The PDP feedback control design and exponential stability for a star-shaped network of open channels

Abstract

This paper investigates the stability of a star-shaped network of open channels under the position and delayed position (PDP) feedback controller. Firstly, we deal with the special case of a single channel, and the well-posedness and exponential stability of the closed-loop system are established by using the semi-group theory and a strict Lyapunov function. The explicit dissipative conditions for system parameters and time-delay are presented. Then, we analyze the delay-independent stability of the closed-loop system by the method of characteristic. Next, we extend the PDP control design to the star-shaped network of $n$ channels with $n-1$ inlet channels and one outlet channel, and the exponential stability can be obtained similarly. Finally, we give some numerical simulations for the special case of three channels.

Introduction

It is well known that in many industrial applications, a simple proportional position feedback is not enough to get satisfactory performance for the controlled ODE/PDE (ordinary differential equation/partial differential equation) systems (see, e.g. Astrom and Murray, ¹ Wang et al., ² and Perrollaz and Rosier ³ ). Additional feedbacks, such as the integral feedback, that is, proportional-integral (PI for short) controller, are needed to enhance the system performance. The main idea of a PI controller is that the system is steered to a given desired equilibrium based on information about the present error and the accumulated past error. In recent years, the topic of controlling hyperbolic systems with PI controller has received a lot of attention (see, e.g. O’Dwyer, ⁴ Bastin et al., ⁵ and Bastin and Coron ⁶ ). For a scalar conservation law with PI boundary control in, ⁷ the stability conditions of feedback parameters were derived based on the Walton-Marshall stability criterion (see Walton and Marshall ⁸ ). The dynamic feedback control law by the PI controller form was proposed such that the star-shaped network and the cascaded network of $2\times 2$ hyperbolic systems are asymptotically stable and the output is also regulated to the desired reference in Trinh et al.,^9,10 respectively. PI controller can also be used to make the nonlinear hyperbolic system stable. In, ¹¹ the local exponential stability was proved by using the method of spectral analysis and the Lyapunov direct method. A necessary and sufficient stability conditions are obtained in Coron and Hayat, ¹² which improves the result in Trinh et al. ¹¹ Recently, the problem of finite-time consensus for multi-agent systems driven by hyperbolic partial differential equations are studied in Wang and Huang, ¹³ and a new fractional power controller is designed to ensure the leaderless and leader-following finite-time consensus.

Till now, there are a growing number of works on the controllability and stability of star-shaped networks, and the control strategies are mainly based on boundary feedback control.^14,15 A system of $N$ Korteweg-de Vries equations coupled by the boundary conditions has appeared in different contexts to describe propagation phenomena, where the configuration is also a star-shaped network, and the boundary inputs can act on a central node and on the $N$ external nodes. By using $(N+1)$ controls, the authors in Ammari and Crépeau ¹⁶ obtained stabilization and controllability results for a KdV system posed on a star-shaped network. And then, the authors in Cerpa et al. ¹⁷ succeed to remove the input acting on the central node and consequently obtain the exact controllability with $N$ inputs. However, several novel control techniques have emerged to apply to the star-shaped systems. In Hugo et al., ¹⁸ by acting with saturated controls, the global well-posedness and stability of the linear and nonlinear Korteweg-de Vries equations on a finite star-shaped network are established. In Wang et al., ¹⁹ a timevarying extended state observer for a star-shaped thermoelastic system is proposed to estimate and cancel the disturbance, thereby largely removing the uncertainties and simplifying the feedback control law design. Based on the linear operator semigroup theory, the well-posedness and exponential stability of the closed-loop system are obtained.

It is found that the control design without considering the delay effect might be not reliable and not safe in applications, where the stability or control performance depends sharply on the delay changes, and then the delay effect on the system dynamics and control performance must be carefully studied.^20–22 Smith and Rubin had shown that in the control of lightly damped oscillatory systems, the time delay when used in the controller may cancel the effect of oscillatory complex poles and produce a deadbeat response in Tallman and Smith ²³ and Rubin. ²⁴ Suh and Bien introduced a control design called “proportional minus delay controller” to improve the performance of the system based on the approximation and numerical simulation in Suh and Bien.^25,26 In the recent two decades, the time-delay is introduced as a control force to stabilize some ODE system and demonstrate some complex dynamic performance. The combination of the position and delayed position (PDP) is used to stabilize the inverted pendulum system, and it shows that there must exist a feedback of the form PDP such that the inverted equilibrium position is asymptotically stable for any positive value of the time-delay, see Atay ²⁷ and Zhao et al. ²⁸ In Hu, ²⁹ a delayed position feedback or a delayed velocity feedback, or both were used to stabilize the periodic vibration of a linear undamped oscillator. A systematic approach for stabilizing a class of linear undamped systems of multiple degrees of freedom with PDP feedbacks is presented in Liu and Hu. ³⁰ The nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied in Mao et al., ³¹ which shows that the delayed couplings play crucial roles in the network dynamics, for example, the enhancement and suppression of the stability, the patterns of the synchronization between networks, and the generation of complicated attractors and multi-stability coexistence. The sufficient and necessary conditions of delay-independent exponential stability for a delayed ring neuron network is presented in Zhao and Wang. ³² And also, several sufficient conditions are established for the stability of the fractional-order delayed neural networks by using the method of linear matrix inequalities. ³³ Furthermore, additional time-delay feedback compensation can also be used to stabilize the PDE system. In Krstic, ³⁴ by using the backstepping transformation and Lyapunov function, the exponential stability of a reaction-diffusion partial differential equation with an arbitrarily long delay at the input are established. For a string equation with time delays in the output feedback loop, it shows that the system is a Riesz spectral system and the spectrum-determined growth condition holds for all delays in Wang et al. ³⁵ And then, when the delay is equal to the even multiples of the wave propagation time, the necessary and sufficient stability conditions for the feedback gain and time delay are developed. It also shows that whenever the delay is an odd multiple of the wave propagation time, the closed-loop system is unstable. Moreover, by using the combination of current states and delay states, the consensus issue of the fractional-order multi-agent systems are studied in Wang et al. ³⁶ Recently, in Hugo et al., ³⁷ the exponential stability of the nonlinear Korteweg-de Vries equation on a finite star-shaped network in the presence of delayed internal feedback are established by using a Lyapunov function and imposing small initial data and a restriction over the lengths. A subcritical gas flow through star-shaped pipe networks is considered in Gugat et al., ³⁸ where the gas flow is modeled by the isothermal Euler equations. By applying the boundary feedback controls with time-varying delays and a novel Lyapunov function with delay terms, the exponential stability of the systems are guaranteed.

