Finite-Region Boundedness and Stabilization of 2D Continuous-Time Roesser Models

Abstract

This paper investigates the finite-region stability (FRS) and stabilization problems for 2D continuous-time systems described by a Roesser model. We first establish a novel set of FRS and finite-region boundedness (FRB) conditions, extending the ${\mathcal{L}}_2$-based concept on finite-time stability from 1D systems to the 2D continuous domain with a new condition based on the generalized state vector of the 2D continuous-time system in contrast with the norm-based condition found in the literature. Sufficient conditions are then derived to guarantee that the system state remains within a predefined quadratic region over a finite-time horizon. Furthermore, the framework is extended to analyze FRB under two distinct classes of external disturbances. Finally, a complete procedure for state feedback stabilization is provided, with all sufficient conditions for FRS and stabilization expressed entirely in terms of numerically tractable Linear Matrix Inequalities (LMIs) enabling controller design that ensures closed-loop finite-region performance under both disturbance classes. The effectiveness and feasibility of the proposed approach are demonstrated through numerical examples.

1. Introduction

Over the past few decades, two-dimensional systems have received considerable attention due to their wide applications in diverse fields such as digital data filtering, image and signal processing [1,2,3], Iterative Learning Control [4], nD filtering [5], and modeling of partial differential equations [6]. It should be noted that the Lyapunov asymptotic stability theory for 2D systems is considered as the most fundamental and important topic. Several contributions have been devoted to the stability analysis of 2D systems, yet only a few studies focused primarily on asymptotic stability, e.g., [7,8,9,10,11]. However, asymptotic stability only guarantees convergence as both $t_1$ and $t_2$ tend to infinity, providing no information or guarantee regarding the crucial transient behavior or state constraints over a finite operational time.

On the other hand, in one-dimensional systems, researchers have tackled finite-time stability as a basic concept in stability analysis since it admits that the state does not exceed a certain bound during a fixed time interval. The notion of short time stability was first introduced in [12], which corresponds to the same concept of finite-time stability defined in [13].

In order to take into account the impact of disturbance input, the notion of finite-time stability has been extended to finite-time boundedness in [14], where the authors have given a more general definition to the concept of finite-time boundedness. In [15], the authors proposed sufficient conditions for finite-time stability and finite-time boundedness for linear continuous systems using the S-variable technique and also by a dual approach in order to derive a state feedback controller.

Despite the fact that most existing references in the literature focus on the study of asymptotic stability, which concerns the stability over an infinite time interval, it was found that finite-time stability is very useful for the study of the behavior of a system during transients to avoid saturation or excitation of nonlinearities. In this context, for two-dimensional systems, the concept of finite-time stability related to 1D systems has been referred to as finite-region stability. There are several existing results on finite-time stability and finite-time boundedness in the literature [16,17,18,19,20,21,22,23,24]. In [16], the authors extended the finite-time stability theory to the class of 2D systems both for continuous and discrete cases in terms of feasibility problems involving Partial Differential Linear Matrix Inequalities (PDLMIs). Sufficient conditions of finite-time stability and finite-time boundedness have been provided in [17] for 2D continuous-discrete systems and have enabled the design of a state feedback controller. Furthermore, the analysis methods to investigate the transient behavior of 2D discrete systems described by the Roesser model for finite-time stability and finite-time boundedness have been proposed in [18] and described by Fornasini Marchesini in his model in [19]. Sufficient conditions of finite-time stability and finite-time boundedness as well as robust finite-time boundedness, using the co-positive-type Lyapunov function method, have been established for 2D positive continuous-discrete systems with interval and polytopic uncertainties in [20], whereas in [21] the authors dealt with 2D continuous-discrete systems subject to energy-bounded disturbance and established sufficient conditions of finite-time stability and finite-time boundedness. In [22], the authors provided a novel sufficient condition for the finite-region stability of linear-time varying-discrete-time 2D systems with the application of Iterative Learning Control. In [23], the authors proposed sufficient conditions for the finite-region stability and stabilization for 2D switched discrete systems with actuator saturation in the Fornasini Marchesini model by using a weighted average dwell time approach. The problems of finite-region passive control and finite-region boundedness for 2D Markovian jump Roesser systems have been addressed in [24]. The authors presented an approach involving an asynchronous $H_{\infty }$ filter design based on a finite region for 2D Markovian Jump systems in Roesser.

In this regard, the aforementioned results addressed the finite-region stability for the 2D systems in the discrete case and in the continuous-discrete case.

Therefore, it is challenging to focus ones’s attention on the study of finite-region stability for 2D continuous systems since the interest is to consider that the system state in both directions must be less than a particular threshold over a finite region. It should be noted that the finite-time stability is different from the classical asymptotic stability notion based on Lyapunov theory, which lies in the convergence of the state vector as time ($t=t_1+t_2$) reaches infinity. Roughly speaking, the Lyapunov asymptotic stability is a powerful tool used to deal with the behavior (state convergence property) of 2D continuous systems within a long time interval, whereas the finite-region stability is a more practical approach than the Lyapunov asymptotic stability since, in some applications of 2D continuous systems (for instance, infinite-dimensional systems), it is desired that the state variables do not exceed a given threshold over a finite region.

The novelty of this paper lies in the different directions taken. First, we provide sufficient novel conditions for the FRS and FRB of 2D continuous systems. The core of our method is the use of a new definition of FRS for 2D continuous-time systems. Second, our new definition of FRS is inspired from the Lyapunov stability of 2D continuous systems, which consists of the stability of the generalized vector along the diagonal ($t=t_1+t_2$). Finally, the obtained conditions (FRS/FRB) involve the solution of feasibility problems constrained by LMIs, which can be easily tractable. Furthermore, the derived conditions are expressed in terms of only one LMI, which is in contrast with existing results in the literature where the conditions (FRS/FRB) are obtained in terms of two LMIs. While related, the results in discrete-time and continuous-discrete systems cannot be directly applied to the 2D continuous-time Roesser model, which is crucial for modeling continuous-flow processes, distributed parameter systems, and certain physical phenomena, where both spatial coordinates evolve continuously. This paper addresses the finite-region stability and finite-region stabilization of 2D continuous systems. The main contributions of this paper can be summarized as follows:

• The concepts of finite-region stability and finite-region boundedness of 2D continuous systems are investigated.
• Sufficient conditions dealing with the finite-region stability and finite-region stabilization of 2D continuous systems are established in terms of LMI conditions.
• Sufficient conditions of finite-region boundedness for 2D continuous systems subject to disturbance generated by an external system and energy-bounded disturbances are given in terms of LMI condition.
• Sufficient finite-region stabilization conditions are derived in order to design the state feedback controller, ensuring that the 2D Roesser continuous model is subject to the different types of disturbances in the closed loop over a finite region.

The remainder of this paper is organized as follows. Section 2 provides the concept of finite-region stability and finite-region boundedness for 2D continuous systems. Section 3 gives sufficient conditions of finite-region stability for 2D continuous systems as well as finite-region boundedness for 2D continuous systems subject to two types of disturbance, namely the disturbance generated by an external system and the energy-bounded disturbance. Section 4 proposes sufficient conditions for the existence of a finite-region stabilizing controller for both types of disturbances. Section 5 presents a numerical example in order to illustrate the effectiveness of the proposed approach. Finally, conclusions are drawn in Section 6.

Notation 1.

In the sequel, $\mathrm{I}\mathrm{R}$ denotes the set of real values. The matrix inequalities must be understood in the sense of Löwner, i.e., $M>0$ (resp., $M<0$, $M\ge 0$, and $M\le 0$) means that the Hermitian matrix M is positive definite (resp., negative definite, positive semi-definite, or negative semi-definite).

