation and Preliminaries

Consider the positive 2D Roesser model with state delays [25]: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣𝑥𝑖,𝑗+1=𝐴^ℎ𝑥𝑖,𝑗_𝑣𝑖,𝑗+𝐴_𝑑𝑥^ℎ𝑖−𝑑_ℎ(𝑖)𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣𝑗+𝐵𝑤𝑖,𝑗,𝑧𝑖,𝑗=𝐻𝑥𝑖,𝑗+𝐿𝑤𝑖,𝑗, (1a)(1b) where 𝑖 and 𝑗 are integers in 𝑍₊, 𝑥^ℎ(𝑖,𝑗) is the horizontal state in𝑅^𝑛1₊, 𝑥^𝑣(𝑖,𝑗) is the vertical state in 𝑅^𝑛2₊, 𝑥(𝑖,𝑗) is the whole state in 𝑅^𝑛₊, 𝑤(𝑖,𝑗)∈𝑅^𝑚1₊ is the 𝑙₁ norm bounded disturbance input, 𝑧(𝑖,𝑗)∈𝑅^𝑚2₊ is the controlled output, and 𝐴, 𝐴_𝑑, 𝐵, 𝐻, 𝐿≽0 are system matrices with compatible dimensions. The matrices are 𝐴𝐴=¹¹𝐴₁₂𝐴₂₁𝐴₂₂,𝐴_𝑑=𝐴_𝑑11𝐴_𝑑12𝐴_𝑑21𝐴_𝑑22𝐵,𝐵=¹𝐵₂. (2)𝑑_ℎ(𝑖) and 𝑑_𝑣(𝑗) are delays along horizontal and vertical directions, respectively. We assume that 𝑑_ℎ(𝑖) and 𝑑_𝑣(𝑗) satisfy 𝑑_ℎ𝐿≤𝑑_ℎ(𝑖)≤𝑑_ℎ𝐻,𝑑_𝑣𝐿≤𝑑_𝑣𝑗≤𝑑_𝑣𝐻, (3) where 𝑑_ℎ𝐿, 𝑑_ℎ𝐻 and 𝑑_𝑣𝐿, 𝑑_𝑣𝐻 denote the lower and upper delay bounds along horizontal and vertical directions, respectively. The boundary conditions are defined by 𝑥^ℎ𝑖,𝑗=ℎ_𝑖𝑗,∀0≤𝑗≤𝑧₁,−𝑑_ℎ𝐻𝑥≤𝑖≤0,_ℎ𝑖,𝑗=0,∀𝑗>𝑧₁,−𝑑_ℎ𝐻𝑥≤𝑖≤0,_𝑣𝑖,𝑗=𝑣_𝑖𝑗,∀0≤𝑖≤𝑧₂,−𝑑_𝑣𝐻𝑥≤𝑗≤0,_𝑣𝑖,𝑗=0,∀𝑖>𝑧₂,−𝑑_𝑣𝐻ℎ≤𝑗≤0,₀₀=𝑣₀₀, (4) where 𝑧₁<∞ and 𝑧₂<∞ are positive integers, ℎ_𝑖𝑗∈𝑅^𝑛1₊ and 𝑣_𝑖𝑗∈𝑅^𝑛2₊ are given vectors.

Definition 1. The 2D positive system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is said to be asymptotically stable if lim_𝑙→∞𝑋_𝑙=0 for all bounded boundary conditions (4), where 𝑋_𝑙=sup𝑥𝑖,𝑗∶𝑖+𝑗=𝑙,𝑖,𝑗≥1. (5)

Definition 2. For 𝛾>0, the system (1a) and (1b) is said to be asymptotically stable with the 𝑙₁-gain index 𝛾, if the following conditions hold.(1)The system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is asymptotically stable.(2)Under zero boundary conditions, that is, ℎ_𝑖𝑗=0, 𝑣_𝑖𝑗=0 in (4), it holds that ^∞∞_𝑖=0𝑗=0𝑧(𝑖,𝑗)₁<𝛾^∞∞_𝑖=0𝑗=0𝑤𝑖,𝑗₁,∀0≠𝑤𝑖,𝑗∈𝑙₁. (6)

Remark 3. From (6), we see that 𝛾 can characterize the disturbance attenuation performance of the system (1a) and (1b). The smaller the 𝛾 is, the better the disturbance attenuation performance is.

3. Main Results

3.1. Stability Analysis

In this subsection, we focus on the problem of delay-dependent asymptotically stability analysis for the positive 2D discrete linear systems with state delays.

