# The Stability Analysis for a 2 × 2 Conservation Law with PI Controller and PDP Controller

Authors: Dong-Xia Zhao; Fang-Xia Bao
Journal: Journal of Mathematics
 (2023)

---

## Abstract

This article is concerned with the stability analysis of a2×2 conservation law with PI and PDP boundary feedback control. In the PI controller case, the authors establish a sufficient and necessary stability condition for feedback parameters by Walton–Marshall criterion. Moreover, in PDP controller case, the characteristic equation is an exponential polynomial which includes two exponential arguments. Based on Ruan’s results on stability for quasi-polynomials and Schur–Cohn criterion on the roots inside the unit circle for a real coefficient polynomial, a complete characterization for system parameters and the feedback delay is presented. Also, the delay-independent and delay-dependent stability intervals are obtained in this article.


---

## Body

## 1. Introduction and Problem Formulation

For the feedback control problem of hyperbolic systems, the PI (proportional-integral) controller may be one of the most widely used controller (see [1–3]). With PI controller, the system is directed to a designated equilibrium based on the present state and the accumulated past error. Moreover, the main approach to stability analysis is the Lyapunov function technique (see [4, 5]). In [6], the authors propose a strict Lyapunov functional for the hyperbolic systems with conservation laws. And then, in [7], the quadratic Lyapunov function is used to demonstrate the local exponential stability in H2 norm. Moreover, the boundary dissipative conditions are presented explicitly. By using the Lyapunov direct method and spectral analysis, the local exponential stability was proved in [8]. Necessary and sufficient stability conditions are obtained in [9], which improves the result in [8].However, there is little reference applying the characteristic method to investigate the stability of a hyperbolic system. The exponential stability of the quasilinear wave equation is established in [10] by studying the solution along the characteristic curves. The feedback stabilization problem of a scalar conservation law with PI controller(1)∂tξt,x+λ¯∂xξt,x=0,t∈0,∞,x∈0,L,ξt,0=kpξt,L+ki∫0tξu,Ldu,is discussed in [11], where λ¯ is a positive constant, kp and ki are feedback parameters, and the characteristic equation is a first-order exponential polynomial:(2)μ−kpμ+kie−L/λ¯μ=0.And then, the sufficient and necessary stability conditions for the parameters are deduced by the Walton–Marshall stability criterion. It is noted that it poses an open problem in dealing with hyperbolic systems of n>1 conservation or balance laws in [11].In this article, we are mainly concerned with the stability analysis of a2×2 hyperbolic system with the form:(3)∂tξ1t,x+λ1∂xξ1t,x=0,∂tξ2t,x−λ2∂xξ2t,x=0,with the following PI controller and initial conditions:(4)ξ1t,0=k0ξ2t,0+k1∫0tξ2s,0ds,ξ2t,L=k2ξ1t,L+k3∫0tξ1s,Lds,(5)ξ10,x=ξ10x,ξ20,x=ξ20x,where ξit,xi=1,2 are the system states over the space interval 0,L and time domain 0,+∞. Here, λ1 and λ2 denote the velocity of the system state, and we assume that λ1 and λ2 are two positive constants. The parameters k0,k1,k2, and k3 denote the proportional and integral control gain, respectively. Furthermore, we ignore the memory term ∫0tξis,⋅ds and substitute it by the delayed position ξit−σ,⋅; i.e., the PDP (position and delayed position) controller is as follows:(6)ξ1t,0=k0ξ2t,0+k1ξ2t−σ,0,ξ2t,L=k2ξ1t,L+k3ξ1t−σ,L,where σ>0 denotes the feedback time delay. Similarly, the PDP controller means that the system is designated to the equilibrium based upon the information of the present state and some past states. Our question is as follows: can system (3) be stabilized by PDP feedback (6)? In [12, 13], the PDP feedback control method was used to stabilize an inverted pendulum system, and it shows that the inverted equilibrium position is asymptotically stable for ∀τ>0. In this paper, the delay-independent stability and instability for systems (3) and (6) can also be analyzed.The contribution of this research is as follows: (i) we construct the characteristic equation for systems (3) and (4) and (3) and (6): a kind of second-order exponential polynomial with single delay and multiple delays, respectively. (ii) We study the root distribution of the characteristic equation and give the sufficient and necessary conditions about the feedback parameters kii=0,1,2,3 such that both the closed-loop systems (3) and (4) and (3) and (6) are asymptotically stable. Additionally, the delay-independent and delay-dependent stability intervals are obtained in this article.The structure of the article is as follows: we deduce the characteristic equation of (3) (4), and then, by Walton–Marshall stability criterion, we obtain the stability conditions for the system parameters kii=0,1,2,3 in Section 2. In Section 3, the sufficient and necessary stability conditions for (3) and (6) are given by Schur–Cohn criterion. Next, we present some numerical simulations in Section 4. Finally, several concluding remarks are presented in Section 5.