As we have known, the delay effect is not only inevitable, but also can improve the system performance and demonstrate some complex dynamic performance. However, there are not any research results in the control of hyperbolic systems with time-delay feedback. Can the delay feedback stabilize the hyperbolic system? Does the delay $\tau$ play an active influence on the stability? In this paper, we shall use the PDP boundary feedback to stabilize a hyperbolic PDE system which can describe the star-shaped network of open channels. Our contribution in this paper is to propose the PDP controller for a hyperbolic system, which means that the system is steered to a given desired equilibrium based on information about the present state and the delayed state. Comparing to the traditional PI controller, which needs the system state of a certain time period from 0 to $t$ , the PDP controller is only composed of the present state at time $t$ and the delayed state at time $t-\tau$ . Moreover, we also analyze the parameters conditions for delay-dependent and delay-independent stability. The organization of this paper is that we rewrite the single hyperbolic system with PDP feedback into an abstract evolution equation in Section 2. In Section 3 and Section 4, we prove the well-posedness and deduce an explicit sufficient dissipative condition for feedback parameters and time-delay, respectively. Section 5 is devoted to the delay-independent stability. It is shown that the system is stable for any positive value of the delay if and only if the feedback parameters satisfy some conditions. In Section 6, we extend the PDP control design to the star-shaped network of $n$ subsystems. Finally, we give some numerical simulations in Section 7.

The mathematical model of a single channel and the PDP control design

Saint-Venant equations and PDP feedback

In this part, we are concerned with a single pool of open-water channel firstly, which can be described by the famous Saint-Venant equations of conservation laws:

$${\partial }_t\left(\begin{array}{c}H\hfill \\ \multicolumn{1}{c}{V}\end{array}\right)+{\partial }_x\left(\begin{array}{c}\mathrm{HV}\hfill \\ \multicolumn{1}{c}{\frac{V^2}{2}+\mathrm{gH}}\end{array}\right)=0,$$ (1)

where $H(t,x)$ is water depth, and $V(t,x)$ is horizonal water velocity at time $t\in [0,+\infty )$ and location $x\in [0,l]$ along the channel, $l$ is the length of the channel, $g$ is the gravity constant. Here, ${\partial }_t,{\partial }_x$ stand for the time and space partial derivative respectively. By linearizing the Saint-Venant equation (1) around a set point $H=H^{*},V=V^{*},$ we can obtain the following equations:

$$\{\begin{array}{c}{\partial }_{t}h+V^{*}{\partial }_{x}h+H^{*}{\partial }_{x}v=0,\hfill \\ \multicolumn{1}{c}{{\partial }_{t}v+g{\partial }_{x}h+V^{*}{\partial }_{x}v=0,}\end{array}$$ (2)

where $h(t,x)=H(t,x)-H^{*},v(t,x)=V(t,x)-V^{*}.$ Then, by making a conversion of Riemann coordinates:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\xi }_1(t,x)=h(t,x)+v(t,x)\sqrt{\frac{H^{*}}{g}},\hfill \\ \multicolumn{1}{c}{{\xi }_2(t,x)=h(t,x)-v(t,x)\sqrt{\frac{H^{*}}{g}},}\end{array}}\end{array}$$ (3)

we can get the characteristic form as follows:

$$\{\begin{array}{c}{\partial }_t{\xi }_1(t,x)+{\lambda }_1{\partial }_x{\xi }_1(t,x)=0,\hfill \\ \multicolumn{1}{c}{{\partial }_t{\xi }_2(t,x)-{\lambda }_2{\partial }_x{\xi }_2(t,x)=0,}\end{array}$$ (4)

which can be regarded as a $2\times 2$ linear and strictly hyperbolic system under the subcritical condition $g{H}^{*}-(V^{*}{)}^2>0$ with the characteristic velocities

$${\lambda }_1=\sqrt{g{H}^{*}}+V^{*}>0,{\lambda }_2=\sqrt{g{H}^{*}}-V^{*}>0.$$ (5)

Now we consider the system (4) under the initial conditions

$${\xi }_1(0,x)={\xi }_1^0(x),{\xi }_2(0,x)={\xi }_2^0(x),$$ (6)

and assume the following boundary conditions:

$${\xi }_1(t,0)=R_1{\xi }_2(t,0)+U(t),{\xi }_2(t,l)=R_2{\xi }_1(t,l),$$ (7)

where $U(t)$ is control input and $R_1,R_2$ are real constant parameters. The main target of this paper is to propose the time-delay controller such that the closed-loop system is asymptotically stable. Specifically, we design the following PDP control law on the boundary with the real feedback parameters $K_p,K_d$ and the time-delay $\tau (>0)$

$$\begin{array}{c}U(t)=K_p(H(t,0)-H^{*})+K_d(H(t-\tau ,0)-H^{*})\hfill \\ =K_{p}h(t,0)+K_{d}h(t-\tau ,0)\hfill \\ =\frac{K_p}{2}({\xi }_1(t,0)+{\xi }_2(t,0))+\frac{K_d}{2}({\xi }_1(t-\tau ,0)+{\xi }_2(t-\tau ,0))\hfill \end{array}$$ (8)

to guarantee the exponential stability of the closed-loop system (4)−(8), and establish the range of feedback parameters $K_p,K_d$ and the time-delay $\tau$ .

• Remark 1 In the original coordinates, the boundary conditions (7) and (8) are expressed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}(1-R_1-K_p)(H(t,0)-H^{*})+(1+R_1)(V(t,0)-V^{*})\hfill \\ \multicolumn{1}{c}{\sqrt{\frac{H^{*}}{g}}=K_d(H(t-\tau ,0)-H^{*}),}\\ \multicolumn{1}{c}{(1-R_2)(H(t,L)-H^{*})=(1+R_2)(V(t,L)-V^{*})\sqrt{\frac{H^{*}}{g}}.}\end{array}}\end{array}$$

Reconstruction of the system (4)−(8) and abstract evolution equation

In view of the well-known change of state variable,

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}u(t,x)={\xi }_2(t-\tau \frac{x}{l},0),x\in (0,l),\hfill \\ \multicolumn{1}{c}{v(t,x)={\xi }_1(t-\tau \frac{x}{l},0),x\in (0,l),}\end{array}}\end{array}$$ (9)

the system (4)−(8) can be written as:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\partial }_t{\xi }_1(t,x)+{\lambda }_1{\partial }_x{\xi }_1(t,x)=0,(t,x)\in (0,\infty )\times (0,l),\hfill \\ \multicolumn{1}{c}{{\partial }_t{\xi }_2(t,x)-{\lambda }_2{\partial }_x{\xi }_2(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{\frac{\tau }{l}{\partial }_{t}u(t,x)+{\partial }_{x}u(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{\frac{\tau }{l}{\partial }_{t}v(t,x)+{\partial }_{x}v(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{{\xi }_1(t,0)=k_0{\xi }_2(t,0)+k_1(u(t,l)+v(t,l)),}\\ \multicolumn{1}{c}{{\xi }_2(t,l)=k_2{\xi }_1(t,l),}\\ \multicolumn{1}{c}{u(t,0)={\xi }_2(t,0),}\\ \multicolumn{1}{c}{v(t,0)={\xi }_1(t,0),}\end{array}}\end{array}$$ (10)