Theorem 4. For given positive constants 𝑑_ℎ𝐿, 𝑑_ℎ𝐻, 𝑑_𝑣𝐿, 𝑑_𝑣𝐻, the positive 2D system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is asymptotically stable if there exist vectors 𝑝, 𝑞, 𝜍₁, 𝜍₂, 𝜁∈𝑅^𝑛₊, such that Φ=diagΦ₁,Φ₂,…,Φ_𝑛,Φ^′₁,Φ^′₂,…,Φ^′_𝑛,Φ₁^′′,Φ₂^′′,…,Φ_𝑛^′′,Φ₁^′′′,Φ₂^′′′,…,Φ_𝑛^′′′<0, (7) where Φ_𝑘=𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_ℎ𝐻−𝑑_ℎ𝐿𝐸_𝑘𝑞+𝐸_𝑘𝜁+𝑑²_ℎ𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_ℎ𝐻𝐸_𝑘𝜍₂+𝑑²_ℎ𝐻𝐸_𝑘𝜍₁,1≤𝑘≤𝑛₁,𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_𝑣𝐻−𝑑_𝑣𝐿𝐸_𝑘𝑞+𝐸_𝑘𝜁+𝑑²_𝑣𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_𝑣𝐻𝐸_𝑘𝜍₂+𝑑²_𝑣𝐻𝐸_𝑘𝜍₁,𝑛₁Φ+1≤𝑘≤𝑛,_′𝑘=𝑎^𝑇_𝑑𝑘𝑝+𝑎^𝑇_𝑑𝑘−𝐸_𝑘𝑞+𝑑²_ℎ𝐻𝑎^𝑇_𝑑𝑘𝜍₂,1≤𝑘≤𝑛₁,𝑎^𝑇_𝑑𝑘𝑝+𝑎^𝑇_𝑑𝑘−𝐸_𝑘𝑞+𝑑²_𝑣𝐻𝑎^𝑇_𝑑𝑘𝜍₂,𝑛₁Φ+1≤𝑘≤𝑛,_𝑘′′=−𝐸_𝑘𝜁+𝑑_ℎ𝐻𝐸_𝑘𝜍₂−𝜍₁,1≤𝑘≤𝑛₁,−𝐸_𝑘𝜁+𝑑_𝑣𝐻𝐸_𝑘𝜍₂−𝜍₁,𝑛₁Φ+1≤𝑘≤𝑛,_{𝑘′′′}=−𝑑_ℎ𝐻𝐸_𝑘𝜍₁,1≤𝑘≤𝑛₁,−𝑑_𝑣𝐻𝐸_𝑘𝜍₁,𝑛₁+1≤𝑘≤𝑛, (8) with 𝑘∈𝑛={1,2,…,𝑛}, 𝐸_𝑘=[0,…^𝑘−1,0,1,0,…^𝑛−𝑘,0], and 𝑎_𝑘(𝑎_𝑑𝑘) represents the 𝑘th column vector of matrix 𝐴(𝐴_𝑑).