For a matrix M, $\mathrm{Sym}\left\{M\right\}$ stands for $M+M^{\top }$.

2. Problem Statement and Preliminaries

This section first gives the system under consideration and then provides the definition of finite-region stability for 2D continuous systems as well as a useful lemma.

Consider the 2D continuous system under the Roesser model [1]:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& Ax(t_1,t_2)+Bu(t_1,t_2)+G\omega (t_1,t_2)\hfill \end{array}$$

(1)

with

$$\begin{array}{ccc}\hfill x(t_1,t_2)& =& \left[\begin{array}{c}{x}_h(t_1,t_2)\\ x_v(t_1,t_2)\end{array}\right],{\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}=\left[\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }{x}_h(t_1,t_2)}{\mathsf{\partial }{t}_1}}\\ {\displaystyle \frac{\mathsf{\partial }{x}_v(t_1,t_2)}{\mathsf{\partial }{t}_2}}\end{array}\right],\omega (t_1,t_2)=\left[\begin{array}{c}{\omega }_h(t_1,t_2)\\ {\omega }_v(t_1,t_2)\end{array}\right],\hfill \\ \hfill A& =& \left[\begin{array}{cc}{A}_{hh}& A_{hv}\\ A_{vh}& A_{vv}\end{array}\right],B=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right],G=\left[\begin{array}{cc}{G}_{hh}& G_{hv}\\ G_{vh}& G_{vv}\end{array}\right]\hfill \end{array}$$

where $x^h(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_h}$ is the horizontal state; $x^v(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_v}$ is the vertical state; and $A_{ij},G_{ij}$, ($i,j=h,v$), $B_i$ ($i=1,2$) are real constant matrices of appropriate dimensions. $\omega (t_1,t_2)$ is the disturbance input. Note that there two types of external signal disturbances:

• Energy-bounded disturbances: $\mathcal{W}\left(d\right)=\left\{\omega (t_1,t_2)\right|{\omega }^{\top }(t_1,t_2)\omega (t_1,t_2)\le d\}$;
• Disturbances generated by an external system: $\mathcal{W}\left(d\right)=\{\omega (t_1,t_2)|{\displaystyle \frac{\mathsf{\partial }\omega (t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}=L\omega (t_1,t_2),{\omega }_h^{\top }(0,t_2){\omega }_h(0,t_2)+{\omega }_v^{\top }(t_1,0){\omega }_v(t_1,0)\le d\}$,

where $L=\left[\begin{array}{cc}{L}_{hh}& L_{hv}\\ L_{vh}& L_{vv}\end{array}\right]$.

The boundary conditions are given by

$$x_h(0,t_2)=x_{h0} \mathrm{and} x_v(t_1,0)=x_{v0}$$

Now, we present the definition of FRS, which corresponds to an extension of the definition of FTS from a 1D system to a 2D system.

Definition 1.

Given positive constants $c_1,c_2,T_f$ and the domain $\mathsf{\Omega }:=[0,T_f]$, the positive definite matrices $R_1:=diag(R_{1_h},R_{1_v})$ and $R_2:=diag(R_{2_h},R_{2_v})$ and $t=t_1+t_2$. The system (1) with $u\equiv 0$ and $\omega \equiv 0$ is said to be finite-region-stable (FRS) with respect to $(\mathsf{\Omega },R_1,R_2)$ if

$$\begin{array}{c}\hfill x_h^{\top }(0,t_2)R_{1_h}{x}_h(0,t_2)+x_v^{\top }(t_1,0)R_{1_v}{x}_v(t_1,0)\le c_1\forall t\in \mathsf{\Omega }\end{array}$$

(2)

which implies, for all $t\in \mathsf{\Omega }$,

$$\begin{array}{c}\hfill {\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s<c_2\end{array}$$

(3)

Remark 1.

The obtained conditions in [16] are time-dependant and, as a consequence, they lead to a finite number of conditions. In [16], the authors assume that the considered time interval is divided into subintervals in which matrix P is constant in order to solve the problem, and this procedure raises many questions. Our result is based on an LMI with a constant P. Moreover, the definition of FRS in our paper overlaps the one used in [16] since we consider the generalized vector which corresponds to a kind of energy over the line $t=t_1+t_2$. Furthermore, the definition adopted in this paper is different from, for instance, the definition used in [16], and especially condition (5), which we will hereafter refer to as the standard condition.

However, condition (3) implies the standard condition in the sense that if (3) holds, then we have

$$T_f\begin{array}{c}\\ \max \\ t=t_1+t_2\end{array}\left(x^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\right)<c_2$$

and the standard condition becomes

$$x^{\top }(t_1,t_2)R_{2}x(t_1,t_2)<{\overline{c}}_2={\displaystyle \frac{c_2}{T_f}}.$$

Next, we consider the situation when the state is subject to some external disturbances. Thus, we give the definition of finite-region boundedness:

Definition 2

([21]). Given positive constants $c_1,c_2,T_f$ and the domain $\mathsf{\Omega }:=[0,T_f]$, the positive definite matrices $R_1:=diag(R_{1_h},R_{1_v})$ and $R_2:=diag(R_{2_h},R_{2_v})$ and $t=t_1+t_2$. The system (1) with $u\equiv 0$ is said to be finite-region-bounded (FRB) with respect to $(\mathsf{\Omega },R_1,R_2,\mathcal{W})$ if

$$\begin{array}{c}\hfill x_h^{\top }(0,t_2)R_{1_h}{x}_h(0,t_2)+x_v^{\top }(t_1,0)R_{1_v}{x}_v(t_1,0)\le c_1\forall t\in \mathsf{\Omega },\end{array}$$

(4)

which implies, for all $t\in \mathsf{\Omega }$,

$$\begin{array}{c}\hfill {\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s<c_2\end{array}$$

(5)

for all $\omega (t_1,t_2)\in \mathcal{W}\left(d\right)$.

Before giving the following lemma, we introduce the dual system of system (1) with $u\equiv 0$ and $\omega \equiv 0$:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& A^{\top }\check{x}(t_1,t_2)\hfill \end{array}$$

(6)

Lemma 1.

Let $\alpha \ge$ be a positive scalar and $P=P^{\top }=diag(P_h,P_v)>0$ be a definite positive matrix. Consider the two quadratic functions $V\left(x(t_1,t_2)\right)$ and $\check{V}\left(x(t_1,t_2)\right)$ (for its dual system (6)) with

$$\begin{array}{ccc}\hfill V\left(x(t_1,t_2)\right)& =& V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill V_h\left(x_h(t_1,t_2)\right)& =& x_h^{\top }(t_1,t_2)P_h{x}_h(t_1,t_2)\hfill \\ \hfill V_v\left(x_v(t_1,t_2)\right)& =& x_v^{\top }(t_1,t_2)P_v{x}_v(t_1,t_2)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \check{V}\left(\check{x}(t_1,t_2)\right)& =& {\check{V}}_h\left({\check{x}}_h(t_1,t_2)\right)+{\check{V}}_v\left({\check{x}}_v(t_1,t_2)\right)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\check{V}}_h\left({\check{x}}_h(t_1,t_2)\right)& =& {\check{x}}_h^{\top }(t_1,t_2)P_h^{-1}{\check{x}}_h(t_1,t_2)\hfill \\ \hfill {\check{V}}_v\left({\check{x}}_v(t_1,t_2)\right)& =& {\check{x}}_v^{\top }(t_1,t_2)P_v^{-1}{\check{x}}_v(t_1,t_2)\hfill \end{array}$$

The two following inequalities are equivalent:

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}-\alpha (V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right))<0\end{array}$$

(7a)

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }{\check{V}}_h\left({\check{x}}_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{\check{V}}_v\left({\check{x}}_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}-\alpha ({\check{V}}_h\left({\check{x}}_h(t_1,t_2)\right)+{\check{V}}_v\left({\check{x}}_v(t_1,t_2)\right))<0\end{array}$$

(7b)

Proof. 