where

$$k_0=\frac{2R_1+K_p}{2-K_p},k_1=\frac{K_d}{2-K_p},k_2=R_2,$$ (11)

and $K_p\ne 2.$ Now we consider Hilbert space

$$\mathcal{H}=L^2(0,l)\times L^2(0,l)\times L^2(0,l)\times L^2(0,l),$$ (12)

equipped with the following inner product:

$$〈X_1,X_2〉={\int }_0^l[f_1\overline{f_2}+g_1\overline{g_2}]\mathrm{dx}+\frac{\tau }{l}{\int }_0^l{h}_1\overline{h_2}\mathrm{dx}+\frac{\tau }{l}{\int }_0^l{q}_1\overline{q_2}\mathrm{dx},$$ (13)

where $X_i=(f_i,g_i,h_i,q_i)\in \mathcal{H}(i=1,2)$ . Define a linear operator $\mathcal{A}:D(\mathcal{A})\subseteq \mathcal{H}\to \mathcal{H}$ by

$$\mathcal{A}X=\left(\begin{array}{cccc}-{\lambda }_1\frac{\partial }{\partial x}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\lambda }_2\frac{\partial }{\partial x}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -\frac{l}{\tau }\frac{\partial }{\partial x}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -\frac{l}{\tau }\frac{\partial }{\partial x}\hfill \end{array}\right)\left(\begin{array}{c}f\hfill \\ \multicolumn{1}{c}{g}\\ \multicolumn{1}{c}{h}\\ \multicolumn{1}{c}{q}\end{array}\right),$$ (14)

$$\begin{array}{c}D(\mathcal{A})=\{(f,g,h,q)\in (H^1(0,l){)}^4|f(0)=k_{0}g(0)+k_1(h(l)+q(l)),\hfill \\ g(l)=k_{2}f(l),h(0)=g(0),q(0)=f(0)\},\hfill \end{array}$$ (15)

then (10) can be written as an evolution equation in $\mathcal{H}$ :

$$\{\begin{array}{c}\stackrel{·}{X}(t)=\mathcal{A}X(t),t>0,\hfill \\ \multicolumn{1}{c}{X(0)=X_0.}\end{array}$$ (16)

where $X(t)=({\xi }_1(t,·),{\xi }_2(t,·),u(t,·),v(t,·))$ .

Well-posedness of the system (16)

Without loss of generality, we set $l=1$ for convenience of subsequent calculation.

• Theorem 1. Let operator $\mathcal{A}$ be given by (14) and (15). Then ${\mathcal{A}}^{-1}$ exists and is compact if the feedback parameters satisfy $(k_0+k_1)k_2+k_1\ne 1$ . Hence, $\sigma (\mathcal{A})$ , the spectrum of $\mathcal{A}$ , consists of isolated eigenvalues of finite algebraic multiplicity only.

Proof. For any $(f_1,g_1,h_1,q_1)\in \mathcal{H}$ , solve $\mathcal{A}(f,g,h,q)=(f_1,g_1,h_1,q_1)$ to obtain

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{A}\left(\begin{array}{c}f\hfill \\ \multicolumn{1}{c}{g}\\ \multicolumn{1}{c}{h}\\ \multicolumn{1}{c}{q}\end{array}\right)=\left(\begin{array}{c}-{\lambda }_{1}f\prime \hfill \\ \multicolumn{1}{c}{{\lambda }_{2}g\prime }\\ \multicolumn{1}{c}{-\frac{1}{\tau }h\prime }\\ \multicolumn{1}{c}{-\frac{1}{\tau }q\prime }\end{array}\right)=\left(\begin{array}{c}{f}_1\hfill \\ \multicolumn{1}{c}{g_1}\\ \multicolumn{1}{c}{h_1}\\ \multicolumn{1}{c}{q_1}\end{array}\right),}\end{array}$$ (17)

we can get

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}-{\lambda }_{1}f\prime =f_1,{\lambda }_{2}g\prime =g_1,\hfill \\ \multicolumn{1}{c}{-\frac{1}{\tau }h\prime =h_1,-\frac{1}{\tau }q\prime =q_1,}\\ \multicolumn{1}{c}{f(0)=k_{0}g(0)+k_1(h(1)+q(1)),}\\ \multicolumn{1}{c}{g(1)=k_{2}f(1),h(0)=g(0),q(0)=f(0).}\end{array}}\end{array}$$ (18)

By using the solution formula of the first-order linear differential equation, we can have the general solution of (18) as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}f(x)=f(0)-\frac{1}{{\lambda }_1}{\int }_0^x{f}_1(s)\mathrm{ds},g(x)=g(0)+\frac{1}{{\lambda }_2}{\int }_0^x{g}_1(s)\mathrm{ds},\hfill \\ \multicolumn{1}{c}{h(x)=h(0)-\tau {\int }_0^x{h}_1(s)\mathrm{ds},q(x)=q(0)-\tau {\int }_0^x{q}_1(s)\mathrm{ds},}\\ \multicolumn{1}{c}{f(0)=q(0)=\frac{1}{(k_0+k_1)k_2+k_1-1}}\\ \multicolumn{1}{c}{((k_0+k_1)(\frac{k_2}{{\lambda }_1}{\int }_0^1{f}_1(s)\mathrm{ds}+\frac{1}{{\lambda }_2}{\int }_0^1{g}_1(s)\mathrm{ds})+k_1\tau {\int }_0^1(h_1(s)+q_1(s))\mathrm{ds}),}\\ \multicolumn{1}{c}{h(0)=g(0)=k_{2}f(0)-\frac{k_2}{{\lambda }_1}{\int }_0^1{f}_1(s)\mathrm{ds}-\frac{1}{{\lambda }_2}{\int }_0^1{g}_1(s)\mathrm{ds}.}\end{array}}\end{array}$$ (19)

Now we get the unique solution $(f,g,h,q)\in D(\mathcal{A})$ of equation (17). Hence, ${\mathcal{A}}^{-1}$ exists and is compact in $\mathcal{H}$ by the Sobolev embedding theorem. Therefore, $\sigma (\mathcal{A})$ consists of the isolated eigenvalues of finite algebraic multiplicity only.

Exponential stability of the system (10)

In this section, we will establish the exponential stability of the system (10) and the range of feedback parameters $k_0,k_1,k_2$ and the time-delay $\tau$ .