Proof. Choose the following copositive Lyapunov-Krasovskii functional candidate: 𝑉𝑖,𝑗=𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗, (9) where 𝑉^ℎ𝑖,𝑗=⁵_𝑘=1𝑉^ℎ_𝑘,𝑉𝑖,𝑗_ℎ1𝑖,𝑗=𝑥^ℎ𝑇𝑖,𝑗𝑝^ℎ,𝑉^ℎ₂𝑖,𝑗=^𝑖_{𝑟=𝑖−𝑑ℎ(𝑖)}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ,𝑉^ℎ₃𝑖,𝑗=^𝑖−1_{𝑟=𝑖−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑟,𝑗𝜁^ℎ,𝑉^ℎ₄𝑖,𝑗=^{−𝑑ℎ𝐿}_{𝑠=−𝑑ℎ𝐻+1}^𝑖−1_{𝑟=𝑖+𝑠}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ,𝑉^ℎ₅𝑖,𝑗=𝑑_ℎ𝐻⁻¹_{𝑠=−𝑑ℎ𝐻}^𝑖−1_{𝑟=𝑖+𝑠}𝜂^ℎ𝑇𝑟,𝑗𝜍^ℎ,𝑉^𝑣𝑖,𝑗=⁵_𝑘=1𝑉^𝑣_𝑘,𝑉𝑖,𝑗_𝑣1𝑖,𝑗=𝑥^𝑣𝑇𝑖,𝑗𝑝^𝑣,𝑉^𝑣₂𝑖,𝑗=^𝑗_{𝑠=𝑗−𝑑𝑣𝑗}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣,𝑉^𝑣₃𝑖,𝑗=^𝑗−1_{𝑡=𝑗−𝑑𝑣𝐻}𝑥^𝑣𝑇(𝑖,𝑡)𝜁^𝑣,𝑉^𝑣₄𝑖,𝑗=^{−𝑑𝑣𝐿}_{𝑠=−𝑑𝑣𝐻+1}^𝑗−1_{𝑡=𝑗+𝑠}𝑥^𝑣𝑇(𝑖,𝑡)𝑞^𝑣,𝑉^𝑣₅𝑖,𝑗=𝑑_𝑣𝐻⁻¹_{𝑠=−𝑑𝑣𝐻}^𝑗−1_{𝑡=𝑗+𝑠}𝜂^𝑣𝑇(𝑖,𝑡)𝜍^𝑣,𝜂^ℎ𝑟,𝑗=𝑥^ℎ𝑇𝑟,𝑗𝛿^ℎ𝑇𝑟,𝑗^𝑇,𝜂^𝑣(𝑖,𝑡)=𝑥^𝑣𝑇(𝑖,𝑡)𝛿^𝑣𝑇(𝑖,𝑡)^𝑇,𝛿^ℎ𝑟,𝑗=𝑥^ℎ𝑟+1,𝑗−𝑥^ℎ,𝛿𝑟,𝑗_𝑣(𝑖,𝑡)=𝑥^𝑣(𝑖,𝑡+1)−𝑥^𝑣(𝑖,𝑡), (10) with 𝑝^ℎ, 𝑞^ℎ, 𝜁^ℎ, 𝜍^ℎ₁, and 𝜍^ℎ₂∈𝑅^𝑛1₊,𝑝^𝑣, 𝑞^𝑣, 𝜁^𝑣, 𝜍^𝑣₁, and 𝜍^𝑣₂∈𝑅^𝑛2₊, 𝜍^ℎ=[𝜍^ℎ₁𝜍^ℎ₂]^𝑇∈𝑅^2𝑛1₊, and 𝜍^𝑣=[𝜍^𝑣₁𝜍^𝑣₂]^𝑇∈𝑅^2𝑛2₊.
Along the trajectory of the system (1a) and (1b), we have Δ𝑉𝑖,𝑗=𝑉^ℎ𝑖+1,𝑗−𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗+1−𝑉^𝑣=𝑖,𝑗_5𝑘=1Δ𝑉^ℎ_𝑘𝑖,𝑗+⁵_𝑘=1Δ𝑉^𝑣_𝑘𝑖,𝑗, (11) where Δ𝑉^ℎ₁𝑖,𝑗=𝑥^ℎ𝑇𝑖+1,𝑗𝑝^ℎ−𝑥^ℎ𝑇𝑖,𝑗𝑝^ℎ,Δ𝑉^ℎ₂𝑖,𝑗=^𝑖+1_{𝑟=𝑖+1−𝑑ℎ()𝑖+1}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ−^𝑖_{𝑟=𝑖−𝑑ℎ(𝑖)}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ=𝑥^ℎ𝑇𝑖+1,𝑗𝑞^ℎ−𝑥^ℎ𝑇𝑖−𝑑_ℎ(𝑖),𝑗𝑞^ℎ+^𝑖_{𝑟=𝑖+1−𝑑ℎ()𝑖+1}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ−^𝑖_{𝑟=𝑖+1−𝑑ℎ(𝑖)}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ≤𝑥^ℎ𝑇𝑖+1,𝑗𝑞^ℎ−𝑥^ℎ𝑇𝑖−𝑑_ℎ(𝑖),𝑗𝑞^ℎ+^𝑖_{𝑟=𝑖+1−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ−^𝑖_{𝑟=𝑖+1−𝑑ℎ𝐿}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ=𝑥^ℎ𝑇𝑖+1,𝑗𝑞^ℎ−𝑥^ℎ𝑇𝑖−𝑑_ℎ(𝑖),𝑗𝑞^ℎ+^{𝑟=𝑖−𝑑ℎ𝐿}_{𝑟=𝑖+1−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ,Δ𝑉^ℎ₃𝑖,𝑗=^𝑖_{𝑟=𝑖+1−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑟,𝑗𝜁^ℎ−^𝑖−1_{𝑟=𝑖−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑟,𝑗𝜁^ℎ=𝑥^ℎ𝑇𝑖,𝑗𝜁^ℎ−𝑥^ℎ𝑇𝑖−𝑑_ℎ𝐻,𝑗𝜁^ℎ,Δ𝑉^ℎ₄𝑖,𝑗=^{−𝑑ℎ𝐿}_{𝑠=−𝑑ℎ𝐻}^𝑖_{+1𝑟=𝑖+1+𝑠}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ−^{−𝑑ℎ𝐿}_{𝑠=−𝑑ℎ𝐻+1}^𝑖−1_{𝑟=𝑖+𝑠}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ=^{−𝑑ℎ𝐿}_{𝑠=−𝑑ℎ𝐻+1}𝑥^ℎ𝑇𝑖,𝑗𝑞^ℎ−𝑥^ℎ𝑇𝑖+𝑠,𝑗𝑞^ℎ=𝑑_ℎ𝐻−𝑑_ℎ𝐿𝑥^ℎ𝑇𝑖,𝑗𝑞^ℎ−^{𝑖−𝑑ℎ𝐿}_{𝑟=𝑖−𝑑ℎ𝐻+1}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ,Δ𝑉^ℎ₅𝑖,𝑗=𝑑_ℎ𝐻⁻¹_{𝑠=−𝑑ℎ𝐻}^𝑖_{𝑟=𝑖+1+𝑠}𝜂^ℎ𝑇𝑟,𝑗𝜍^ℎ−𝑑_ℎ𝐻⁻¹_{𝑠=−𝑑ℎ𝐻}^𝑖−1_{𝑟=𝑖+𝑠}𝜂^ℎ𝑇𝑟,𝑗𝜍^ℎ=𝑑_ℎ𝐻⁻¹_{𝑠=−𝑑ℎ𝐻}𝜂^ℎ𝑇𝑖,𝑗𝜍^ℎ−𝜂^ℎ𝑇𝑖+𝑠,𝑗𝜍^ℎ=𝑑²_ℎ𝐻𝜂^ℎ𝑇𝑖,𝑗𝜍^ℎ−𝑑_ℎ𝐻^𝑖−1_{𝑟=𝑖−𝑑ℎ𝐻}𝜂^ℎ𝑇𝑟,𝑗𝜍^ℎ=𝑑²_ℎ𝐻𝑥^ℎ𝑇𝑖,𝑗𝑥^ℎ𝑇𝑖+1,𝑗−𝑥^ℎ𝑇𝜍𝑖,𝑗^ℎ1𝜍^ℎ₂−𝑑∑_ℎ𝐻^𝑖−1_{𝑟=𝑖−𝑑ℎ𝐻}𝑥^ℎ𝑇𝑖,𝑗𝑥^ℎ𝑇𝑖,𝑗−𝑥^ℎ𝑇𝑖−𝑑_ℎ𝐻×𝜍,𝑗_ℎ1𝜍^ℎ₂,Δ𝑉^𝑣₁𝑖,𝑗=𝑥^𝑣𝑇𝑖,𝑗+1𝑝^𝑣−𝑥^𝑣𝑇𝑖,𝑗𝑝^𝑣,Δ𝑉^𝑣₂𝑖,𝑗=^𝑗+1_{𝑠=𝑗+1−𝑑𝑣𝑗+1}