Note that condition (7a) can be written explicitly as

$$\begin{array}{cc}& x^{\top }(t_1,t_2)\left(PA+A^{\top }P-\alpha P\right)x(t_1,t_2)\hfill \\ =& x^{\top }(t_1,t_2)P\left(A{P}^{-1}+P^{-1}{A}^{\top }-\alpha P^{-1}\right)Px(t_1,t_2)<0\hfill \end{array}$$

(8)

whereas, similarly, condition (7b) for the dual system reads

$$\begin{array}{cc}& {\check{x}}^{\top }(t_1,t_2)\left(A{P}^{-1}+P^{-1}{A}^{\top }-\alpha P^{-1}\right)\check{x}(t_1,t_2)\hfill \\ =& {\check{x}}^{\top }(t_1,t_2)P^{-1}\left(PA+A^{\top }P-\alpha P\right)P^{-1}\check{x}(t_1,t_2)<0\hfill \end{array}$$

(9)

from which we have the equivalence between the two statements; condition (8) holds if and only if (9) is satisfied, which means that (7a) and (7b) are equivalent. □

3. Stability Analysis

3.1. Finite-Region Stability

This section is devoted to the analysis of finite-region stability of 2D continuous systems. The following theorem presents a sufficient condition for system (1) with $u\equiv 0$ and $\omega \equiv 0$.

Theorem 1.

Given $\alpha \ge 0$, the 2D continuous system (1) with $u\equiv 0$ and $\omega \equiv 0$ is finite-region-stable with respect to $(c_1,c_2,T_f,R_1,R_2)$ if there exist a symmetric positive definite matrix $P\in {\mathrm{I}\mathrm{R}}^{n\times n}$ and some matrices $X_1$ and $X_2$ such that

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}+\mathrm{Sym}\left\{A{X}_1\right\}& P^{-1}-X_1^{\top }+A{X}_2\\ P^{-1}-X_1+X_2^{\top }{A}^{\top }& -\mathrm{Sym}\left\{X_2\right\}\end{array}\right]<0\end{array}$$

(10)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\sqrt{2}}{\alpha }}\left({\mathsf{e}}^{\alpha T_f}-1\right){\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[R_1\right]}}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[P\right]}}{\displaystyle \frac{c_1}{c_2}}\le 1\end{array}$$

(11)

Proof. 

For the dual system, we consider the positive definite function $\check{V}$. Notice that one can write

$$\begin{array}{ccc}\hfill \left[\begin{array}{cc}{A}^{\top }& -I\end{array}\right]\left[\begin{array}{c}\check{x}(t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]& =& 0\hfill \end{array}$$

(12)

Let

$$\begin{array}{c}\hfill {\check{x}}^{\top }(t_1,t_2)\left(A{P}^{-1}+P^{-1}{A}^{\top }-\alpha P^{-1}\right)\check{x}(t_1,t_2)<0\end{array}$$

and then we have

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}{P}^{-1}+P^{-1}{\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}-\alpha \check{x}(t_1,t_2)P^{-1}\check{x}(t_1,t_2)=}\hfill \\ \hfill {\left[\begin{array}{c}\check{x}(t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]}^{\top }\left[\begin{array}{cc}-\alpha P^{-1}& P^{-1}\\ P^{-1}& 0\end{array}\right]\left[\begin{array}{c}\check{x}(t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }\check{x}(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]<0\hfill \end{array}$$

(13)

Combining (12) and (13) and using the Finsler lemma [25], one can deduce that there exist matrices $X_1$ and $X_2$ such as

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}& P^{-1}\\ P^{-1}& 0\end{array}\right]+\left[\begin{array}{c}{X}_1^{\top }\\ X_2^{\top }\end{array}\right]\left[\begin{array}{cc}{A}^{\top }& -I\end{array}\right]+\left[\begin{array}{c}A\\ -I\end{array}\right]\left[\begin{array}{cc}{X}_1& X_2\end{array}\right]<0\end{array}$$

(14)

or in a compact form

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}+X_1^{\top }{A}^{\top }+A{X}_1& P^{-1}-X_1^{\top }+A{X}_2\\ P^{-1}-X_1+X_2^{\top }{A}^{\top }& -X_2-X_2^{\top }\end{array}\right]<0\end{array}$$

(15)

with $P^{-1}=\left[\begin{array}{cc}{P}_h^{-1}& 0\\ 0& P_v^{-1}\end{array}\right].$

• In order to prove (11), note that condition (15) guarantees what follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}-\alpha (V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right))<0\end{array}$$

which can be written as

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}<\alpha (V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right))\end{array}$$

(16)

Integrating inequality (16) along the line $t_1+t_2=t$ gives

$$\begin{array}{ccc}\hfill {\int }_{t_1+t_2=t}\left({\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(·,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,·)\right)}{\mathsf{\partial }{t}_2}}\right)\mathrm{d}s& \le & \alpha W\left(t\right)\hfill \end{array}$$

with

$$\begin{array}{c}\hfill W\left(t\right)={\int }_{t_1+t_2=t}(V_h\left(x_h(·,t_2)\right)+V_v\left(x_v(t_1,·)\right))\mathrm{d}s\end{array}$$

(17)

which corresponds to the ‘energy’ stored on the line $\{(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^2|t_1+t_2=t\}$. Using the same reasoning as in [26], one can write

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[{\int }_{t_1+t_2=t}(V_h\left(x_h(·,t_2)\right)+V_v\left(x_v(t_1,·)\right))\mathrm{d}s\right]=& {\int }_{t_1+t_2=t}\left({\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(·,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,·)\right)}{\mathsf{\partial }{t}_2}}\right)\mathrm{d}s\hfill \\ \hfill & +\sqrt{2}(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \hfill \end{array}$$

(18)

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}W\left(t\right)-\alpha W\left(t\right)-\sqrt{2}(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\le 0\end{array}$$

(19)

Note that $W\left(0\right)=0$, so the solution to (19) for $t\le T_f$ satisfies

$$\begin{array}{cc}\hfill W\left(t\right)& \le W\left(0\right){\mathsf{e}}^{\alpha t}+{\int }_0^t{\mathsf{e}}^{\alpha (t-\tau )}\sqrt{2}\times (V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \hfill \\ \hfill & \le {\mathsf{e}}^{\alpha t}\sqrt{2}{\int }_0^t{\mathsf{e}}^{-\alpha \tau }(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \hfill \end{array}$$

(20)

The condition above holds for $t\le T_f$. Moreover, from the definition of $W\left(t\right)$, we deduce that $W\left(t\right)\le W\left(T_f\right)$; then one can write

$$\begin{array}{c}\hfill W\left(t\right)\le \sqrt{2}{\mathsf{e}}^{\alpha T_f}{\int }_0^{T_f}{\mathsf{e}}^{-\alpha \tau }(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \end{array}$$

(21)

Notice that

$$\begin{array}{cc}\hfill {\displaystyle (V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))}& \le {\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}(x_h^{\top }(0,t)R_{1_h}{x}_h(0,t)+x_v^{\top }(t,0)R_{1_v}{x}_v(t,0))\hfill \\ \hfill & \le c_1{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\hfill \end{array}$$

(22)

which allows for $W\left(t\right)$ to be bound as follows:

$$\begin{array}{c}\hfill W\left(t\right)\le \sqrt{2}{\mathsf{e}}^{\alpha T_f}{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)\end{array}$$

(23)

Note also that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\lambda }_{min}\left[P\right]}{{\lambda }_{max}\left[R_2\right]}}{\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s\le W\left(t\right)\end{array}$$

(24)

which allows us to write

$$\begin{array}{ccc}\hfill {\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s\le {\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}W\left(t\right)& & \\ \hfill \le {\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}\sqrt{2}{\mathsf{e}}^{\alpha T_f}{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)\le c_2\end{array}$$

(25)

which states the finite-region stability. □

Remark 2.