## 2. Stability of the Closed-Loop System (3) and (4)

### 2.1. The Characteristic Equation

Given the formal solution of (4) can be calculated as follows:(7)ξ1t,x=yt−1λ1x,ξ2t,x=zt+1λ2x.Substitute (7) into the PI boundary (4), and assume yt=ceμtc≠0 is a solution. By a series of calculation, we can get the characteristic equation of the closed-loop system (3) with PI control (4):(8)Δμ=μ2−k0μ+k1k2μ+k3e−τμ=0,where(9)τ=a+b>0,Lλ1=a>0,Lλ2=b>0.Remark 1.
We can also obtain the characteristic equation (8) as follows. Consider the characteristic problem for system (3)(10)μξ1=−λ1ξ1′,μξ2=λ2ξ2′,whose fundamental solution matrix under PI boundary (4) is(11)μ−k0μ+k1k2μ+k3e−μL/λ1−μeμL/λ2.
Hence, let the determinant of (11) equal to 0, we can have (8).


### 2.2. Walton–Marshall Stability Criterion

In this section, firstly, we state the Walton–Marshall stability criterion. It gives the sufficient and necessary conditions under which all the roots of the characteristic equation with an exponential argument(12)fμ+gμe−τμ=0,have negative real parts, where τ≥0 denotes time delay, μ∈C, fμ is a p−order polynomial with real coefficients, gμ is a q−order polynomial with real coefficients, and p≥q.The Walton–Marshall stability criterion ([14, 15] (subsection 5.6)) can be summarized in three steps as follows:Step 1: We show the stability ifτ=0.Step 2: We increaseτ from zero to q+ and analyze how the eigenvalues move with the increasing τ. The process can be divided into two parts:(i)
Ifp>q, for an infinitely small τ, the new roots must appear at infinity; otherwise, e−τμ would be near 1 and there cannot be any new roots. Hence, if p>q, (12) can be satisfied for large μ if and only if e−τμ is large, i.e., Reμ<0. So, in this situation, we can omit this step.(ii)
Ifp=q, we study the potential crossings of the imaginary axis, i.e., we analyze the purely imaginary roots μ=iω,ω∈R. Given to conjugate symmetry of (12), we get that if μ=iω is a solution, then −μ=−iω is also a solution. Substitute μ=±iω into (12), we get a real polynomial in ω2, and for the sake of simplicity, we denote it by(13)Pω2=fiωf−iω−giωg−iω.The location of new roots is determined by the sign ofPω2 for large ω. A necessary condition for exponential stability is(14)Pω2>0,for largeω∈R,which can be found in [15] (subsection 5.6).Step 3 We study the positive real roots ofPω2=0. Moreover, according to fiω+giωe−iωτ=0, the corresponding critical values for time-delayτ can be calculated:(15)cosτω=Re−fiωgiω,sinτω=Imfiωgiω.When we got a value ofτ such that a pure imaginary root appears, we study the roots’ behavior as τ changes. To be specific, we study whether, as τ is increasing, this root crosses the imaginary axis from the right half plane into the left plane (a stabilizing root), or vice versa, if it goes from the left half plane into the right half plane (a destabilizing root). And then, the stability switching would be analyzed. A nonnegative root y of P is stabilizing if the first derivative P′y<0 and destabilizing if P′y>0. If P′y=0, one should consider the higher-order derivatives.