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣−^𝑗_{𝑠=𝑗−𝑑𝑣𝑗}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣=𝑥^𝑣𝑇𝑖,𝑗+1𝑞^𝑣−𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝑗𝑞^𝑣+^𝑗_{𝑠=𝑗+1−𝑑𝑣𝑗+1}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣−^𝑗_{𝑠=𝑗+1−𝑑𝑣𝑗}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣≤𝑥^𝑣𝑇𝑖,𝑗+1𝑞^𝑣−𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝑗𝑞^𝑣+^𝑗_{𝑠=𝑗+1−𝑑𝑣𝐻}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣−^𝑗_{𝑠=𝑗+1−𝑑𝑣𝐿}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣=𝑥^𝑣𝑇𝑖,𝑗+1𝑞^𝑣−𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝑗𝑞^𝑣+^{𝑗−𝑑𝑣𝐿}_{𝑡=𝑗+1−𝑑𝑣𝐻}𝑥^𝑣𝑇(𝑖,𝑡)𝑞^𝑣,Δ𝑉^𝑣₃𝑖,𝑗=^𝑗_{𝑠=𝑗+1−𝑑𝑣𝐻}𝑥^ℎ𝑇(𝑖,𝑠)𝜁^𝑣−^𝑗−1_{𝑠=𝑗−𝑑𝑣𝐻}𝑥^ℎ𝑇(𝑖,𝑎)𝜁^𝑣=𝑥^𝑣𝑇𝑖,𝑗𝜁^𝑣−𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝐻𝜁^𝑣,Δ𝑉^𝑣₄𝑖,𝑗=^{−𝑑𝑣𝐿}_{𝑠=−𝑑𝑣𝐻}^𝑗_{+1𝑡=𝑗+1+𝑠}𝑥^𝑣𝑇(𝑖,𝑡)𝑞^𝑣−^{−𝑑𝑣𝐿}_{𝑠=−𝑑𝑣𝐻+1}^𝑗−1_{𝑡=𝑗+𝑠}𝑥^𝑣𝑇(𝑖,𝑡)𝑞^𝑣=^{−𝑑𝑣𝐿}_{𝑠=−𝑑𝑣𝐻+1}𝑥^𝑣𝑇𝑖,𝑗𝑞^𝑣−𝑥^𝑣𝑇𝑖,𝑗+𝑠𝑞^𝑣=𝑑_𝑣𝐻−𝑑_𝑣𝐿𝑥^ℎ𝑇𝑖,𝑗𝑞^𝑣−^{𝑗−𝑑𝑣𝐿}_{𝑡=𝑗−𝑑𝑣𝐻+1}𝑥^𝑣𝑇(𝑖,𝑡)𝑞^𝑣,Δ𝑉^𝑣₅𝑖,𝑗=𝑑_𝑣𝐻⁻¹_{𝑠=−𝑑𝑣𝐻}^𝑗_{𝑡=𝑗+1+𝑠}𝜂^𝑣𝑇(𝑖,𝑡)𝜍^𝑣−𝑑_𝑣𝐻⁻¹_{𝑠=−𝑑𝑣𝐻}^𝑗−1_{𝑡=𝑗+𝑠}𝜂^𝑣𝑇(𝑖,𝑡)𝜍^𝑣=𝑑_𝑣𝐻⁻¹_{𝑠=−𝑑𝑣𝐻}𝜂^𝑣𝑇𝑖,𝑗𝜍^𝑣−𝜂^𝑣𝑇𝑖,𝑗+𝑠𝜍^𝑣=𝑑²_𝑣𝐻𝜂^𝑣𝑇𝑖,𝑗𝜍^𝑣−𝑑_𝑣𝐻^𝑗−1_{𝑡=𝑗−𝑑𝑣𝐻}𝜂^𝑣𝑇(𝑖,𝑡)𝜍^𝑣=𝑑²_𝑣𝐻𝑥^𝑣𝑇𝑖,𝑗𝑥^𝑣𝑇𝑖,𝑗+1−𝑥^𝑣𝑇×𝜍𝑖,𝑗_𝑣1𝜍^𝑣₂−𝑑∑_𝑣𝐻^𝑗−1_{𝑡=𝑗−𝑑𝑣𝐻}𝑥^𝑣𝑇𝑖,𝑗𝑥^𝑣𝑇𝑖,𝑗−𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝐻×𝜍^𝑣₁𝜍^𝑣₂. (12) Substitute the previously mentioned formulations into (11), and take 𝑝𝑝=^ℎ𝑝^𝑣𝑞,𝑞=^ℎ𝑞^𝑣𝜁,𝜁=^ℎ𝜁^𝑣,𝜍₁=𝜍^ℎ₁𝜍^ℎ₁,𝜍₂=𝜍^ℎ₂𝜍^ℎ₂,𝐷_𝐻=𝑑_ℎ𝐻𝐼_𝑛100𝑑_𝑣𝐻𝐼_𝑛2,𝐷_𝐿=𝑑_ℎ𝐿𝐼_𝑛100𝑑_𝑣𝐿𝐼_𝑛2,𝑥𝑖,𝑗=𝑥^ℎ𝑇𝑖,𝑗𝑥^𝑣𝑇𝑖,𝑗^𝑇,𝑥_𝑑𝑖,𝑗=𝑥^ℎ𝑇𝑖−𝑑_ℎ(𝑖),𝑗𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣(𝑗)^𝑇,𝑥_𝐻𝑖,𝑗=𝑥^ℎ𝑇𝑖−𝑑_ℎ𝐻,𝑗𝑥^𝑣𝑇𝑖,𝑗−𝑑_𝑣𝐻^𝑇,𝑥_𝑠𝑖,𝑗=∑^𝑖−1_{𝑟=𝑖−𝑑ℎ𝐻+1}𝑥^ℎ𝑇𝑟,𝑗∑^𝑗−1_{𝑡=𝑗−𝑑𝑣𝐻+1}𝑥^𝑣𝑇(𝑖,𝑡)^𝑇. (13) Then we have Δ𝑉𝑖,𝑗=𝑥^𝑇𝑖,𝑗𝐴^𝑇−𝐼_𝑛𝑝+𝐴^𝑇+𝐷_𝐻−𝐷_𝐿𝑞+𝜁+𝐷²_𝐻𝐴^𝑇−𝐼_𝑛𝜍₂+𝜍₁−𝐷_𝐻𝜍₂+𝑥^𝑇_𝑑𝑖,𝑗𝐴^𝑇_𝑑𝑝+𝐴^𝑇_𝑑−𝐼_𝑛𝑞+𝐷²_𝐻𝐴^𝑇_𝑑𝜍₂+𝑥^𝑇_𝐻𝑖,𝑗−𝜁+𝐷_𝐻𝜍₂−𝜍₁+𝑥^𝑇_𝑠𝑖,𝑗−𝐷_𝐻𝜍₁. (14) If condition (7) holds, one obtains 𝐴^𝑇−𝐼_𝑛𝑝+𝐴^𝑇+𝐷_𝐻−𝐷_𝐿𝑞+𝜁+𝐷²_𝐻𝐴^𝑇−𝐼_𝑛𝜍₂+𝜍₁−𝐷_𝐻𝜍₂𝐴≺0,_𝑇𝑑𝑝+𝐴^𝑇_𝑑−𝐼_𝑛𝑞+𝐷²_𝐻𝐴^𝑇_𝑑𝜍₂≺0,−𝜁+𝐷_𝐻𝜍₂−𝜍₁≺0,−𝐷_𝐻𝜍₁≺0. (15) It follows that Δ𝑉(𝑖,𝑗)<0, which means that 𝑉^ℎ𝑖+1,𝑗+𝑉^𝑣𝑖,𝑗+1<𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗. (16) Summing up both sides of (16) from 𝐷 to 0 with respect to 𝑖 and from 0 to 𝐷 with respect to 𝑗, for any nonnegative integer 𝐷≥max(𝑧₁,𝑧₂), one gets 𝑉^ℎ(1,𝐷)+𝑉^𝑣(0,𝐷+1)+𝑉^ℎ(2,𝐷−1)+𝑉^𝑣(1,𝐷)+⋯+𝑉^ℎ(𝐷+1,0)+𝑉^𝑣(𝐷,1)=_{𝑖+𝑗=𝐷+1}𝑉^ℎ𝑖,𝑗+_{𝑖+𝑗=𝐷+1}𝑉^𝑣𝑖,𝑗=_{𝑖+𝑗=𝐷+1}𝑉𝑖,𝑗≤𝑉^ℎ(0,𝐷)+𝑉^𝑣(0,𝐷)+𝑉^ℎ(1,𝐷−1)+𝑉^𝑣(1,𝐷−1)+⋯+𝑉^ℎ(𝐷,0)+𝑉^𝑣()=𝐷,0_{𝑖+𝑗=𝐷}𝑉𝑖,𝑗. (17) Then from (9), we can conclude that lim_{𝑖+𝑗→∞}𝑥𝑖,𝑗=0, (18) which implies that the system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is asymptotically stable.
This completes the proof.