If there exist ${\lambda }_1$ and ${\lambda }_2$ such that the three conditions below hold:

$$\begin{array}{ccc}\hfill P& <& {\lambda }_1{R}_1\hfill \end{array}$$

(26a)

$$\begin{array}{ccc}\hfill {\lambda }_2{R}_2& <& P\hfill \end{array}$$

(26b)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}& <& {\displaystyle \frac{\alpha }{\sqrt{2}{\mathsf{e}}^{\alpha T_f}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)}}{\displaystyle \frac{c_2}{c_1}}\hfill \end{array}$$

(26c)

then condition (25) is satisfied.

Indeed, (26a) implies

$${\lambda }_{max}\left[P\right]<{\lambda }_1{\lambda }_{min}\left[R_1\right]$$

whereas (26b) yields

$${\lambda }_2{\lambda }_{max}\left[R_2\right]<{\lambda }_{min}\left[P\right]$$

Combining both conditions, ones gets ${\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}<{\lambda }_1$ and ${\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}<{\displaystyle \frac{1}{{\lambda }_2}}$, which yield

$${\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}{\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}<{\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}<{\displaystyle \frac{\alpha }{\sqrt{2}{\mathsf{e}}^{\alpha T_f}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)}}{\displaystyle \frac{c_2}{c_1}}$$

in other words, (25) is satisfied.

3.2. Finite-Region Boundedness for Disturbances Generated by an External System

In this section, we investigate the finite-region boundedness for the 2D continuous system (1) with $u(t_1,t_2)=0$ and where the disturbance input $\omega (t_1,t_2)$ belongs to the set

$$\begin{array}{c}\hfill \mathcal{W}\left(d\right)=\{\omega (t_1,t_2)|{\displaystyle \frac{\mathsf{\partial }\omega (t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}=L\omega (t_1,t_2),{\omega }_h^{\top }(0,t_2){\omega }_h(0,t_2)+{\omega }_v^{\top }(t_1,0){\omega }_v(t_1,0)\le d\}\end{array}$$

(27)

Let

$$\begin{array}{ccc}\hfill z(t_1,t_2)& =& \left[\begin{array}{c}\hfill z_h(t_1,t_2)\\ \hfill z_v(t_1,t_2)\end{array}\right]=\left[\begin{array}{c}\hfill x_h(t_1,t_2)\\ \hfill {\omega }_h(t_1,t_2)\\ \hfill x_v(t_1,t_2)\\ \hfill {\omega }_v(t_1,t_2)\end{array}\right]\hfill \end{array}$$

(28)

then it becomes

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }z(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& \underset{\mathbf{A}}{\underbrace{\left[\begin{array}{cccc}{A}_{hh}& G_{hh}& A_{hv}& G_{hv}\\ 0& L_{hh}& 0& L_{hv}\\ A_{vh}& G_{vh}& A_{vv}& G_{vv}\\ 0& L_{vh}& 0& L_{vv}\end{array}\right]}}\left[\begin{array}{c}{x}_h(t_1,t_2)\\ {\omega }_h(t_1,t_2)\\ x_v(t_1,t_2)\\ {\omega }_v(t_1,t_2)\end{array}\right]\hfill \end{array}$$

(29)

or, otherwise,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }z(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& \mathbf{A}z(t_1,t_2)\hfill \end{array}$$

(30)

where $\mathbf{A}$ can be identified from (29).

Let

$$\begin{array}{c}\hfill V_h(t_1,t_2)=z_h^T(t_1,t_2)\left[\begin{array}{cc}{P}^h& 0\\ 0& S_h\end{array}\right]z_h(t_1,t_2),V_v(t_1,t_2)=z_v^T(t_1,t_2)\left[\begin{array}{cc}{P}^v& 0\\ 0& S_v\end{array}\right]z_v(t_1,t_2)\end{array}$$

(31)

where $z_h(t_1,t_2)$ and $z_v(t_1,t_2)$ are defined in (28).

Firstly, we present a sufficient condition for finite-region boundedness for the system (1).

Theorem 2.

Given $\alpha \ge 0$, the 2D continuous system (1) with $u\equiv 0$ is finite-region-stable with respect to $(c_1,c_2,T_f,{\overline{R}}_1,{\overline{R}}_2,d)$ if there exist symmetric positive definite matrices $P_h\in {\mathrm{I}\mathrm{R}}^{n_h\times n_h}$, $P_v\in {\mathrm{I}\mathrm{R}}^{n_v\times n_v}$, $S_h\in {\mathrm{I}\mathrm{R}}^{n_{{\omega }_h}\times n}$ and $S_v\in {\mathrm{I}\mathrm{R}}^{n_{{\omega }_v}\times n}$ and some matrices ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$, such that

$$\left[\begin{array}{cc}\hfill -\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{\mathbf{A}{\mathbf{X}}_1\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+\mathbf{A}{\mathbf{X}}_2\\ \hfill {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_2^{\top }{\mathbf{A}}^{\top }\hfill & -\mathrm{Sym}\left\{{\mathbf{X}}_2\right\}\end{array}\right]<0$$

(32)

with $\mathbf{P}=\mathrm{diag}\left\{P_h,S_h,P_v,S_v\right\}$ and

$$\begin{array}{c}\hfill \sqrt{2}{\mathsf{e}}^{\alpha T_f}\left(1-{\mathsf{e}}^{-\alpha T_f}\right){\displaystyle \frac{\left(c_1+d\right)}{c_2\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[{\overline{R}}_2\right]}{{\lambda }_{min}\left[\mathbf{P}\right]}}{\displaystyle \frac{{\lambda }_{max}\left[\mathbf{P}\right]}{{\lambda }_{min}\left[{\overline{R}}_1\right]}}\le 1\end{array}$$

(33)

Proof. 

Since the disturbance belongs to $\mathcal{W}\left(d\right)$ and the initial condition for $x(t_1,t_2)$ complies with (4), then the initial conditions for system (30) can be bounded as

$$\begin{array}{c}\hfill z_h^{\top }(0,t_2){\overline{R}}_{1_h}{z}_h(0,t_2)+z_v^{\top }(t_1,0){\overline{R}}_{1_v}{z}_v(t_1,0)\le d+c_1\forall t\in \mathsf{\Omega },\end{array}$$

(34)

and ${\overline{R}}_{i_h}=\mathrm{diag}\left\{R_{1_h},I\right\}$, $i=1,2$.

At this step, we can apply Theorem 1 to system (30) and write condition (10) for system (30) as

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{\mathbf{A}{\mathbf{X}}_1\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+\mathbf{A}{\mathbf{X}}_2\\ {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_2^{\top }{\mathbf{A}}^{\top }& -\mathrm{Sym}\left\{{\mathbf{X}}_2\right\}\end{array}\right]<0\end{array}$$

which is exactly condition (15).