### 2.3. Stability Conditions for System Parameters

For characteristic equation (8), we assume(16)fμ=μ2,gμ=−k0μ+k1k2μ+k3,where ki≠0i=0,1,2,3, and p=q=2. Now, we analyze according to the three steps of Walton–Marshall stability criterion.Step 1.
Ifτ=0, (8) is equivalent to(17)k0k2−1μ2+k0k3+k1k2μ+k1k3=0,which is a quadratic equation in μ. The discriminant of (17) is(18)Δ1=k0k3−k1k22+4k1k3.
Hence, the system without delay is asymptotically stable if and only if one of the following cases holds.Case 1Δ1≥0, k0k2<1, and −k0k3+k1k2−Δ1>0Case 2Δ1<0, k0k2<1, and k0k3+k1k2<0Remark 2.
Since in step 2, we havek0k2<1 holds. Here, the case k0k2≥1 can be omitted.Step 2.
Letμ=iω is a pure imaginary root of (8), by (13), we have that(19)Pω2=ω4−k12+k02ω2k32+k22ω2=1−k02k22ω4−k12k22+k02k32ω2−k12k32.
It is obvious thatPω2>0, for large ω∈R if and only if(20)k0k2<1.Step 3.
Assumey=ω2, we can have(21)Py=1−k02k22y2−k12k22+k02k32y−k12k32,and the discriminant is(22)Δ2=k12k22+k02k322+41−k02k22k12k32>0,under the condition (20). Thus, we can have one positive root y1 and one negative root y2:(23)y1=k12k22+k02k32+Δ221−k02k22>0,y2=k12k22+k02k32−Δ221−k02k22<0.
Sincey=ω2>0, so we only need to consider y1:=ω02. Noticed that(24)P′y=21−k02k22y−k12k22+k02k32,and(25)P′y1=Δ2>0,the root is destabilizing. By (15), time delay τ˜ satisfies(26)cosτ˜ω=−ω2−k0k2ω2+k1k3−k0k2ω2+k1k32+k0k3+k1k22ω2,sinτ˜ω=−ω3k0k3+k1k2−k0k2ω2+k1k32+k0k3+k1k22ω2,and then,(27)tanτ˜ω=ωk0k3+k1k2−k0k2ω2+k1k3.
Assumeτ0 is the smallest τ˜, so we have(28)τ0=1ω0arctanω0k0k3+k1k2−k0k2ω2+k1k3+jπ,where(29)j=0,ifω0k0k3+k1k2−k0k2ω2+k1k3>0,1,ifω0k0k3+k1k2−k0k2ω2+k1k3<0.
In conclusion, we can have the following results.Theorem 1.
Supposeki≠0. The closed-loop system (3) and (4) are asymptotically stable for theL2-norm if and only if the feedback parameters satisfyk0k2<1 and one of the following conditions:(i)
k0k3−k1k22+4k1k3≥0, k0k3+k1k2<0, and k1k3<0(ii)
k0k3−k1k22+4k1k3<0, and k0k3+k1k2<0
Furthermore,τsatisfiesτ∈0,τ0, whereτ0is given by (28).Remark 3.
Ifk0k2=0,k1k3≠0, from the characteristic equation (8), we have(30)fμ=μ2,gμ=−k0k3+k1k2μ−k1k3,where p=2,q=1. Hence, p>q. By Walton–Marshall stability criterion, we can have results as follows.Theorem 2.
Let us assumek0k2=0,k1k3≠0, and(31)Δ3=k0k3+k1k22+4k1k3.The closed-loop system (3) and (4) is asymptotically stable for ∀τ>0 if and only if the feedback parameters satisfy one of the following conditions:(i)
Δ3<0, and k0k3+k1k2<0(ii)
Δ3≥0, and k1k3<0,k0k3+k1k2<0Remark 4.
Ifk1k3=0, the characteristic equation (8) changes to(32)Δμ=μ2eτμ−k0k2μ2+k0k3+k1k2μ=0,which meansμ=0must be an eigenvalue, and then, systems (3) and (4) cannot be stable.


## 2.1. The Characteristic Equation

Given the formal solution of (4) can be calculated as follows:(7)ξ1t,x=yt−1λ1x,ξ2t,x=zt+1λ2x.Substitute (7) into the PI boundary (4), and assume yt=ceμtc≠0 is a solution. By a series of calculation, we can get the characteristic equation of the closed-loop system (3) with PI control (4):(8)Δμ=μ2−k0μ+k1k2μ+k3e−τμ=0,where(9)τ=a+b>0,Lλ1=a>0,Lλ2=b>0.Remark 1.
We can also obtain the characteristic equation (8) as follows. Consider the characteristic problem for system (3)(10)μξ1=−λ1ξ1′,μξ2=λ2ξ2′,whose fundamental solution matrix under PI boundary (4) is(11)μ−k0μ+k1k2μ+k3e−μL/λ1−μeμL/λ2.
Hence, let the determinant of (11) equal to 0, we can have (8).