• Theorem 2. Let us assume $k_i\ne 0(i=0,1,2).$ The closed-loop system (10) is exponentially stable if the system parameters satisfy

$$\begin{array}{c}{k}_0{k}_1>0,\frac{1+e^{-\tau }}{k_2^2(1+e^{\tau })}\ge \frac{{\lambda }_2}{{\lambda }_1},\frac{1+e^{-\tau }}{4k_1(k_0+k_1)}\ge {\lambda }_1+1,\hfill \\ k_0(k_0+k_1)\le \frac{{\lambda }_2-1}{{\lambda }_1+1}.\hfill \end{array}$$ (20)

Proof. Let us choose the following function as Lyapunov functional:

$$V(t)=V_1(t)+V_2(t),$$ (21)

where,

$$\begin{array}{c}{V}_1(t)={\int }_0^1({\xi }_1^2(t,x)+{\xi }_2^2(t,x))\mathrm{dx}+\tau {\int }_0^1{u}^2(t,x)\mathrm{dx}\hfill \\ +\tau {\int }_0^1{v}^2(t,x)\mathrm{dx},\hfill \end{array}$$ (22)

$$\begin{array}{c}\multicolumn{1}{c}{V_2(t)={\int }_0^1(e^{-\tau x}{\xi }_1^2(t,x)+e^{\tau x}{\xi }_2^2(t,x))\mathrm{dx}}\\ +\tau {\int }_0^1{e}^{-\tau x}(u^2(t,x)+v^2(t,x))\mathrm{dx}.\hfill \end{array}$$ (23)

Obviously, there exist positive constants $L_1$ and $L_2$ such that for any $t\ge 0$ , we have

$$L_1{V}_1(t)\le V(t)\le L_2{V}_1(t).$$ (24)

Differentiating (22) with respect to $t$ and using the boundary conditions in (10), we get

$$\begin{array}{c}{\stackrel{·}{V}}_1(t)=2{\int }_0^1[{\xi }_1({\partial }_t{\xi }_1)+{\xi }_2({\partial }_t{\xi }_2)]\mathrm{dx}+2\tau {\int }_0^{1}u({\partial }_{t}u)\mathrm{dx}+2\tau {\int }_0^{1}v({\partial }_{t}v)\mathrm{dx}\hfill \\ =-2{\lambda }_1{\int }_0^1{\xi }_1(t,x){\xi }_{1x}(t,x)\mathrm{dx}+2{\lambda }_2{\int }_0^1{\xi }_2(t,x){\xi }_{2x}(t,x)\mathrm{dx}\hfill \\ -2{\int }_0^{1}u(t,x)u_x(t,x)\mathrm{dx}-2{\int }_0^{1}v(t,x)v_x(t,x)\mathrm{dx}\hfill \\ =-{\lambda }_1{\xi }_1^2(t,1)+{\lambda }_1{\xi }_1^2(t,0)+{\lambda }_2{\xi }_2^2(t,1)-{\lambda }_2{\xi }_2^2(t,0)\hfill \\ -u^2(t,1)+u^2(t,0)-v^2(t,1)+v^2(t,0).\hfill \end{array}$$ (25)

Differentiating (23) with respect to $t$ , and using integration by parts and the boundary conditions in (10), we get

$$\begin{array}{c}{\stackrel{·}{V}}_2(t)=2{\int }_0^1[e^{-\tau x}{\xi }_1({\partial }_t{\xi }_1)+e^{\tau x}{\xi }_2({\partial }_t{\xi }_2)]\mathrm{dx}\hfill \\ +2\tau {\int }_0^1{e}^{-\tau x}[u({\partial }_{t}u)+v({\partial }_{t}v)]\mathrm{dx}\hfill \\ =-{\lambda }_1{\int }_0^1{e}^{-\tau x}d({\xi }_1^2(t,x))+{\lambda }_2{\int }_0^1{e}^{\tau x}d({\xi }_2^2(t,x))\hfill \\ -{\int }_0^1{e}^{-\tau x}d(u^2(t,x))-{\int }_0^1{e}^{-\tau x}d(v^2(t,x))\hfill \\ =-{\lambda }_1{e}^{-\tau }{\xi }_1^2(t,1)+{\lambda }_1{\xi }_1^2(t,0)+{\lambda }_2{e}^{\tau }{\xi }_2^2(t,1)-{\lambda }_2{\xi }_2^2(t,0)\hfill \\ -e^{-\tau }{u}^2(t,1)+u^2(t,0)-e^{-\tau }{v}^2(t,1)+v^2(t,0)\hfill \\ -\tau {\lambda }_1{\int }_0^1{e}^{-\tau x}{\xi }_1^2(t,x)\mathrm{dx}-\tau {\lambda }_2{\int }_0^1{e}^{\tau x}{\xi }_2^2(t,x)\mathrm{dx}\hfill \\ -\tau {\int }_0^1{e}^{-\tau x}{u}^2(t,x)\mathrm{dx}-\tau {\int }_0^1{e}^{-\tau x}{v}^2(t,x)\mathrm{dx}.\hfill \end{array}$$ (26)

Then,

$$\begin{array}{c}\stackrel{·}{V}={\stackrel{·}{V}}_1+{\stackrel{·}{V}}_2\le [-{\lambda }_1-{\lambda }_1{e}^{-\tau }+{\lambda }_2{k}_2^2(1+e^{\tau })]{\xi }_1^2(t,1)\hfill \\ +[2-2{\lambda }_2+k_0^2(2+2{\lambda }_1)+k_0{k}_1(2+2{\lambda }_1)]{\xi }_2^2(t,0)\hfill \\ +[2k_1^2(2+2{\lambda }_1)-1-e^{-\tau }+2k_0{k}_1(2+2{\lambda }_1)](u^2(t,1)+v^2(t,1))\hfill \\ -\tau {\lambda }_1{\int }_0^1{e}^{-\tau x}{\xi }_1^2(t,x)\mathrm{dx}-\tau {\lambda }_2{\int }_0^1{e}^{\tau x}{\xi }_2^2(t,x)\mathrm{dx}\hfill \\ -\tau {\int }_0^1{e}^{-\tau x}(u^2(t,x)+v^2(t,x))\mathrm{dx}.\hfill \end{array}$$ (27)