When 𝑑_ℎ𝐻=𝑑_ℎ𝐿=𝑑_ℎ, and 𝑑_𝑣𝐻=𝑑_𝑣𝐿=𝑑_𝑣, the system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is reduced to the following system: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣𝑥𝑖,𝑗+1=𝐴^ℎ𝑥𝑖,𝑗_𝑣𝑖,𝑗+𝐴_𝑑𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣, (19) where 𝑑_ℎ and 𝑑_𝑣 are constant delays along horizontal and vertical directions, respectively, and the boundary conditions are defined in (4). Then we can get the following result.

Corollary 5. For given positive constants 𝑑_ℎ and 𝑑_𝑣, the positive 2D system (19) is asymptotically stable if there exist vectors 𝑝 and 𝑞∈𝑅^𝑛₊, such that Φ=diagΦ₁,Φ₂,…,Φ_𝑛,Φ^′₁,Φ^′₂,…,Φ^′_𝑛<0, (20) where Φ_𝑘=𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘Φ𝑞,_′𝑘=𝑎^𝑇_𝑑𝑘𝑝+𝑎^𝑇_𝑑𝑘−𝐸_𝑘𝑞, (21) with 𝑘∈𝐾={1,2,…,𝑛}, 𝐸_𝑘=[0,…^𝑘−1,0,1,0,…^𝑛−𝑘,0], 𝑝, 𝑞∈𝑅^𝑛₊, and 𝑎_𝑘(𝑎_𝑑𝑘) represents the 𝑘th column vector of matrix 𝐴(𝐴_𝑑).

Proof. Choose the following copositive Lyapunov-Krasovskii functional candidate for the system (19): 𝑉𝑖,𝑗=𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗, (22) where 𝑉^ℎ𝑖,𝑗=²_𝑘=1𝑉^ℎ_𝑘𝑖,𝑗,𝑉^ℎ₁𝑖,𝑗=𝑥^ℎ𝑇𝑖,𝑗𝑝^ℎ,𝑉^ℎ₂𝑖,𝑗=^𝑖_{𝑟=𝑖−𝑑ℎ}𝑥^ℎ𝑇𝑟,𝑗𝑞^ℎ,𝑉^𝑣𝑖,𝑗=²_𝑘=1𝑉^𝑣_𝑘𝑖,𝑗,𝑉^𝑣₁𝑖,𝑗=𝑥^𝑣𝑇𝑖,𝑗𝑝^𝑣,𝑉^𝑣₂𝑖,𝑗=^𝑗_{𝑠=𝑗−𝑑𝑣}𝑥^𝑣𝑇(𝑖,𝑠)𝑞^𝑣, (23) with 𝑝^ℎ,𝑞^ℎ∈𝑅^𝑛1₊, 𝑝^𝑣,𝑞^𝑣∈𝑅^𝑛2₊. Then following the proof line of Theorem 4, the corollary can be obtained.

3.2. 𝑙₁-Gain Analysis

The following theorem establishes sufficient condition of the asymptotical stability with 𝑙₁-gain performance for the system (1a) and (1b).