## 2.2. Walton–Marshall Stability Criterion

In this section, firstly, we state the Walton–Marshall stability criterion. It gives the sufficient and necessary conditions under which all the roots of the characteristic equation with an exponential argument(12)fμ+gμe−τμ=0,have negative real parts, where τ≥0 denotes time delay, μ∈C, fμ is a p−order polynomial with real coefficients, gμ is a q−order polynomial with real coefficients, and p≥q.The Walton–Marshall stability criterion ([14, 15] (subsection 5.6)) can be summarized in three steps as follows:Step 1: We show the stability ifτ=0.Step 2: We increaseτ from zero to q+ and analyze how the eigenvalues move with the increasing τ. The process can be divided into two parts:(i)
Ifp>q, for an infinitely small τ, the new roots must appear at infinity; otherwise, e−τμ would be near 1 and there cannot be any new roots. Hence, if p>q, (12) can be satisfied for large μ if and only if e−τμ is large, i.e., Reμ<0. So, in this situation, we can omit this step.(ii)
Ifp=q, we study the potential crossings of the imaginary axis, i.e., we analyze the purely imaginary roots μ=iω,ω∈R. Given to conjugate symmetry of (12), we get that if μ=iω is a solution, then −μ=−iω is also a solution. Substitute μ=±iω into (12), we get a real polynomial in ω2, and for the sake of simplicity, we denote it by(13)Pω2=fiωf−iω−giωg−iω.The location of new roots is determined by the sign ofPω2 for large ω. A necessary condition for exponential stability is(14)Pω2>0,for largeω∈R,which can be found in [15] (subsection 5.6).Step 3 We study the positive real roots ofPω2=0. Moreover, according to fiω+giωe−iωτ=0, the corresponding critical values for time-delayτ can be calculated:(15)cosτω=Re−fiωgiω,sinτω=Imfiωgiω.When we got a value ofτ such that a pure imaginary root appears, we study the roots’ behavior as τ changes. To be specific, we study whether, as τ is increasing, this root crosses the imaginary axis from the right half plane into the left plane (a stabilizing root), or vice versa, if it goes from the left half plane into the right half plane (a destabilizing root). And then, the stability switching would be analyzed. A nonnegative root y of P is stabilizing if the first derivative P′y<0 and destabilizing if P′y>0. If P′y=0, one should consider the higher-order derivatives.


## 2.3. Stability Conditions for System Parameters

For characteristic equation (8), we assume(16)fμ=μ2,gμ=−k0μ+k1k2μ+k3,where ki≠0i=0,1,2,3, and p=q=2. Now, we analyze according to the three steps of Walton–Marshall stability criterion.Step 1.
Ifτ=0, (8) is equivalent to(17)k0k2−1μ2+k0k3+k1k2μ+k1k3=0,which is a quadratic equation in μ. The discriminant of (17) is(18)Δ1=k0k3−k1k22+4k1k3.
Hence, the system without delay is asymptotically stable if and only if one of the following cases holds.Case 1Δ1≥0, k0k2<1, and −k0k3+k1k2−Δ1>0Case 2Δ1<0, k0k2<1, and k0k3+k1k2<0Remark 2.
Since in step 2, we havek0k2<1 holds. Here, the case k0k2≥1 can be omitted.Step 2.
Letμ=iω is a pure imaginary root of (8), by (13), we have that(19)Pω2=ω4−k12+k02ω2k32+k22ω2=1−k02k22ω4−k12k22+k02k32ω2−k12k32.
It is obvious thatPω2>0, for large ω∈R if and only if(20)k0k2<1.Step 3.
Assumey=ω2, we can have(21)Py=1−k02k22y2−k12k22+k02k32y−k12k32,and the discriminant is(22)Δ2=k12k22+k02k322+41−k02k22k12k32>0,under the condition (20). Thus, we can have one positive root y1 and one negative root y2:(23)y1=k12k22+k02k32+Δ221−k02k22>0,y2=k12k22+k02k32−Δ221−k02k22<0.
Sincey=ω2>0, so we only need to consider y1:=ω02. Noticed that(24)P′y=21−k02k22y−k12k22+k02k32,and(25)P′y1=Δ2>0,the root is destabilizing. By (15), time delay τ˜ satisfies(26)cosτ˜ω=−ω2−k0k2ω2+k1k3−k0k2ω2+k1k32+k0k3+k1k22ω2,sinτ˜ω=−ω3k0k3+k1k2−k0k2ω2+k1k32+k0k3+k1k22ω2,and then,(27)tanτ˜ω=ωk0k3+k1k2−k0k2ω2+k1k3.
Assumeτ0 is the smallest τ˜, so we have(28)τ0=1ω0arctanω0k0k3+k1k2−k0k2ω2+k1k3+jπ,where(29)j=0,ifω0k0k3+k1k2−k0k2ω2+k1k3>0,1,ifω0k0k3+k1k2−k0k2ω2+k1k3<0.
In conclusion, we can have the following results.Theorem 1.
Supposeki≠0. The closed-loop system (3) and (4) are asymptotically stable for theL2-norm if and only if the feedback parameters satisfyk0k2<1 and one of the following conditions:(i)
k0k3−k1k22+4k1k3≥0, k0k3+k1k2<0, and k1k3<0(ii)
k0k3−k1k22+4k1k3<0, and k0k3+k1k2<0
Furthermore,τsatisfiesτ∈0,τ0, whereτ0is given by (28).Remark 3.
Ifk0k2=0,k1k3≠0, from the characteristic equation (8), we have(30)fμ=μ2,gμ=−k0k3+k1k2μ−k1k3,where p=2,q=1. Hence, p>q. By Walton–Marshall stability criterion, we can have results as follows.Theorem 2.
Let us assumek0k2=0,k1k3≠0, and(31)Δ3=k0k3+k1k22+4k1k3.The closed-loop system (3) and (4) is asymptotically stable for ∀τ>0 if and only if the feedback parameters satisfy one of the following conditions:(i)
Δ3<0, and k0k3+k1k2<0(ii)
Δ3≥0, and k1k3<0,k0k3+k1k2<0Remark 4.
Ifk1k3=0, the characteristic equation (8) changes to(32)Δμ=μ2eτμ−k0k2μ2+k0k3+k1k2μ=0,which meansμ=0must be an eigenvalue, and then, systems (3) and (4) cannot be stable.