Since the system parameters satisfy the condition (20), so we have

$$\begin{array}{c}\stackrel{·}{V}(t)\le -\tau {\lambda }_1{\int }_0^1{e}^{-\tau x}{\xi }_1^2(t,x)\mathrm{dx}-\tau {\lambda }_2{\int }_0^1{e}^{\tau x}{\xi }_2^2(t,x)\mathrm{dx}\hfill \\ -\tau {\int }_0^1{e}^{-\tau x}(u^2(t,x)+v^2(t,x))\mathrm{dx}\hfill \\ \le -\tau {\lambda }_1{e}^{-\tau }{\int }_0^1{\xi }_1^2(t,x)\mathrm{dx}-\tau {\lambda }_2{\int }_0^1{\xi }_2^2(t,x)\mathrm{dx}\hfill \\ -\tau e^{-\tau }{\int }_0^1(u^2(t,x)+v^2(t,x))\mathrm{dx}\hfill \\ =-{\epsilon }_1{\int }_0^1{\xi }_1^2(t,x)\mathrm{dx}-{\epsilon }_2{\int }_0^1{\xi }_2^2(t,x)\mathrm{dx}\hfill \\ -{\epsilon }_3\tau {\int }_0^1({\xi }_1^2(t-\tau x,0)+{\xi }_2^2(t-\tau x,0))\mathrm{dx}\hfill \\ \le -\epsilon V_1(t),\hfill \end{array}$$ (28)

where,

$$\epsilon =min\{{\epsilon }_i,1\le i\le 3\},$$ (29)

and

$${\epsilon }_1=\tau {\lambda }_1{e}^{-\tau },{\epsilon }_2=\tau {\lambda }_2,{\epsilon }_3=e^{-\tau }.$$ (30)

Obviously, combining with the inequality (24), there exists a positive constant $\zeta =\frac{\epsilon }{L_2}>0$ such that $\stackrel{·}{V}(t)\le -\zeta V(t)$ , so the system (10) is exponentially stable.

• Remark 2. From the parameters’ sufficient conditions in Theorem 2, we find that with the increase of time delay $\tau$ from $\tau =0$ to $\tau >0$ , the range of system parameters becomes smaller. Even so, it is reasonable to take time delay into account in the control design. Furthermore, we prove that the system can always be stabilized regardless of the value of time delay. The delay-independent stability conditions are presented in Section 5.

$\tau -$ independent stability analysis

In this part, we will discuss how does the feedback time-delay $\tau$ in the controller $U(t)$ of (8) can affect the stability of the system (4)–(7).

Firstly, we consider the eigenvalue problem $\mathcal{A}X=\mu X$ of operator $\mathcal{A}$ , where $X=(f,g,h,q)\in D(\mathcal{A})$ , that is

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}-{\lambda }_{1}f\prime =\mu f,{\lambda }_{2}g\prime =\mu g,\hfill \\ \multicolumn{1}{c}{-\frac{1}{\tau }h\prime =\mu h,-\frac{1}{\tau }q\prime =\mu q,}\\ \multicolumn{1}{c}{f(0)=k_{0}g(0)+k_1(h(1)+q(1)),}\\ \multicolumn{1}{c}{g(1)=k_{2}f(1),h(0)=g(0),q(0)=f(0).}\end{array}}\end{array}$$ (31)

By the simple calculation, we can get that $f(x),g(x),h(x),q(x)$ have the following general solution

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}f(x)=c_1{e}^{-\frac{\mu }{{\lambda }_1}x},g(x)=c_2{e}^{\frac{\mu }{{\lambda }_2}x},\hfill \\ \multicolumn{1}{c}{h(x)=c_3{e}^{-\mu \tau x},q(x)=c_4{e}^{-\mu \tau x}.}\end{array}}\end{array}$$ (32)

Combined with boundary conditions in (31), we have:

$$\{\begin{array}{c}{c}_1=k_0{c}_2+k_1(c_3+c_4)e^{-\mu \tau },\\ c_2{e}^{\frac{\mu }{{\lambda }_2}}=k_2{c}_1{e}^{-\frac{\mu }{{\lambda }_1}},c_3=c_2,c_4=c_1.\end{array}$$ (33)

So we can get the characteristic equation of the closed-loop system (10):

$$\Delta (\mu )\stackrel{.}{=}{e}^{a\mu }-k_0{k}_2-k_1(k_2+e^{a\mu })e^{-\mu \tau }=0,$$ (34)

where $a=\frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}>0.$ Obviously, the equation (34) is equivalent to

$$1-k_0{k}_2{e}^{-a\mu }-k_1(k_2{e}^{-a\mu }+1)e^{-\mu \tau }=0.$$ (35)

Now we state a basic result on zeros of the following $n-$ order exponential polynomial:

$$\begin{array}{c}P(\lambda ,e^{-\lambda {\tau }_1},\cdots ,e^{-\lambda {\tau }_m})={\lambda }^n+p_1^{(0)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(0)}\lambda +p_n^{(0)}\hfill \\ +[p_1^{(1)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(1)}\lambda +p_n^{(1)}]e^{-\lambda {\tau }_1}+\cdots \hfill \\ +[p_1^{(m)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(m)}\lambda +p_n^{(m)}]e^{-\lambda {\tau }_m},\hfill \end{array}$$ (36)

where ${\tau }_i\ge 0(i=1,2,\cdots ,m)$ and $p_j^{(i)}(i=0,1,\cdots ,m;j=1,2,\cdots ,n)$ are constants.

• Lemma 1. As $({\tau }_1,{\tau }_2,\cdots ,{\tau }_m)$ vary, the sum of the orders of the zeros of $P(\lambda ,e^{-\lambda {\tau }_1},\cdots ,e^{-\lambda {\tau }_m})$ in the open right half plane can change only if a zero appears on or crosses the imaginary access ³⁹ .

• Remark 1. Obviously, Lemma 1 can be used to discuss the roots of (35) by regarding the delay $\tau$ as a parameter.

• Lemma 2. Assume $\tau =0.$ All the roots of (35) have negative real parts, if and only if the feedback parameters satisfy

$$0<\left|\frac{k_2(k_0+k_1)}{1-k_1}\right|<1.$$ (37)

Proof. Since $\tau =0,$ the equation (35) becomes

$$e^{a\mu }=\frac{k_2(k_0+k_1)}{1-k_1}.$$ (38)

• (i) If $0<\frac{k_2(k_0+k_1)}{1-k_1}<1$ , we have

$$\mu =\frac{\ln \frac{k_2(k_0+k_1)}{1-k_1}+2k\pi i}{a},$$ (39)

where $i^2=-1,k=\pm 1,\pm 2,\cdots .$

• (ii) If $-1<\frac{k_2(k_0+k_1)}{1-k_1}<0$ , we have

$$\mu =\frac{\left(\ln \frac{-k_2(k_0+k_1)}{1-k_1}\right)+(\pi +2k\pi )i}{a}.$$ (40)

Both (39) and (40) have negative real part.

• (iii) For all other cases, there must be a root with positive part or zero root for (38). So we have finished the proof.