Similarly, following the same development as for the proof of Theorem 1, one obtains inequality (25) expressed for system (30) as

$$\begin{array}{ccc}\hfill {\int }_{t_1+t_2=t}{z}^{\top }(t_1,t_2){\overline{R}}_{2}z(t_1,t_2)\mathrm{d}s\le {\displaystyle \frac{{\lambda }_{max}\left[{\overline{R}}_2\right]}{{\lambda }_{min}\left[\mathbf{P}\right]}}\overline{W}\left(t\right)& & \\ \hfill \le {\displaystyle \frac{{\lambda }_{max}\left[{\overline{R}}_2\right]}{{\lambda }_{min}\left[\mathbf{P}\right]}}\sqrt{2}{\mathsf{e}}^{\alpha T_f}{\displaystyle \frac{\left(c_1+d\right)}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[\mathbf{P}\right]}{{\lambda }_{min}\left[{\overline{R}}_1\right]}}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)\end{array}$$

(35)

where $\overline{W}\left(t\right)$ is given by (17) expressed with $z(t_1,t_2)$ instead of $x(t_1,t_2)$, or explicitly

$$\begin{array}{c}\hfill \overline{W}\left(t\right)={\int }_{t_1+t_2=t}(V_h\left(z_h(·,t_2)\right)+V_v\left(z_v(t_1,·)\right))\mathrm{d}s\end{array}$$

(36)

with $V_h(z_h(t_1,t_2)$ and $V_v\left(z_v(t_1,t_2)\right)$, given by (31).

Moreover, since we have

$$\begin{array}{c}\hfill {\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s\le {\int }_{t_1+t_2=t}{z}^{\top }(t_1,t_2){\overline{R}}_{2}z(t_1,t_2)\mathrm{d}s\end{array}$$

a sufficient condition for finite-region boundedness is to ensure that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\lambda }_{max}\left[{\overline{R}}_2\right]}{{\lambda }_{min}\left[\mathbf{P}\right]}}\sqrt{2}{\mathsf{e}}^{\alpha T_f}{\displaystyle \frac{\left(c_1+d\right)}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[\mathbf{P}\right]}{{\lambda }_{min}\left[{\overline{R}}_1\right]}}\left(1-{\mathsf{e}}^{-\alpha T_f}\right)\le c_2\end{array}$$

□

3.3. Finite-Region Boundedness for Energy-Bounded Disturbances

In this section, we will deal with finite-region boundedness for 2D continuous systems with energy-bounded disturbances:

$$\begin{array}{c}\hfill \mathcal{W}\left(d\right)=\left\{\omega (t_1,t_2)\right|{\omega }^{\top }(t_1,t_2)\omega (t_1,t_2)\le d\}\end{array}$$

(37)

The following theorem presents a sufficient condition for system (1) with $u=0$.

Theorem 3.

The 2D continuous system (1) with $u=0$ is finite-region-bounded with respect to $(c_1,c_2,T_f,R_1,R_2,d)$ if there exist symmetric positive definite matrices $P\in {\mathrm{I}\mathrm{R}}^{n\times n}$ and $S\in {\mathrm{I}\mathrm{R}}^{n_{\omega }\times n_{\omega }}$, and some auxiliary matrices $T_1$, $T_2$, and $T_3$ such that the following condition holds:

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha P& 0& P\\ 0& -\gamma S& 0\\ P& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right]\left[\begin{array}{ccc}A& G& -I\end{array}\right]\right\}<0\end{array}$$

(38)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}{\displaystyle \frac{\sqrt{2}}{c_2}}\left(T_f{\displaystyle \frac{d\gamma }{\alpha }}{\lambda }_{max}\left[S\right]+{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\right)\left({\mathsf{e}}^{\alpha T_f}-1\right)\le 1\end{array}$$

(39)

Proof. 

By taking into account the impact of the disturbance on the system (1) with $u=0$, condition (7a) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}-\alpha (V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right))\\ \hfill -\gamma {\omega }^{\top }(t_1,t_2)S\omega (t_1,t_2)<0\end{array}$$

(40)

or explicitly

$$\begin{array}{c}\hfill 2x^{\top }(t_1,t_2)P{\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}-\alpha x^{\top }(t_1,t_2)Px(t_1,t_2)-\gamma {\omega }^{\top }(t_1,t_2)S\omega (t_1,t_2)<0\end{array}$$

which can be written as

$$\begin{array}{c}\hfill {\left[\begin{array}{c}x(t_1,t_2)\\ \omega (t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]}^{\top }\left[\begin{array}{ccc}-\alpha P& 0& P\\ 0& -\gamma S& 0\\ P& 0& 0\end{array}\right]\left[\begin{array}{c}x(t_1,t_2)\\ \omega (t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]<0\end{array}$$

(41)

where $x(t_1,t_2)$ and $\omega (t_1,t_2)$ satisfy

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}A& G& -I\end{array}\right]\left[\begin{array}{c}x(t_1,t_2)\\ \omega (t_1,t_2)\\ {\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}\end{array}\right]=0\end{array}$$

(42)

Combining (41) and (42) and applying the Finsler lemma, one can state that there exists a matrix $T={\left[\begin{array}{ccc}{T}_1^{\top }& T_2^{\top }& T_3^{\top }\end{array}\right]}^{\top }$ such that (38) holds.

For the proof of the second condition of the Theorem, one has to integrate both sides of (40) to get

$$\begin{array}{c}{\displaystyle {\int }_{t_1+t_2=t}\left({\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}\right) \mathrm{d}s<}\hfill \\ \alpha {\int }_{t_1+t_2=t}(V_h\left(x_h(t_1,t_2)\right)+V_v\left(x_v(t_1,t_2)\right))\mathrm{d}s+\gamma {\int }_{t_1+t_2=t}{\omega }^{\top }(t_1,t_2)S\omega (t_1,t_2) \mathrm{d}s\hfill \end{array}$$

Using $W\left(t\right)$ defined by (17), one gets

$$\begin{array}{c}{\displaystyle {\int }_{t_1+t_2=t}\left({\displaystyle \frac{\mathsf{\partial }{V}_h\left(x_h(t_1,t_2)\right)}{\mathsf{\partial }{t}_1}}+{\displaystyle \frac{\mathsf{\partial }{V}_v\left(x_v(t_1,t_2)\right)}{\mathsf{\partial }{t}_2}}\right) \mathrm{d}s<}\hfill \\ \alpha W\left(t\right)+\gamma {\int }_{t_1+t_2=t}{\omega }^{\top }(t_1,t_2)S\omega (t_1,t_2)\mathrm{d}s\hfill \end{array}$$

and using (18), one obtains

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}W\left(t\right)-\alpha W\left(t\right)& \le & \gamma {\int }_{t_1+t_2=t}{\omega }^{\top }(t_1,t_2)S\omega (t_1,t_2)\mathrm{d}s+\sqrt{2}(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\hfill \\ & \le & \gamma \sqrt{2}{\lambda }_{max}\left[S\right]T_{f}d+\sqrt{2}(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\hfill \end{array}$$

(43)

Note that $W\left(0\right)=0$ so the solution to (43) for $t\le T_f$ satisfies

$$\begin{array}{cc}\hfill W\left(t\right)& \le W\left(0\right){\mathsf{e}}^{\alpha t}+{\int }_0^t{\mathsf{e}}^{\alpha (t-\tau )}\sqrt{2}\times (V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \hfill \\ \hfill & \le \gamma {\displaystyle \frac{\sqrt{2}}{\alpha }}{\lambda }_{max}\left[S\right]T_{f}d\left({\mathsf{e}}^{\alpha t}-1\right)+{\mathsf{e}}^{\alpha t}\sqrt{2}{\int }_0^t{\mathsf{e}}^{-\alpha \tau }(V_h\left(x_{h0}(·,t)\right)+V_v\left(x_{v0}(t,·)\right))\mathrm{d}\tau \hfill \end{array}$$

(44)

and similarly to (21), (22), and (23), one can bound $W\left(t\right)$ as follows:

$$\begin{array}{ccc}\hfill W\left(t\right)& \le & \sqrt{2}\left(T_f{\displaystyle \frac{d\gamma }{\alpha }}{\lambda }_{max}\left[S\right]+{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\right)\left({\mathsf{e}}^{\alpha T_f}-1\right)\hfill \end{array}$$

(45)

At this step, we use (24) to get

$$\begin{array}{c}\hfill {\int }_{t_1+t_2=t}{x}^{\top }(t_1,t_2)R_{2}x(t_1,t_2)\mathrm{d}s\le {\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}\sqrt{2}\left(T_f{\displaystyle \frac{d\gamma }{\alpha }}{\lambda }_{max}\left[S\right]+{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\right)\left({\mathsf{e}}^{\alpha T_f}-1\right)\le c_2\end{array}$$

(46)

which states the finite-region stability.