Theorem 6. For given positive constants 𝑑_ℎ𝐿, 𝑑_ℎ𝐻, 𝑑_𝑣𝐿, 𝑑_𝑣𝐻, and 𝛾, the positive 2D system (1a) and (1b) is asymptotically stable with the 𝑙₁-gain index 𝛾 if there exist vectors 𝑝∈𝑅^𝑛₊, 𝑞∈𝑅^𝑛₊, 𝜍₁∈𝑅^𝑛₊, 𝜍₂∈𝑅^𝑛₊, and 𝜁∈𝑅^𝑛₊, such that Φ=diagΦ₁,Φ₂,…,Φ_𝑛,Φ^′₁,Φ^′₂,…,Φ^′_𝑛,Φ₁^′′,Φ₂^′′,…,Φ_𝑛^′′,Φ₁^′′′,Φ₂^′′′,…,Φ_𝑛^′′′,Τ₁,Τ₂,…,Τ_𝑚<0, (24) where Φ^′_𝑘, Φ_𝑘^′′, and Φ_𝑘^′′′ are denoted as in Theorem 4, and Φ_𝑘=𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_ℎ𝐻−𝑑_ℎ𝐿𝐸_𝑘𝑞+𝐸_𝑘𝜁+𝑑²_ℎ𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_ℎ𝐻𝐸_𝑘𝜍₂1≤𝑘≤𝑛₁,+𝑑²_ℎ𝐻𝐸_𝑘𝜍₁+ℎ_𝑘1,𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_𝑣𝐻−𝑑_𝑣𝐿𝐸_𝑘𝑞+𝐸_𝑘𝜁+𝑑²_𝑣𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_𝑣𝐻𝐸_𝑘𝜍₂𝑛₁+1≤𝑘≤𝑛,+𝑑²_𝑣𝐻𝐸_𝑘𝜍₁+ℎ_𝑘1,𝑇_𝜀=𝑏^𝑇_𝜀𝑝+𝑏^𝑇_𝜀𝑞+𝑏^𝑇_𝜀𝐷²_ℎ𝐻𝜍₂+𝑙_𝜀1,𝐷−𝛾,1≤𝜀≤𝑚_𝐻=diag𝑑_ℎ𝐻𝐼_𝑛1,𝑑_𝑣𝐻𝐼_𝑛2, (25) with 𝑘∈𝑛={1,2,…,𝑛}, 𝜀∈𝑚={1,2,…,𝑚}, 𝐸_𝑘=[0,…^𝑘−1,0,1,0,…^𝑛−𝑘,0], 𝑎_𝑘, 𝑎_𝑑𝑘, 𝑏_𝑘, and ℎ_𝑘 represent the 𝑘th column vector of matrices 𝐴, 𝐴_𝑑, 𝐵, and 𝐻, respectively, and 𝑙_𝜀 represents the 𝜀th column vector of matrix 𝐿.