## 3. Stability of System (3) with PDP Control (6)

In this part, we will analyze the stability for the hyperbolic system (3) under PDP (position and delayed position) feedback instead of PI controller:(33)∂tξ1t,x+λ1∂xξ1t,x=0,∂tξ2t,x−λ2∂xξ2t,x=0,ξ1t,0=k0ξ2t,0+k1ξ2t−σ,0,ξ2t,L=k2ξ1t,L+k3ξ1t−σ,L,where x∈0,L,σ>0 denotes the feedback delay. By a simple calculation as in Section 2, we can get the characteristic equation of the closed-loop system (33) as(34)eμτ+2σ=k0k2e2μσ+k1k2+k0k3eμσ+k1k3,where τ=L/λ1+L/λ2.Remark 5.
Here, the Walton–Marshall stability criterion in Section2.2 cannot be applied, since there are two delays: τ and σ.


### 3.1. Delay-Independent Stability and Instability Results

According to Corollary 2.4 in [16], we can get the following delay-independent stability and instability results.Let(35)Δ=4k1k3+k0k22k1k2+k0k32−16k0k1k2k3k02k22+k12k32+k12k22+k02k32−1,and(36)y1,2=−2k1k3+k0k2k1k2+k0k3±Δ8k0k1k2k3.Theorem 3.
Assume the feedback parameters satisfy the following conditions:(37)k0k2+k1k3+k1k2+k0k3<1.Then,(i)
IfΔ<0, system (33) is always asymptotically stable for ∀τ,σ>0(ii)
IfΔ≥0, and y1,2>1, system (33) is always asymptotically stable for ∀τ,σ>0(iii)
IfΔ≥0, and y1≤1 (or y2≤1), system (33) is always unstable for ∀τ,σ>0Proof.
Firstly, we consider the limit caseσ=0, and then, the characteristic equation (34) turns to(38)eμτ=k0k2+k1k3+k1k2+k0k3.
Hence, all roots of (38) have negative real parts if and only if the condition (37) holds.
As the delay increasesσ>0, let μ=iω is a pure imaginary root of (34). Hence, we have that(39)eiωτ+2σ=k0k2e2iωσ+k1k2+k0k3eiωσ+k1k3.
Separating the real part and imaginary part, we get(40)cosωτ+2σ=k0k2cos2ωσ+k1k2+k0k3cosωσ+k1k3,sinωτ+2σ=k0k2sin2ωσ+k1k2+k0k3sinωσ.
By squaring both sides of the (40), we have(41)1=k02k22+k1k2+k0k32+k12k32+2k0k1k2k3cos2ωσ+2k1k3+k0k2k1k2+k0k3cosωσ,and then,(42)4k0k1k2k3cos2ωσ+2k1k3+k0k2k1k2+k0k3cosωσ+k02k22+k12k32+k12k22+k02k32−1=0,which is a quadratic equation of cosωσ.
Lety=cosωσ, and y1,2 denotes the two roots of (42). If (40) has no real root (i.e., Δ<0), or the absolute values of both roots y1,2 are bigger than 1 (i.e. Δ≥0 and y1,2>1), it means that the characteristic equation has no pure imaginary roots. Hence, system (33) is always stable for any positive value of the delay σ, if and only if (37) holds. On the other hand, if Δ≥0, and y1≤1 (or y2≤1), no matter what the time delay σ>0 is, the characteristic equation (34) must have pure imaginary root. Hence, system (33) is always unstable for any positive value of the delay σ.