• Theorem 3. Assume $0<\left|\frac{k_2(k_0+k_1)}{1-k_1}\right|<1$ . All the roots of (35) have negative real parts for any $\tau \in (0,+\infty )$ , if and only if the feedback parameters satisfy the condition:

$$\begin{array}{c}|-(4k_1^2{k}_2(1+k_2^2)-2k_2(1-k_0)(1-k_0{k}_2^2))\pm 2k_2(k_0+1)(1+k_0{k}_2^2)|\hfill \\ >8k_2^2|k_1^2+k_0|.\hfill \end{array}$$ (41)

Proof. Assume $\tau \ne 0$ . Let $\mu =i\omega$ is a pure imaginary root of (35), we have

$$1-k_0{k}_2{e}^{-\mathrm{ia}\omega }=k_1(k_2{e}^{-\mathrm{ia}\omega }+1)e^{-i\omega \tau },$$

that is,

$$e^{-i\omega \tau }=\frac{1-k_0{k}_2{e}^{-\mathrm{ia}\omega }}{k_1(k_2{e}^{-\mathrm{ia}\omega }+1)}.$$ (42)

Separating the real part and imaginary part, we can have

$$\{\begin{array}{c}\cos \omega \tau =\frac{k_2(1-k_0)\cos a\omega +1-k_0{k}_2^2}{k_1((1+k_2\cos a\omega {)}^2+k_2^2\stackrel{2}{\sin }a\omega )},\hfill \\ \multicolumn{1}{c}{-\sin \omega \tau =\frac{k_2(1+k_0)\sin a\omega }{k_1((1+k_2\cos a\omega {)}^2+k_2^2\stackrel{2}{\sin }a\omega )}.}\end{array}$$ (43)

By squaring both sides of (43), we have

$$\begin{array}{c}{k}_1^2((1+k_2\cos a\omega {)}^2+k_2^2\stackrel{2}{\sin }a\omega {)}^2=(k_2(1-k_0)\cos a\omega \hfill \\ +1-k_0{k}_2^2{)}^2+(k_2(1+k_0)\sin a\omega {)}^2,\hfill \end{array}$$ (44)

that is,

$$\begin{array}{l}{k}_1^2({(1+k_2^2)}^2+4k_2(1+k_2^2)\cos a\omega +4k_2^2{\cos }^{2}a\omega )\\ =k_2^2(1+k_0^2+2k_0)+{(1-k_0{k}_2^2)}^2-4k_0{k}_2^2{\cos }^{2}a\omega \\ +2k_2(1-k_0)(1-k_0{k}_2^2)\cos a\omega .\end{array}$$ (45)

Assume $\rho =\cos a\omega$ , the equation (45) can be transformed into a quadratic equation of $\rho$ as follows:

$$\begin{array}{c}4k_2^2(k_1^2+k_0){\rho }^2+(4k_1^2{k}_2(1+k_2^2)-2k_2(1-k_0)(1-k_0{k}_2^2))\rho \hfill \\ +k_1^2{(1+k_2^2)}^2-k_2^2{(1+k_0)}^2-{(1-k_0{k}_2^2)}^2=0.\hfill \end{array}$$

that is,

$$\begin{array}{c}4k_2^2(k_1^2+k_0){\rho }^2+(4k_1^2{k}_2(1+k_2^2)-2k_2(1-k_0)(1-k_0{k}_2^2))\rho \hfill \\ +k_1^2-1+k_2^2(2k_1^2-1-k_0^2)=0.\hfill \end{array}$$ (46)

By a simple calculation, we can get the discriminant of (46) is

$$\Delta =4k_2^2(k_0+1{)}^2(1+k_0{k}_2^2{)}^2\ge 0.$$ (47)

Hence, the equation (46) has two real roots ${\rho }_1$ and ${\rho }_2$ . If the module of ${\rho }_1$ and ${\rho }_2$ is bigger than 1, that is,

$$\begin{array}{c}\multicolumn{1}{c}{\frac{|-(4k_1^2{k}_2(1+k_2^2)-2k_2(1-k_0)(1-k_0{k}_2^2))\pm 2k_2(k_0+1)(1+k_0{k}_2^2)|}{8k_2^2|k_1^2+k_0|}>1.}\end{array}$$ (48)

So the equation (46) has no root, and then the characteristic equation (35) has no pure imaginary root. Otherwise, there will be infinitely many roots for $\omega$ , and then the stability will not be able to retain for any $\tau >0.$ Therefore, all the roots of (35) have negative real part for any $\tau \in (0,+\infty )$ if and only if the condition (41) holds.

PDP controller design for a star-shaped network of channels

In this section, we extend the PDP controller and control results for a single channel to a star-shaped network of $n$ channels. The connection between channels is depicted in Figure 1 with $n-1$ inlet channels and one outlet channel. Without loss of generality, the length of each channel is $l$ and each channel is modeled by two PDE hyperbolic equation (1). The following notations are used for the next: $i=1,2,\cdots ,n,$ and $j=1,2,\cdots ,n-1;$ $H_i(x,t)$ is water level in the $i$ th channel, $H_{i0}(t)=H_i(0,t)$ and $H_{\mathrm{il}}(t)=H_i(l,t)$ , $V_i(x,t)$ is water velocity in the $i$ th channel, $V_{i0}(t)=V_i(0,t)$ and $V_{\mathrm{il}}(t)=V_i(l,t)$ . The network is thus governed by $2n$ equations:

$${\partial }_t\left(\begin{array}{c}{H}_i\hfill \\ \multicolumn{1}{c}{V_i}\end{array}\right)+{\partial }_x\left(\begin{array}{c}{H}_i{V}_i\hfill \\ \multicolumn{1}{c}{\frac{V_i^2}{2}+g{H}_i}\end{array}\right)=0,i=1,2,\cdots ,n.$$ (49)

Figure 1.

Star-shaped networks of $n$ channels.

By applying the similar Riemann coordinates for the linearized model around the set-point $H_i^{*},V_i^{*},$ we can have the characteristic forms as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\partial }_t{\xi }_{i1}(t,x)+{\lambda }_{i1}{\partial }_x{\xi }_{i1}(t,x)=0,\hfill \\ \multicolumn{1}{c}{{\partial }_t{\xi }_{i2}(t,x)-{\lambda }_{i2}{\partial }_x{\xi }_{i2}(t,x)=0,}\end{array}}\end{array}$$ (50)

where

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\xi }_{i1}(t,x)=(H_i-H_i^{*})+(V_i-V_i^{*})\sqrt{\frac{H^{*}}{g}},\hfill \\ \multicolumn{1}{c}{{\xi }_{i2}(t,x)=(H_i-H_i^{*})-(V_i-V_i^{*})\sqrt{\frac{H^{*}}{g}},}\end{array}}\end{array}$$ (51)

and

$${\lambda }_{i1}=\sqrt{{\mathrm{gH}}_i^{*}}+V_i^{*}>0,{\lambda }_{i2}=\sqrt{{\mathrm{gH}}_i^{*}}-V_i^{*}>0,$$ (52)

under the subcritical condition ${\mathrm{gH}}_i^{*}>(V_i^{*}{)}^2,i=1,2,\cdots ,n.$