□

4. State Feedback Synthesis

This section is dedicated to the synthesis of a state feedback controller for 2D continuous systems. Consider the state feedback $u(t_1,t_2)=Kx(t_1,t_2)$.

The closed loop system can be written as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }x(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& A_{c}x(t_1,t_2)+G\omega (t_1,t_2)\hfill \end{array}$$

(47)

with

$$\begin{array}{ccc}\hfill A_c& =& A+BK\hfill \\ \hfill A_c& =& \left[\begin{array}{cc}{A}_{hh}& A_{hv}\\ A_{vh}& A_{vv}\end{array}\right]+\left[\begin{array}{c}{B}_h\\ B_v\end{array}\right]\left[\begin{array}{cc}{K}_1& K_2\end{array}\right],\hfill \end{array}$$

4.1. Finite-Region Stabilization

In this subsection, we will assume that $\omega (t_1,t_2)=0$ in (47), and the objective is to design a state feedback gain K such that the closed loop system (47) with $\omega (t_1,t_2)=0$ is finite-region-stable.

The following theorem presents sufficient conditions for finite-region stability for the closed loop system (47).

Theorem 4.

Given $\alpha \ge 0$, the 2D continuous system in the closed loop (47) is finite-region-stable with respect to $(c_1,c_2,T_f,R_1,R_2)$ if there exist a symmetric definite positive matrix $P\in {\mathrm{I}\mathrm{R}}^{n\times n}$ and a matrix $X_1$ such that

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\mathsf{\Phi }}_1& {\mathsf{\Phi }}_2\\ {\mathsf{\Phi }}_2^{\top }& -\mathrm{Sym}\left\{X_1\right\}\end{array}\right]<0\end{array}$$

(48)

and

$$\begin{array}{c}\hfill {ϵ}_1={\displaystyle \frac{\sqrt{2}}{\alpha }}\left({\mathsf{e}}^{\alpha T_f}-1\right){\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[R_1\right]}}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[P\right]}}{\displaystyle \frac{c_1}{c_2}}\le 1\end{array}$$

(49)

where

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1& =-\alpha P^{-1}+A{X}_1+BY+X_1^{\top }{A}^{\top }+Y^{\top }{B}^{\top }=-\alpha P^{-1}+\mathrm{Sym}\left\{A{X}_1+BY\right\}\hfill \\ \hfill {\mathsf{\Phi }}_2& =P^{-1}-X_1^{\top }+A{X}_1+BY\hfill \end{array}$$

Then, the control gain K is given by $K=Y{X}_1^{-1}$.

Proof. 

By applying Equation (10) for the closed loop system (47), one gets

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}+\mathrm{Sym}\left\{A_c{X}_1\right\}& P^{-1}-X_1^{\top }+A_c{X}_2\\ P^{-1}-X_1+X_2^{\top }{A}_c^{\top }& -\mathrm{Sym}\left\{X_2\right\}\end{array}\right]<0\end{array}$$

(50)

which can be written as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}+A{X}_1+BK{X}_1+X_1^{\top }{A}^{\top }+X_1^{\top }{K}^{\top }{B}^{\top }& P^{-1}-X_1^{\top }+A{X}_2+BK{X}_2\\ P^{-1}-X_1+X_2^{\top }{A}^{\top }+X_2^{\top }{K}^{\top }{B}^{\top }& -X_2-X_2^{\top }\end{array}\right]<0\end{array}$$

(51)

The inequality contains nonlinear terms $K{X}_1$ and $K{X}_2$.

In order to ensure linearity, we take $X_1=X_2$ and $Y=K{X}_2=K{X}_1$.

Thus, the condition becomes

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha P^{-1}+A{X}_1+BY+X_1^{\top }{A}^{\top }+Y^{\top }{B}^{\top }& P^{-1}-X_1^{\top }+A{X}_1+BY\\ P^{-1}-X_1+X_1^{\top }{A}^{\top }+Y^{\top }{B}^{\top }& -X_1-X_1^{\top }\end{array}\right]<0\end{array}$$

(52)

which can be written in a compact form as condition (48) in Theorem 4.

□

4.2. Finite-Region Stabilization Under Disturbances Generated by an External System

In this section, we investigate the finite-region stabilization issue for the 2D continuous system (1) under disturbances generated by an external system (27). Consider the closed loop system

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }z(t_1,t_2)}{\mathsf{\partial }{t}_1\mathsf{\partial }{t}_2}}& =& \mathbf{A}z(t_1,t_2)+Bu(t_1,t_2)\hfill \\ & =& {\mathbf{A}}_{c}z(t_1,t_2)\hfill \end{array}$$

(53)

with

$$\begin{array}{c}\hfill {\mathbf{A}}_c=\mathbf{A}+B\mathbf{K}=\left[\begin{array}{cccc}{A}_{hh}+B_h{\mathbf{K}}_1& G_{hh}& A_{hv}+B_h{\mathbf{K}}_2& G_{hv}\\ 0& L_{hh}& 0& L_{hv}\\ A_{vh}+B_v{\mathbf{K}}_1& G_{vh}& A_{vv}+B_v{\mathbf{K}}_2& G_{vv}\\ 0& L_{vh}& 0& L_{vv}\end{array}\right]\end{array}$$

The following theorem presents a sufficient condition that ensures that system (53) is finite-region-bounded with respect to $(c_1,c_2,T_f,{\overline{R}}_1,{\overline{R}}_2,d)$.

Theorem 5.

Given $\alpha \ge 0$, the 2D continuous system (1) is finite-region-stable with respect to $(c_1,c_2,T_f,{\overline{R}}_1,{\overline{R}}_2,d)$ if there exist symmetric positive definite matrices $P_h\in {\mathrm{I}\mathrm{R}}^{n_h\times n_h}$, $P_v\in {\mathrm{I}\mathrm{R}}^{n_v\times n_v}$, $S_h\in {\mathrm{I}\mathrm{R}}^{n_{{\omega }_h}\times n}$ and $S_v\in {\mathrm{I}\mathrm{R}}^{n_{{\omega }_v}\times n}$ and a matrix $\mathbf{X}$ such that

$$\left[\begin{array}{cc}\hfill -\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{\mathbf{A}{\mathbf{X}}_1+B\mathbf{Y}\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+\mathbf{A}{\mathbf{X}}_1+B\mathbf{Y}\\ \hfill {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_1^{\top }{\mathbf{A}}^{\top }+Y^{\top }{B}^{\top }& -\mathrm{Sym}\left\{{\mathbf{X}}_1\right\}\end{array}\right]<0$$

(54)

with $\mathbf{P}=\mathrm{diag}\left\{P_h,S_h,P_v,S_v\right\}$ and

$$\begin{array}{c}\hfill {ϵ}_2=\sqrt{2}{\mathsf{e}}^{\alpha T_f}\left(1-{\mathsf{e}}^{-\alpha T_f}\right){\displaystyle \frac{\left(c_1+d\right)}{c_2\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[{\overline{R}}_2\right]}{{\lambda }_{min}\left[\mathbf{P}\right]}}{\displaystyle \frac{{\lambda }_{max}\left[\mathbf{P}\right]}{{\lambda }_{min}\left[{\overline{R}}_1\right]}}\le 1\end{array}$$

(55)

The controller $\mathbf{K}$ is given by $\mathbf{K}=\mathbf{Y}{\mathbf{X}}_1^{-1}$.