### 3.2. Schur–Cohn Criterion and Stability Analysis

In this section, we discuss a special case:τ is an integral multiple of σ. Let(43)τσ=k,where k>0 is an integral number. Then, the characteristic (34) can be written as follows:(44)1−k0k2e−kμσ−k1k2+k0k3e−k+1μσ−k1k3e−k+2μσ=0.Remark 6.
It should be noted that the Walton–Marshall stability criterion for systems with multiple commensurate delays(45)∑j=0nAjμe−jμτ=0,cannot be applied here, where τ>0 denotes the delay, and Akμk=0,1,…,n are polynomials in μ with real coefficients. In fact, in (41), Ajμ are constant, and then, the recursive procedure for finding the pure imaginary roots in Walton–Marshall stability criterion cannot be used.On the other hand, denote(46)eμσ=z,and then, the characteristic equation (34) can be written as follows:(47)Fz=z2+k−k0k2z2−k1k2+k0k3z−k1k3=0,which is a real 2+k− order polynomial in z. Hence, Reμ<0 in (34) if and only if z<1 in (47).Next, we shall apply the Schur–Cohn criterion (see, e.g., [17] on pp. 34–36 or Proposition 5.3 of [18] on p. 27) to check that the roots of the polynomial Fz are inside the unit circle.Theorem 4.
Suppose that all the roots ofFzin (43) lie inside the unit circle. Then,(48)1−k0k2−k1k2+k0k3−k1k3>0,and(49)1+−1k−k0k2+k1k2+k0k3−k1k3>0.More accurately, there are two cases:Case 1: ifk is an even number, then(50)1−k0k2+k1k3>k1k2+k0k3.Case 2: ifk is an odd number, then(51)1−k1k2+k0k3>k0k2+k1k3.Proof.
By Schur–Cohn Criterion, we have(52)F1>0,−12+kF−1>0.
Substitute this into (43), we can get (44) and (45), respectively.
Furthermore, whenk is an even number, (45) becomes(53)1+−k0k2+k1k2+k0k3−k1k3>0.
Combining (44) with (48), we have (46) directly. Similarly, if k is an odd number, we get (47).
Next, for the givenk, we show the necessary and sufficient conditions of the stability of the feedback parameters kii=0,1,2,3.Theorem 5.
(i)
Fork=1, all the roots of Fz lie inside the unit circle if and only if(54)1−k1k2+k0k3>k0k2+k1k3,1−k1k2+k0k3−k1k3k0k2+k1k3>0,1+k1k2+k0k3−k1k3−k0k2+k1k3>0.(ii)
Fork=2, all the roots of Fz lie inside the unit circle if and only if(55)1−k0k2+k1k3>k1k2+k0k3,k1k3<1,1−k1k31−k0k2−k1k2+k0k32−k1k3k0k2+k1k31−k1k3>0,1+k1k31+k0k2−k1k2+k0k32−k1k3k1k3−k0k21+k1k3>0.Proof.
Ifk=1,(56)Fz=z3−k0k2z2−k1k2+k0k3z−k1k3=0.
By Theorem4, we have the first inequality of (54): 1−k1k2+k0k3>k0k2+k1k3. Noting that(57)Δ2±=10−k0k21±0−k1k3−k1k3−k1k2+k0k3=1∓k1k3−k0k2∓k1k31∓k1k2+k0k3,all the determinants of all inners are(58)detΔ2±=1∓k1k2+k0k3−k1k3±k0k2+k1k3.
SodetΔ2±>0 yields the last two equalities of (54).
Ifk=2,(59)Fz=z4−k0k2z2−k1k2+k0k3z−k1k3=0.
SinceF1>0, and F−1>0, we have the first inequality of (55) by Theorem 4. Noting that(60)Δ3±=100010−k0k201±00−k1k30−k1k3−k1k2+k0k3−k1k3−k1k2+k0k3−k0k2=10∓k1k301∓k1k3∓k1k2+k0k3−k0k2∓k1k3∓k1k2+k0k31∓k0k2.
So the determinants of all innersdetΔ1±>0 and detΔ3±>0 yield the last three inequalities in (55).