We consider the system (50) with the following boundary conditions:

$${\xi }_{j1}(t,0)=R_{j1}{\xi }_{j2}(t,0)+U_j(t),{\xi }_{i2}(t,l)=R_{i2}{\xi }_{i1}(t,l),$$ (53)

where $R_{j1}$ and $R_{i2}$ are arbitrary constants, the condition at the junction is:

$${\xi }_{n1}(t,0)=R_{n1}{\xi }_{n2}(t,0)+\sum _{j=1}^{n-1}{\alpha }_j{\xi }_{j1}(t,l),$$ (54)

where,

$$R_{n1}=\frac{{\lambda }_{n2}}{{\lambda }_{n1}},{\alpha }_j=\frac{{\lambda }_{j1}-R_{j2}{\lambda }_{j2}}{{\lambda }_{n1}}.$$ (55)

Furthermore, the PDP feedback controller $U_j(t)$ are given as follows:

$$\begin{array}{c}{U}_j(t)=K_{\mathrm{jp}}(H_j(t,0)-H_j^{*})+K_{\mathrm{jd}}(H_j(t-{\tau }_j,0)-H_j^{*})\hfill \\ =K_{\mathrm{jp}}{h}_j(t,0)+K_{\mathrm{jd}}{h}_j(t-{\tau }_j,0)\hfill \\ =\frac{K_{\mathrm{jp}}}{2}({\xi }_{j1}(t,0)+{\xi }_{j2}(t,0))+\frac{K_{\mathrm{jd}}}{2}({\xi }_{j1}(t-{\tau }_j,0)+{\xi }_{j2}(t-{\tau }_j,0)),\hfill \end{array}$$ (56)

where $K_{\mathrm{jp}}$ and $K_{\mathrm{jd}}(j=1,2,\cdots ,n-1)$ are real position and delayed position feedback parameters, respectively.

It is similar to the process used in Section 2.2, the closed-loop system (50)–(56) is governed by:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\partial }_t{\xi }_{i1}(t,x)+{\lambda }_{i1}{\partial }_x{\xi }_{i1}(t,x)=0,(t,x)\in (0,\infty )\times (0,l),\hfill \\ \multicolumn{1}{c}{{\partial }_t{\xi }_{i2}(t,x)-{\lambda }_{i2}{\partial }_x{\xi }_{i2}(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{\frac{{\tau }_j}{l}{\partial }_t{u}_j(t,x)+{\partial }_x{u}_j(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{\frac{{\tau }_j}{l}{\partial }_t{v}_j(t,x)+{\partial }_x{v}_j(t,x)=0,(t,x)\in (0,\infty )\times (0,l),}\\ \multicolumn{1}{c}{{\xi }_{j1}(t,0)=k_{j0}{\xi }_{j2}(t,0)+k_{j1}(u_j(t,l)+v_j(t,l)),}\\ \multicolumn{1}{c}{{\xi }_{n1}(t,0)=k_{n0}{\xi }_{n2}(t,0)+\sum _{j=1}^{n-1}{\alpha }_j{\xi }_{j1}(t,l),}\\ \multicolumn{1}{c}{{\xi }_{i2}(t,l)=k_{i2}{\xi }_{i1}(t,l),}\\ \multicolumn{1}{c}{u_j(t,0)={\xi }_{j2}(t,0),}\\ \multicolumn{1}{c}{v_j(t,0)={\xi }_{j1}(t,0),}\end{array}}\end{array}$$ (57)

where

$$k_{j0}=\frac{2R_{j1}+K_{\mathrm{jp}}}{2-K_{\mathrm{jp}}},k_{j1}=\frac{K_{\mathrm{jd}}}{2-K_{\mathrm{jp}}},k_{i2}=R_{i2},k_{n0}=R_{n1}.$$ (58)

Inspired from the proof of the Theorem 1, we can construct the following Lyapunov function for the closed-loop system (57):

$$V(t)=\sum _{j=1}^{n-1}{V}_j+V_n,$$ (59)

where

$$V_j=V_{j1}+V_{j2},$$ (60)

$$\begin{array}{c}\multicolumn{1}{c}{V_{j1}(t)={\int }_0^l({\xi }_{j1}^2(t,x)+{\xi }_{j2}^2(t,x))\mathrm{dx}+\frac{{\tau }_j}{l}{\int }_0^l{u}_j^2(t,x)\mathrm{dx}}\\ +\frac{{\tau }_j}{l}{\int }_0^l{v}_j^2(t,x)\mathrm{dx},\hfill \end{array}$$ (61)

$$\begin{array}{c}\multicolumn{1}{c}{V_{j2}(t)={\int }_0^l(e^{-{\tau }_{j}x}{\xi }_{j1}^2(t,x)+e^{{\tau }_{j}x}{\xi }_{j2}^2(t,x))\mathrm{dx}}\\ +\frac{{\tau }_j}{l}{\int }_0^l{e}^{-{\tau }_{j}x}(u_j^2(t,x)+v_j^2(t,x))\mathrm{dx},\hfill \end{array}$$ (62)

$$V_n(t)={\int }_0^l(q{\xi }_{n1}^2(t,x)e^{-\mu x}+{\xi }_{n2}^2(t,x)e^{\mu x})\mathrm{dx},$$ (63)

and $q,\mu$ are undetermined parameters.

Similar to Section 4, we have that there must exist some positive numbers ${\zeta }_j$ such that

$${\stackrel{·}{V}}_j(t)\le -{\zeta }_j{V}_j(t)+[-{\lambda }_{j1}-{\lambda }_{j1}{e}^{-{\tau }_{j}l}+{\lambda }_{j2}{k}_{j2}^2(1+e^{{\tau }_{j}l})]{\xi }_{j1}^2(t,l),$$ (64)

where $j=1,2,\cdots ,n-1$ .