Proof. It is an obvious fact that (24) implies the following inequality: Ψ=diagΨ₁,Ψ₂,…,Ψ_𝑛,Ψ^′₁,Ψ^′₂,…,Ψ^′_𝑛,Ψ₁^′′,Ψ₂^′′Ψ,…,_𝑛′′,Ψ₁^′′′,Ψ₂^′′′,…,Ψ_𝑛^′′′<0, (26) where Ψ_𝑘=𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_ℎ𝐻−𝑑_ℎ𝐿𝐸_𝑘𝑞+𝜁_𝑘+𝑑²_ℎ𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_ℎ𝐻𝐸_𝑘𝜍₂+𝑑²_ℎ𝐻𝐸_𝑘𝜍₁,1≤𝑘≤𝑛₁,𝑎^𝑇_𝑘−𝐸_𝑘𝑝+𝑎^𝑇_𝑘+𝑑_𝑣𝐻−𝑑_𝑣𝐿𝐸_𝑘𝑞+𝜁_𝑘+𝑑²_𝑣𝐻𝑎^𝑇_𝑘−𝐸_𝑘−𝑑_𝑣𝐻𝐸_𝑘𝜍₂+𝑑²_𝑣𝐻𝐸_𝑘𝜍₁,𝑛₁Ψ+1≤𝑘≤𝑛,_′𝑘=𝑎^𝑇_𝑑𝑘𝑝+𝑎^𝑇_𝑑𝑘−𝐸_𝑘𝑞+𝑑²_ℎ𝐻𝑎^𝑇_𝑑𝑘𝜍₂,1≤𝑘≤𝑛₁,𝑎^𝑇_𝑑𝑘𝑝+𝑎^𝑇_𝑑𝑘−𝐸_𝑘𝑞+𝑑²_𝑣𝐻𝑎^𝑇_𝑑𝑘𝜍₂,𝑛₁Ψ+1≤𝑘≤𝑛,_𝑘′′=−𝜁_𝑘+𝑑_ℎ𝐻𝐸_𝑘𝜍₂−𝜍₁,1≤𝑘≤𝑛₁,−𝜁_𝑘+𝑑_𝑣𝐻𝐸_𝑘𝜍₂−𝜍₁,𝑛₁Ψ+1≤𝑘≤𝑛,_{𝑘′′′}=−𝑑_ℎ𝐻𝐸_𝑘𝜍₁,1≤𝑘≤𝑛₁,−𝑑_𝑣𝐻𝐸_𝑘𝜍₁,𝑛₁+1≤𝑘≤𝑛, (27) with 𝑘∈𝑛={1,2,…,𝑛}, 𝐸_𝑘=[0,…^𝑘−1,0,1,0,…^𝑛−𝑘,0], and 𝑎_𝑘(𝑎_𝑑𝑘) represents the 𝑘th column vector of matrix 𝐴(𝐴_𝑑).
By Theorem 4, we can obtain that the system (1a) and (1b) with 𝑤(𝑖,𝑗)=0 is asymptotically stable. Now we are in a position to prove that the system (1a) and (1b) has a prescribed 𝑙₁-gain index 𝛾 for any nonzero 𝑤(𝑖,𝑗)∈𝑙₁. To establish the 𝑙₁-gain performance, we choose the same copositive Lyapunov-Krasovskii functional candidate as in (9) for the system (1a) and (1b). Following the proof line of Theorem 4, we can get that Δ𝑉𝑖,𝑗+𝑧(𝑖,𝑗)₁−𝛾𝑤𝑖,𝑗₁=𝑥^𝑇𝑖,𝑗𝐴^𝑇−𝐼_𝑛𝑝+𝐴^𝑇+𝐷_𝐻−𝐷_𝐿𝑞+𝜁+𝐷²_𝐻𝐴^𝑇−𝐼_𝑛𝜍₂+𝜍₁−𝐷_𝐻𝜍₂+𝑥^𝑇_𝑑𝑖,𝑗𝐴^𝑇_𝑑𝑝+𝐴^𝑇_𝑑−𝐼_𝑛𝑞+𝐷²_𝐻𝐴^𝑇_𝑑𝜍₂+𝑥^𝑇_𝐻𝑖,𝑗−𝜁+𝐷_𝐻𝜍₂−𝜍₁+𝑥^𝑇_𝑠𝑖,𝑗−𝐷_𝐻𝜍₁+𝑤^𝑇𝑖,𝑗𝐵^𝑇𝑝+𝑞+𝐷²_𝐻𝜍₂+𝐻𝑥𝑖,𝑗₁+𝐿𝑤𝑖,𝑗₁−𝛾𝑤𝑖,𝑗₁. (28) According to the definition of 𝑙₁ norm, one obtains=ℎ𝐻𝑥𝑖,𝑗^1,1ℎ_1,2⋯ℎ_1,𝑛1ℎ_2,1ℎ_2,2⋯ℎ_2,𝑛1ℎ⋮⋮⋯⋮_𝑝,1ℎ_𝑝,2⋯ℎ_𝑝,𝑛1ℎ_1,𝑛1+1ℎ_1,𝑛1+2⋯ℎ_1,𝑛ℎ_2,𝑛1+1ℎ_2,𝑛1+2⋯ℎ_2,𝑛ℎ⋮⋮⋯⋮_𝑝,𝑛1+1ℎ_𝑝,𝑛1+2⋯ℎ_𝑝,𝑛𝑥^ℎ₁⋮𝑥𝑖,𝑗_ℎ𝑛1𝑥𝑖,𝑗_𝑣1⋮𝑥𝑖,𝑗_𝑣𝑛2=ℎ𝑖,𝑗_1,1𝑥^ℎ₁𝑖,𝑗+⋯+ℎ_1,𝑛1𝑥^ℎ_𝑛1𝑖,𝑗+ℎ_1,𝑛1+1𝑥^𝑣₁𝑖,𝑗+⋯+ℎ_1,𝑛𝑥^𝑣_𝑛2ℎ𝑖,𝑗_2,1𝑥^ℎ₁𝑖,𝑗+⋯+ℎ_2,𝑛1𝑥^ℎ_𝑛1𝑖,𝑗+ℎ_2,𝑛1+1𝑥^𝑣₁𝑖,𝑗+⋯+ℎ_2,𝑛𝑥^𝑣_𝑛2⋮ℎ𝑖,𝑗_𝑝,1𝑥^ℎ₁𝑖,𝑗+⋯+ℎ_𝑝,𝑛1𝑥^ℎ_𝑛1𝑖,𝑗+ℎ_𝑝,𝑛1+1𝑥^𝑣₁𝑖,𝑗+⋯+ℎ_𝑝,𝑛𝑥^𝑣_𝑛2,𝑖,𝑗𝐻𝑥𝑖,𝑗₁=^𝑝_𝑘=1ℎ_𝑘,1𝑥^ℎ₁𝑖,𝑗+⋯+ℎ_𝑘,𝑛1𝑥^ℎ_𝑛1𝑖,𝑗+ℎ_𝑘,𝑛1+1𝑥^𝑣₁𝑖,𝑗+⋯+ℎ_𝑘,𝑛𝑥^𝑣_𝑛2=𝑖,𝑗_𝑝𝑘=1ℎ_𝑘,1𝑥^ℎ₁𝑖,𝑗+⋯+^𝑝_𝑘=1ℎ_𝑘,𝑛1𝑥^ℎ_𝑛1𝑖,𝑗+^𝑝_𝑘=1ℎ_𝑘,𝑛1+1𝑥^𝑣₁𝑖,𝑗+⋯+^𝑝_𝑘=1ℎ_𝑘,𝑛𝑥^𝑣_𝑛2𝑖,𝑗=𝑥^𝑇𝑖,𝑗ℎ₁₁ℎ₂₁⋯ℎ_𝑛1^𝑇, (29)where ℎ_𝑘 represents the 𝑘th column vector and ℎ_𝑖,𝑗 represents the entry located at (𝑖,𝑗) of matrix 𝐻. Then, similarly 𝐿𝑤𝑖,𝑗₁=𝑤^𝑇𝑖,𝑗𝑙₁₁𝑙₂₁⋯𝑙_𝑚1^𝑇,𝛾𝑤𝑖,𝑗₁=𝑤^𝑇𝑖,𝑗𝛾𝛾⋯𝛾^𝑇, (30) where 𝑙_𝑘 represents the 𝑘th column vector of matrix 𝐿.
Substituting (29)-(30) into (28) leads to Δ𝑉𝑖,𝑗+𝑧(𝑖,𝑗)₁−𝛾𝑤𝑖,𝑗₁=𝑥^𝑇𝑖,𝑗𝐴^𝑇−𝐼_𝑛𝑝+𝐴^𝑇+𝐷_𝐻−𝐷_𝐿𝑞+𝜁+𝐷²_𝐻𝐴^𝑇−𝐼_𝑛𝜍₂+𝜍₁−𝐷_𝐻𝜍₂+ℎ₁₁ℎ₂₁⋯ℎ_𝑛1^𝑇+𝑥^𝑇_𝑑𝑖,𝑗𝐴^𝑇_𝑑𝑝+𝐴^𝑇_𝑑−𝐼_𝑛𝑞+𝐷²_𝐻𝐴^𝑇_𝑑𝜍₂+𝑥^𝑇_𝐻𝑖,𝑗−𝜁+𝐷_𝐻𝜍₂−𝜍₁+𝑥^𝑇_𝑠𝑖,𝑗−𝐷_𝐻𝜍₁+𝑤^𝑇𝑖,𝑗𝐵^𝑇𝑝+𝑞+𝐷²_𝐻𝜍₂+𝑙₁₁𝑙₂₁⋯𝑙_𝑚1^𝑇−𝛾𝛾⋯𝛾^𝑇. (31) If condition (24) holds, we have 𝑉^ℎ𝑖+1,𝑗−𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗+1−𝑉^𝑣+𝑧𝑖,𝑗𝑖,𝑗₁−𝛾𝑤𝑖,𝑗₁<0. (32) We know that Δ𝑉𝑖,𝑗=𝑉^ℎ𝑖+1,𝑗−𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗+1−𝑉^𝑣𝑖,𝑗. (33) For any positive scalars 𝑘_ℎ, and𝑘_𝑣∈𝑍₊, it can be verified that Δ𝑉𝑖,𝑗=^𝑘ℎ𝑘_𝑖=0^𝑣_𝑗=0^𝑘ℎ𝑘_𝑖=0^𝑣_𝑗=0𝑉^ℎ𝑖+1,𝑗−𝑉^ℎ+𝑖,𝑗_{𝑘ℎ𝑘𝑖=0𝑣𝑗=0}𝑉^𝑣𝑖,𝑗+1−𝑉^𝑣=𝑖,𝑗_{𝑘𝑣𝑗=0}𝑉^ℎ𝑇𝑘_ℎ+1,𝑗−𝑉^ℎ𝑇+0,𝑗_{𝑘ℎ𝑖=0}𝑉^𝑣𝑇𝑖,𝑘_𝑣+1−𝑉^𝑣𝑇(𝑖,0). (34) When 𝑘_ℎ and 𝑘_𝑣=∞, we have ^∞∞_𝑖=0𝑗=0𝑧𝑖,𝑗₁−𝛾𝑤𝑖,𝑗₁<^∞∞_𝑖=0𝑗=0Δ𝑉𝑖,𝑗. (35) The existence of solution for LMI (24) implies that the positive 2D system (1a) and (1b) is asymptotically stable. Together with the zero boundary conditions, one can get ^∞∞_𝑖=0𝑗=0Δ𝑉𝑖,𝑗=0. (36) Applying (36) to (35), one has ^∞∞_𝑖=0𝑗=0𝑧𝑖,𝑗₁<𝛾^∞∞_𝑖=0𝑗=0𝑤𝑖,𝑗₁. (37) By Definition 2, the positive 2D system (1a) and (1b) is asymptotically stable and has the 𝑙₁-gain index 𝛾.
This completes the proof.