## 3.1. Delay-Independent Stability and Instability Results

According to Corollary 2.4 in [16], we can get the following delay-independent stability and instability results.Let(35)Δ=4k1k3+k0k22k1k2+k0k32−16k0k1k2k3k02k22+k12k32+k12k22+k02k32−1,and(36)y1,2=−2k1k3+k0k2k1k2+k0k3±Δ8k0k1k2k3.Theorem 3.
Assume the feedback parameters satisfy the following conditions:(37)k0k2+k1k3+k1k2+k0k3<1.Then,(i)
IfΔ<0, system (33) is always asymptotically stable for ∀τ,σ>0(ii)
IfΔ≥0, and y1,2>1, system (33) is always asymptotically stable for ∀τ,σ>0(iii)
IfΔ≥0, and y1≤1 (or y2≤1), system (33) is always unstable for ∀τ,σ>0Proof.
Firstly, we consider the limit caseσ=0, and then, the characteristic equation (34) turns to(38)eμτ=k0k2+k1k3+k1k2+k0k3.
Hence, all roots of (38) have negative real parts if and only if the condition (37) holds.
As the delay increasesσ>0, let μ=iω is a pure imaginary root of (34). Hence, we have that(39)eiωτ+2σ=k0k2e2iωσ+k1k2+k0k3eiωσ+k1k3.
Separating the real part and imaginary part, we get(40)cosωτ+2σ=k0k2cos2ωσ+k1k2+k0k3cosωσ+k1k3,sinωτ+2σ=k0k2sin2ωσ+k1k2+k0k3sinωσ.
By squaring both sides of the (40), we have(41)1=k02k22+k1k2+k0k32+k12k32+2k0k1k2k3cos2ωσ+2k1k3+k0k2k1k2+k0k3cosωσ,and then,(42)4k0k1k2k3cos2ωσ+2k1k3+k0k2k1k2+k0k3cosωσ+k02k22+k12k32+k12k22+k02k32−1=0,which is a quadratic equation of cosωσ.
Lety=cosωσ, and y1,2 denotes the two roots of (42). If (40) has no real root (i.e., Δ<0), or the absolute values of both roots y1,2 are bigger than 1 (i.e. Δ≥0 and y1,2>1), it means that the characteristic equation has no pure imaginary roots. Hence, system (33) is always stable for any positive value of the delay σ, if and only if (37) holds. On the other hand, if Δ≥0, and y1≤1 (or y2≤1), no matter what the time delay σ>0 is, the characteristic equation (34) must have pure imaginary root. Hence, system (33) is always unstable for any positive value of the delay σ.


## 3.2. Schur–Cohn Criterion and Stability Analysis

In this section, we discuss a special case:τ is an integral multiple of σ. Let(43)τσ=k,where k>0 is an integral number. Then, the characteristic (34) can be written as follows:(44)1−k0k2e−kμσ−k1k2+k0k3e−k+1μσ−k1k3e−k+2μσ=0.Remark 6.
It should be noted that the Walton–Marshall stability criterion for systems with multiple commensurate delays(45)∑j=0nAjμe−jμτ=0,cannot be applied here, where τ>0 denotes the delay, and Akμk=0,1,…,n are polynomials in μ with real coefficients. In fact, in (41), Ajμ are constant, and then, the recursive procedure for finding the pure imaginary roots in Walton–Marshall stability criterion cannot be used.On the other hand, denote(46)eμσ=z,and then, the characteristic equation (34) can be written as follows:(47)Fz=z2+k−k0k2z2−k1k2+k0k3z−k1k3=0,which is a real 2+k− order polynomial in z. Hence, Reμ<0 in (34) if and only if z<1 in (47).Next, we shall apply the Schur–Cohn criterion (see, e.g., [17] on pp. 34–36 or Proposition 5.3 of [18] on p. 27) to check that the roots of the polynomial Fz are inside the unit circle.Theorem 4.
Suppose that all the roots ofFzin (43) lie inside the unit circle. Then,(48)1−k0k2−k1k2+k0k3−k1k3>0,and(49)1+−1k−k0k2+k1k2+k0k3−k1k3>0.More accurately, there are two cases:Case 1: ifk is an even number, then(50)1−k0k2+k1k3>k1k2+k0k3.Case 2: ifk is an odd number, then(51)1−k1k2+k0k3>k0k2+k1k3.Proof.
By Schur–Cohn Criterion, we have(52)F1>0,−12+kF−1>0.
Substitute this into (43), we can get (44) and (45), respectively.
Furthermore, whenk is an even number, (45) becomes(53)1+−k0k2+k1k2+k0k3−k1k3>0.
Combining (44) with (48), we have (46) directly. Similarly, if k is an odd number, we get (47).
Next, for the givenk, we show the necessary and sufficient conditions of the stability of the feedback parameters kii=0,1,2,3.Theorem 5.
(i)
Fork=1, all the roots of Fz lie inside the unit circle if and only if(54)1−k1k2+k0k3>k0k2+k1k3,1−k1k2+k0k3−k1k3k0k2+k1k3>0,1+k1k2+k0k3−k1k3−k0k2+k1k3>0.(ii)
Fork=2, all the roots of Fz lie inside the unit circle if and only if(55)1−k0k2+k1k3>k1k2+k0k3,k1k3<1,1−k1k31−k0k2−k1k2+k0k32−k1k3k0k2+k1k31−k1k3>0,1+k1k31+k0k2−k1k2+k0k32−k1k3k1k3−k0k21+k1k3>0.Proof.
Ifk=1,(56)Fz=z3−k0k2z2−k1k2+k0k3z−k1k3=0.
By Theorem4, we have the first inequality of (54): 1−k1k2+k0k3>k0k2+k1k3. Noting that(57)Δ2±=10−k0k21±0−k1k3−k1k3−k1k2+k0k3=1∓k1k3−k0k2∓k1k31∓k1k2+k0k3,all the determinants of all inners are(58)detΔ2±=1∓k1k2+k0k3−k1k3±k0k2+k1k3.
SodetΔ2±>0 yields the last two equalities of (54).
Ifk=2,(59)Fz=z4−k0k2z2−k1k2+k0k3z−k1k3=0.
SinceF1>0, and F−1>0, we have the first inequality of (55) by Theorem 4. Noting that(60)Δ3±=100010−k0k201±00−k1k30−k1k3−k1k2+k0k3−k1k3−k1k2+k0k3−k0k2=10∓k1k301∓k1k3∓k1k2+k0k3−k0k2∓k1k3∓k1k2+k0k31∓k0k2.
So the determinants of all innersdetΔ1±>0 and detΔ3±>0 yield the last three inequalities in (55).