Now we analyze the time derivative of $V_n$ along the solution of the system (57):

$$\begin{array}{l}{\stackrel{.}{V}}_n(t)={\int }_0^l(2q{\xi }_{n1}\frac{\partial {\xi }_{n1}}{\partial t}{e}^{-\mu x}+2{\xi }_{n2}\frac{\partial {\xi }_{n2}}{\partial t}{e}^{\mu x})dx\hfill \\ ={\int }_0^l(2q{\xi }_{n1}(-{\lambda }_{n1})\frac{\partial {\xi }_{n1}}{\partial x}{e}^{-\mu x}+2{\xi }_{n2}{\lambda }_{n2}\frac{\partial {\xi }_{n2}}{\partial x}{e}^{\mu x})dx\hfill \\ ={\int }_0^l(q(-{\lambda }_{n1})e^{-\mu x})d{\xi }_{n1}^2+{\int }_0^l({\lambda }_{n2}{e}^{\mu x})d{\xi }_{n2}^2\hfill \\ =q(-{\lambda }_{n1})[e^{-\mu x}{\xi }_{n1}^2{|}_0^l+\mu {\int }_0^l{\xi }_{n1}^2{e}^{-\mu x}dx]+{\lambda }_{n2}[e^{\mu x}{\xi }_{n2}^2{|}_0^l-\mu {\int }_0^l{\xi }_{n2}^2{e}^{\mu x}dx]\hfill \\ ={\xi }_{n1}^2(t,l)(-{\lambda }_{n1}q{e}^{-\mu l}+{\lambda }_{n2}{e}^{\mu l}{k}_{n2}^2)+{\lambda }_{n1}q{(k_{n0}{\xi }_{n2}(t,0)+\sum _{j=1}^{n-1}{\alpha }_j{\xi }_{j1}(t,l))}^2\hfill \\ -{\lambda }_{n2}{\xi }_{n2}^2(t,0)-{\lambda }_{n1}q\mu {\int }_0^l{\xi }_{n1}^2{e}^{-\mu x}dx-{\lambda }_{n2}\mu {\int }_0^l{\xi }_{n2}^2{e}^{\mu x}dx\hfill \\ \le {\xi }_{n1}^2(t,l)(-{\lambda }_{n1}q{e}^{-\mu l}+{\lambda }_{n2}{e}^{\mu l}{k}_{n2}^2)+{\lambda }_{n1}q{(k_{n0}{\xi }_{n2}(t,0)+\sum _{j=1}^{n-1}{\alpha }_j{\xi }_{j1}(t,l))}^2\hfill \\ -{\lambda }_{n2}{\xi }_{n2}^2(t,0)-\mu \min \{{\lambda }_{n1},{\lambda }_{n2}\}{V}_n(t)\hfill \\ \le -\mu \min \{{\lambda }_{n1},{\lambda }_{n2}\}{V}_n(t)+{\xi }_{n1}^2(t,l)(-{\lambda }_{n1}q{e}^{-\mu l}+{\lambda }_{n2}{e}^{\mu l}{k}_{n2}^2)-{\lambda }_{n2}{\xi }_{n2}^2(t,0)\hfill \\ +2q{\lambda }_{n1}(k_{n0}^2{\xi }_{n2}^2(t,0)+(n-1)\sum _{j=1}^{n-1}{\alpha }_j^2{\xi }_{j1}{(t,l)}^2)\hfill \\ =-\mu \min \{{\lambda }_{n1},{\lambda }_{n2}\}{V}_n(t)+{\xi }_{n1}^2(t,l)(-{\lambda }_{n1}q{e}^{-\mu l}+{\lambda }_{n2}{e}^{\mu l}{k}_{n2}^2)\hfill \\ +(2q{\lambda }_{n1}{k}_{n0}^2-{\lambda }_{n2}){\xi }_{n2}^2(t,0)+2q{\lambda }_{n1}(n-1)\sum _{j=1}^{n-1}{\alpha }_j^2{\xi }_{j1}^2(t,l).\hfill \end{array}$$ (65)

Combining with (64) and (65), we have the following results.

• Theorem 4. Let us assume $k_{i0}\ne 0,k_{i2}\ne 0,k_{j1}\ne 0(i=1,2,\cdots ,n,j=1,2,\cdots ,n-1).$ The closed-loop system (57) is exponentially stable

$$\begin{array}{c}\stackrel{·}{V}(t)\le -\delta V(t),\hfill \end{array}$$ (66)

if the system parameters satisfy

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\lambda }_{j2}{k}_{j2}^2(1+e^{{\tau }_{j}l})+2q{\lambda }_{n1}(n-1){\alpha }_j^2\le {\lambda }_{j1}(1+e^{-{\tau }_{j}l}),\hfill \\ \multicolumn{1}{c}{1-{\lambda }_{j2}+k_{j0}(k_{j0}+k_{j1})(1+{\lambda }_{j1})\le 0,}\\ \multicolumn{1}{c}{4k_{j1}(k_{j0}+k_{j1})(1+{\lambda }_{j1})\le 1+e^{-{\tau }_{j}l},}\\ \multicolumn{1}{c}{k_{j0}{k}_{j1}>0,}\end{array}}\end{array}$$ (67)

and $q,\mu$ satisfy

$$\frac{{\lambda }_{n2}{k}_{n2}^2{e}^{2\mu l}}{{\lambda }_{n1}}\le q\le \frac{{\lambda }_{n2}}{2{\lambda }_{n1}{k}_{n0}^2},$$ (68)

where

$$\delta =min\{{\zeta }_j,\mu min\{{\lambda }_{n1},{\lambda }_{n2}\}\}.$$ (69)

Numerical simulations

In this section, we make some numerical simulations for a star-shaped network with $n(=3)$ channels and the following data:

• – The length of each channel $l=10m$ ;

• – The steady-state water depth and velocity are $H^{*}=2m,V^{*}=0.5m/\sec ;$

• – The characteristic velocities ${\lambda }_{i1}=5m/\sec ,{\lambda }_{i2}=4m/\sec ,i=1,2,3;$

• – The time-delay ${\tau }_j=0.1,j=1,2.$

Then by the conditions (67) and (68) in Theorem 4, the stability is guaranteed if the parameters are chosen as

$$\begin{array}{l}\mu =0.1,q=0.4,k_{30}=0.8,k_{32}=0.2,{\alpha }_j=0.35,\\ k_{j2}=0.625,k_{j1}=k_{j0}=0.1(j=1,2).\end{array}$$ (70)

Figure 2 shows the stability of the outlet channel in system (57) with the initial conditions ${\xi }_{31}(0,x)=\cos (0.2\pi x)-\frac{1}{2},{\xi }_{32}(0,x)=\sin (0.2\pi x)+\frac{1}{2},$ and Figure 3 shows the stability of the PDP controller $U_j(t)$ in (56) with $K_{\mathrm{jp}}=-1.27,K_{\mathrm{jd}}=0.327(j=1,2)$ .

Figure 2.

The convergence of the state ${\xi }_{31}(t,x)$ and ${\xi }_{32}(t,x)$ in system.

Figure 3.

The controller $U(t)$ .

Conclusions

In this paper, we construct the position and delayed position (PDP) feedback controller for a single channel system and the star-shaped network of open channels. Then we prove the well-posedness of the closed-loop system by using the semi-group theory, and give the explicit sufficient dissipative conditions for the system parameters and the time-delay value. Next, the delay-independent stability of the closed-loop system is presented by the method of characteristic. Finally, we give some numerical simulations for the special case of three channels.