Proof. 

By replacing $\mathbf{A}$ by ${\mathbf{A}}_c$ in condition (15), one obtains

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{{\mathbf{A}}_c{\mathbf{X}}_1\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+{\mathbf{A}}_c{\mathbf{X}}_2\\ {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_2^{\top }{\mathbf{A}}_c^{\top }& -\mathrm{Sym}\left\{{\mathbf{X}}_2\right\}\end{array}\right]<0\end{array}$$

(56)

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{(\mathbf{A}+B\mathbf{K}){\mathbf{X}}_1\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+(\mathbf{A}+B\mathbf{K}){\mathbf{X}}_2\\ {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_2^{\top }{(\mathbf{A}+B\mathbf{K})}^{\top }& -\mathrm{Sym}\left\{{\mathbf{X}}_2\right\}\end{array}\right]<0\end{array}$$

(57)

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\alpha {\mathbf{P}}^{-1}+\mathrm{Sym}\left\{\mathbf{A}{\mathbf{X}}_2+B\mathbf{K}{\mathbf{X}}_2\right\}& {\mathbf{P}}^{-1}-{\mathbf{X}}_1^{\top }+\mathbf{A}{\mathbf{X}}_1+B\mathbf{K}{\mathbf{X}}_1\\ {\mathbf{P}}^{-1}-{\mathbf{X}}_1+{\mathbf{X}}_2^{\top }{\mathbf{A}}^{\top }+{\mathbf{X}}_2^{\top }{\mathbf{K}}^{\top }{B}^{\top }& -\mathrm{Sym}\left\{{\mathbf{X}}_2\right\}\end{array}\right]<0\end{array}$$

(58)

which gives nonlinear terms $\mathbf{K}{\mathbf{X}}_1$ and $\mathbf{K}{\mathbf{X}}_2$. In order to obtain linear terms, we take ${\mathbf{X}}_1={\mathbf{X}}_2$ and put $\mathbf{Y}=\mathbf{K}\mathbf{X}$. This results in condition (54), which concludes the proof. □

4.3. Finite-Region Stabilization for Energy-Bounded Disturbances

This section focuses on the finite-region stabilization issue for the 2D continuous system (1) under energy-bounded disturbances (37).

Theorem 6.

The 2D continuous system (1) is finite-region-bounded with respect to $(c_1,c_2,T_f,R_1,R_2,d)$ if there exist two matrices X and Z and two symmetric positive definite matrices $P=X^{-\top }\overline{P}{X}^{-1}\in {\mathrm{I}\mathrm{R}}^{n\times n}$ and $S=Z^{-\top }\overline{S}{Z}^{-1}\in {\mathrm{I}\mathrm{R}}^{n_{\omega }\times n_{\omega }}$ and a matrix Y such that the following condition holds:

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha \overline{P}& 0& \overline{P}\\ 0& -\gamma \overline{S}& 0\\ \overline{P}& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{I}_{n,n}\\ I_{n_{\omega },n}\\ I_{n,n}\end{array}\right]\left[\begin{array}{ccc}AX+BY& GZ& -X\end{array}\right]\right\}<0\end{array}$$

(59)

and

$$\begin{array}{c}\hfill {ϵ}_3={\displaystyle \frac{{\lambda }_{max}\left[R_2\right]}{{\lambda }_{min}\left[P\right]}}{\displaystyle \frac{\sqrt{2}}{c_2}}\left(T_f{\displaystyle \frac{d\gamma }{\alpha }}{\lambda }_{max}\left[S\right]+{\displaystyle \frac{c_1}{\alpha }}{\displaystyle \frac{{\lambda }_{max}\left[P\right]}{{\lambda }_{min}\left[R_1\right]}}\right)\left({\mathsf{e}}^{\alpha T_f}-1\right)\le 1\end{array}$$

(60)

The controller K is given by $K=Y{X}^{-1}$.

Remark 3.

From the Finsler lemma, matrices X and Z should be full square matrices. However, since we have to recover matrices P and S from $\overline{P}$ and $\overline{S}$, one has to ensure that P is a bloc diagonal matrix. The easy way to ensure this is to assume that X and $\overline{P}$ have the same structure. The same applies for Z and $\overline{S}$.

Proof. 

Replacing $\mathbf{A}$ by ${\mathbf{A}}_c$ in condition (38) of Theorem 3, one obtains

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha P& 0& P\\ 0& -\gamma S& 0\\ P& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right]\left[\begin{array}{ccc}{A}_c& G& -I_n\end{array}\right]\right\}<0\end{array}$$

(61)

which can be written as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha P& 0& P\\ 0& -\gamma S& 0\\ P& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right]\left[\begin{array}{ccc}(A+BK)& G& -I\end{array}\right]\right\}<0\end{array}$$

(62)

In order to obtain the gain K, we have to assume that $T_1=T_3=X^{-\top }$ and $T_2=Z^{-\top }{I}_{n_{\omega },n}$. So one gets

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha P& 0& P\\ 0& -\gamma S& 0\\ P& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{X}^{-\top }\\ Z^{-\top }{I}_{n_{\omega },n}\\ X^{-\top }\end{array}\right]\left[\begin{array}{ccc}(A+BK)& G& -I\end{array}\right]\right\}<0\end{array}$$

Multiplying, appropriately, both sides of the expression above by $\mathrm{diag}\left\{X,Z,X\right\}$ and its transpose, ones gets

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}-\alpha X^{\top }PX& 0& X^{\top }PX\\ 0& -\gamma Z^{\top }SZ& 0\\ X^{\top }PX& 0& 0\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{I}_n\\ I_{n_{\omega },n}\\ I_n\end{array}\right]\left[\begin{array}{ccc}(A+BK)X& GZ& -X\end{array}\right]\right\}<0\end{array}$$

The change in variables $\overline{P}=X^{\top }PX$, $\overline{S}=Z^{\top }SZ$ and $Y=KX$ allows one to obtain (59), which concludes the proof. □

5. Numerical Example

In this section, we consider numerical examples in order to illustrate the relevance of the proposed methods.

Example 1.