## 4. Simulation Results

In this part, we will give some simulation results by Matlab software.Example 1.
Let the feedback parametersk0=−2,k1=−1,k2=k3=1/4in (4). Then, we havek0k2=1/2<1and(61)k0k3−k1k22+4k1k3=−15/16<0,k0k3+k1k2=−3/4<0,which satisfy the conditions of Theorem 1 (ii). Furthermore, by a series of calculations, we can get the critical valueτ0≈2.1616. Therefore, systems (3) and (4) is asymptotically stable ifτ=L/λ1+L/λ2<τ0. Hence, we can chooseλ1=0.9,λ2=0.8,L=0.8, and thenτ≈1.8889<τ0. Figure 1shows the stability of (3) and (4) with the initial conditionsξ10,x=−cos2πx/L+1,ξ20,x=sin2πx/L.Figure 1
The convergence of the stateξ1t,x and ξ2t,x for system (3) with boundary conditions (4).Example 2.
Let the feedback parametersk0=0.3,k1=0.1,k2=0.2,k3=0.4in (33). Then, we havek0k2+k1k3+k1k2+k0k3=0.24<1andΔ≈−0.036848<0in (35). Hence, by Theorem 3 (i), system (33) is always stable for∀σ>0. Figure 2shows the stability of (33) with the parametersλ1=10,λ2=3,L=1,σ=0.5and the initial conditionsξ10,x=12cos2πx+6sin2πx,ξ20,x=12cos2πx.Figure 2
The convergence of the stateξ1t,x and ξ2t,x for system (33).


## 5. Concluding Remarks

In this paper, the PI and PDP boundary feedback controllers are applied to a2×2 hyperbolic system with conservation law. By using Walton–Marshall stability criterion and Schur–Cohn criterion, the sufficient and necessary stability and instability conditions for feedback parameters are established individually. And especially, the delay-dependent and delay-independent stabilities are analyzed.It is worth noting that a2×2 hyperbolic system with balance laws(62)∂tξ1t,x+λ1∂xξ1t,x=γξ1t,x,∂tξ2t,x−λ2∂xξ2t,x=δξ1t,x,which can depict the phenomenon of biological population affected by age structure in population ecology and describe the traffic dynamics in a freeway segment, which can also be discussed by the methods of the Walton–Marshall stability criterion and Schur–Cohn criterion, where the right-hand side of (62) can be regarded as a nonhomogeneous disturbance term.In the future, our objective is to study the stability of a2×2 coupled hyperbolic system(63)∂tξ1t,x+λ1∂xξ1t,x+γξ1t,x+δξ2t,x=0,∂tξ2t,x−λ2∂xξ2t,x+γξ1t,x+δξ2t,x=0,which can describe the dynamics of a single open-channel system with nonzero bottom slope and bottom friction. The main difficulty is to find the characteristic equation under PI and PDP boundary feedback controller and choose a suitable method to analyze the stability and instability conditions for system parameters and the time delay.



---

*Source: 4382517-2023-02-22.xml*