Remark 7. In Theorem 6, the disturbance attenuation performance of positive 2D linear systems is analyzed, and sufficient conditions for the existence of 𝑙₁-gain performance for positive 2D system to (1a) and (1b) are proposed in terms of LMIs which are computationally tractable. This is also the major contribution of our paper.

4. Numerical Example

Consider the positive 2D system with delays in the Roesser model (1a) and (1b), where ,𝐴𝐴=0.100.20⋮0.20.000.30⋮0.10⋯⋯⋯⋯0.000.10⋮0.40_𝑑=,𝑑0.100.01⋮0.050.100.02⋮0.05⋯⋯⋯⋯0.030.12⋮0.03𝐵=0.20.10.1,𝐻=0.100.2,𝐿=0.1,_ℎ(𝑖)=4+2sin𝜋𝑖2,𝑑_𝑣𝑗=5+2sin𝜋𝑗2,𝑤𝑖,𝑗=𝑒^{−(𝑖+0.5𝑗)}, (38) where state dimensions are 𝑛_ℎ=2 and 𝑛_𝑣=1. The boundary conditions are given by 𝑥^ℎ𝑖,𝑗=0.10.1,∀0≤𝑗≤52,−𝑑_ℎ𝐻𝑥≤𝑖≤0,_𝑣𝑖,𝑗=0.1,∀0≤𝑖≤52,−𝑑_𝑣𝐻≤𝑗≤0. (39) In this example, we can get 𝑑_ℎ𝐿=2, 𝑑_ℎ𝐻=6, 𝑑_𝑣𝐿=3, and 𝑑_𝑣𝐻=7. Given 𝛾=4.5, then by using the LMI Control Toolbox [37] to solve the inequalities in Theorem 6, we can get the following solutions: 𝑝=1.73101.68421.7364^𝑇,𝑞=3.20772.87683.3026^𝑇,𝜁=1.03341.10750.9957^𝑇,𝜍₁=0.36750.40420.3552^𝑇,𝜍₂=0.03850.04890.0330^𝑇. (40)

Figures 1, 2, and 3 show the state responses of the system; it can be seen that the corresponding positive 2D system is asymptotically stable. Furthermore, by computing, under zero boundary conditions, we have ∑^∞_𝑖=0∑^∞_𝑗=0‖𝑧(𝑖,𝑗)‖₁=4.0206 ∑^∞_𝑖=0∑^∞_𝑗=0‖𝑤(𝑖,𝑗)‖₁=1.0977. It is obvious that the prescribed 𝑙₁-gain performance level 𝛾=4.5 is satisfied.

Figure 1 

State response of 𝑥ℎ1(𝑖,𝑗).

Figure 2 

State response of 𝑥ℎ2(𝑖,𝑗).

Figure 3 

State response of 𝑥𝑣(𝑖,𝑗).

5. Conclusions

This paper has addressed the delay-dependent stability analysis with 𝑙₁-gain performance for positive 2D systems with state delays in the Roesser model. A sufficient condition for the existence of the delay-dependent asymptotic stability of positive 2D linear systems with time delays has been established. Copositive-type Lyapunov function method has been used to get a computationally tractable LMI-based sufficient criterion which ensures that the system is asymptotically stable and has a prescribed 𝑙₁-gain performance. A numerical example has been given to illustrate the efficiency of the results. Furthermore, our future work will be devoted to the 𝑙₁-gain control problem for positive 2D systems with delays.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant nos. 60974027 and 61273120.