Consider the 2D continuous Roesser model (1) without disturbances where

$$\begin{array}{c}{A}_{hh}=\left[\begin{array}{cc}\hfill 1.0& \hfill -0.5\\ \hfill 0.0& \hfill -2.0\end{array}\right],A_{hv}=\left[\begin{array}{cc}\hfill 0.1& \hfill -1.0\\ \hfill 0.0& \hfill 0.1\end{array}\right],A_{vh}=\left[\begin{array}{cc}\hfill -1.0& \hfill 0.0\\ \hfill 0.0& \hfill 0.1\end{array}\right],A_{vv}=\left[\begin{array}{cc}\hfill 0.0& \hfill -3.0\\ \hfill 1.0& \hfill -0.6\end{array}\right],\hfill \\ \begin{array}{c}\hfill B_1=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right],B_2=\left[\begin{array}{cc}\hfill 0& \hfill 0\\ \hfill 1& \hfill 1\end{array}\right]\end{array}\hfill \end{array}$$

For the feasibility of conditions of Theorem 4, we use the following parameters: $R_1=10I,R_2=I,c_1=2.5,c_2=10,T_f=1.1940,N=20,\alpha =3.5$. By applying Theorem 4, the condition (48) is feasible and the eigenvalues of the this LMI are all negatives.The solution proposed by the solver is

$$\begin{array}{c}\hfill P=\left[\begin{array}{cccc}\hfill 1.8354& \hfill 0.0537& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.0537& \hfill 2.4070& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.0000& \hfill 0.0000& \hfill 1.5095& \hfill 0.0050\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.0050& \hfill 2.3256\end{array}\right]\end{array}$$

The obtained matrix P satisfies condition (49) with ${ϵ}_1=0.9998652$. The resulting state feedback controller is given by

$$\begin{array}{c}\hfill K=\left[\begin{array}{cccc}\hfill -0.5382& \hfill 0.9872& \hfill -0.2467& \hfill -0.0537\\ \hfill 0.5225& \hfill -0.6607& \hfill -0.3940& \hfill -0.4369\end{array}\right]\end{array}$$

Example 2.

Consider the 2D continuous Roesser model under disturbances generated by an external system (29), where

$$\begin{array}{c}\hfill A_{hh}=\left[\begin{array}{cc}\hfill 1& \hfill -0.5\\ \hfill 0& \hfill -2\end{array}\right],A_{hv}=\left[\begin{array}{cc}\hfill 0.1& \hfill -1\\ \hfill 0& \hfill 0.1\end{array}\right],A_{vh}=\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill 0.1\end{array}\right],A_{vv}=\left[\begin{array}{cc}\hfill 0& \hfill -3\\ \hfill 1& \hfill -0.6\end{array}\right],\end{array}$$

$$\begin{array}{c}\hfill G_{hh}=\left[\begin{array}{cc}\hfill 0.1& \hfill -0.5\\ \hfill 0& \hfill -2\end{array}\right],G_{hv}=\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill -0.1\end{array}\right],G_{vh}=\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill -0.1\end{array}\right],G_{vv}=\left[\begin{array}{cc}\hfill 0& \hfill -3\\ \hfill 1& \hfill -0.6\end{array}\right],\end{array}$$

$$\begin{array}{c}\hfill L_{hh}=\left[\begin{array}{cc}\hfill 0.3& \hfill -0.5\\ \hfill 0& \hfill -2\end{array}\right],L_{hv}=\left[\begin{array}{cc}\hfill 0.3& \hfill -1\\ \hfill 0& \hfill 0.1\end{array}\right],L_{vh}=\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill -0.1\end{array}\right],L_{vv}=\left[\begin{array}{cc}\hfill 0& \hfill -3\\ \hfill 1& \hfill -0.6\end{array}\right],\end{array}$$

$$\begin{array}{c}\hfill B_1=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right],B_2=\left[\begin{array}{cc}\hfill 0& \hfill 0\\ \hfill 1& \hfill 1\end{array}\right]\end{array}$$

For the feasibility of conditions of Theorem 6, we have used the following parameters: $R_1=10I,R_2=I,c_1=2.5,c_2=10,T_f=1.0880,N=20,\alpha =3.5$.

By applying Theorem 6, condition (54) was found feasible.

The solution proposed by the solver is

$$\begin{array}{c}\hfill P=\left[\begin{array}{cccccccc}\hfill 3.2428& \hfill 0.2170& \hfill -0.0665& \hfill -0.1176& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.2170& \hfill 2.9521& \hfill 0.1645& \hfill -0.0786& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000\\ \hfill -0.0665& \hfill 0.1645& \hfill 3.2395& \hfill 0.0001& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000\\ \hfill -0.1176& \hfill -0.0786& \hfill 0.0001& \hfill 3.4726& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 2.5936& \hfill -0.2322& \hfill -0.0807& \hfill 0.0152\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill -0.2322& \hfill 3.6619& \hfill 0.1946& \hfill 0.3340\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill -0.1641& \hfill -0.0807& \hfill 0.1946& \hfill 2.7456\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0000& \hfill 0.0152& \hfill 0.3340& \hfill -0.1681& \hfill 3.7704\end{array}\right]\end{array}$$

Here as well, the obtained matrix P satisfies condition (55) with ${ϵ}_2=0.9998618$.

The obtained state feedback controller is given as follows:

$$\begin{array}{c}\hfill K=\left[\begin{array}{cccccccc}\hfill -0.7882& \hfill 0.4500& \hfill -0.3378& \hfill 0.2979& \hfill -0.4746& \hfill 0.1443& \hfill -0.3943& \hfill 0.7477\\ \hfill 0.1867& \hfill -0.6003& \hfill 0.0585& \hfill 0.5068& \hfill -0.0665& \hfill -1.4522& \hfill -0.6026& \hfill -1.1629\end{array}\right]\end{array}$$

Example 3.

Consider the 2D continuous Roesser model under energy-bounded disturbances (1), where the system matrices are defined in the previous example.

For the feasibility of conditions of Theorem 6, we have used the following parameters: $R_1=10I,R_2=I,c_1=2.5,c_2=10,T_f=0.02421,N=20,\alpha =3.5$ and $\gamma =3.38$.

By applying Theorem 6, condition (59) was found feasible.

The solution proposed by the solver is

$$\begin{array}{c}\hfill P=\left[\begin{array}{cccc}\hfill 0.4336& \hfill 0.1222& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.1222& \hfill 0.5326& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.1437& \hfill -0.1356\\ \hfill 0.0000& \hfill 0.0000& \hfill -0.1356& \hfill 3.0026\end{array}\right],S=\left[\begin{array}{cccc}\hfill 8.9346& \hfill 3.0178& \hfill 0.0000& \hfill 0.0000\\ \hfill 8.9346& \hfill 3.0178& \hfill 0.0000& \hfill 0.0000\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.6520& \hfill 0.2505\\ \hfill 0.0000& \hfill 0.0000& \hfill 0.2505& \hfill 3.9159\end{array}\right]\end{array}$$

These matrices satisfy condition (60) with ${ϵ}_3=0.9998576$.

The obtained state feedback controller is given as follows:

$$\begin{array}{c}\hfill K=\left[\begin{array}{cccc}\hfill -0.4443& \hfill 0.7320& \hfill -0.2221& \hfill -0.2760\\ \hfill 0.4968& \hfill -0.4387& \hfill -0.4075& \hfill -0.8818\end{array}\right]\end{array}$$

6. Conclusions

This paper addresses the finite-region stability and finite-region boundedness for 2D continuous systems described by a Roesser model. The first contribution of the paper is a sufficient condition for finite-region stability based on a new definition which uses an integral condition along the characteristic lines. Then, the proposed condition is applied to two distinct classes of disturbances to establish the condition of finite-region boundedness in order to guarantee that the state remains bounded even in the presence of external disturbances or energy-bounded disturbances. Finally, we derive a sufficient condition for the finite-region stabilization of 2D continuous systems. This synthesis procedure not only stabilizes the system but guarantees that the resulting closed-loop dynamics strictly adhere to the defined finite-region performance. All the conditions provided in the paper involves the solution of feasibility problems constrained by LMIs, which can be solved via widely available software. Some numerical examples illustrate the effectiveness of the derived conditions. The limitation of the study lies on the derived conditions which are sufficient but not necessary and could be conservative. Future research may include the extension of the FRS/FRB framework to 2D switched systems, nonlinear Roesser models, and the design of robust controllers under polytopic uncertainties and the design of an observer-based dynamic output feedback controller for FRS/FRB.