## The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances

Gao Caixia,, Zhao Dongxia,*

School of Mathematics, North University of China, Taiyuan 030051

 基金资助: 山西省基础研究计划(20210302123046)国家自然科学基金青年基金(12001343)

 Fund supported: Fundamental Research Program of Shanxi Province(20210302123046)National Natural Science Foundation of China(12001343)

Abstract

For the linearized ARZ traffic flow model, the existing literatures are usually based on the following assumptions: First, the equilibrium state of the system is exactly equal to the speed of free flow; Second, the traffic flow entering the upstream section is exactly equal to the mathematical expectation of traffic demand; Third, the boundary feedback doesn't consider the impact of time delay factors. In this paper, a PDE-PDE infinite-dimensional coupled closed-loop system with model drift term and boundary disturbance term is established without these constraints by combining the time-delay boundary control strategy. Specifically, the closed-loop system is transformed into an abstract evolution equation by using operator semigroup theory. The well-posedness of the closed-loop system is proved by combining the admissible theory of linear system solutions and control operators. The weighting ISS-Lyapunov function is constructed, and the input-to-state stability(ISS) of the closed-loop system is proved. The dissipative conditions of the feedback parameters are obtained. The effectiveness of the proposed controller and the feasibility of the parameter conditions are further verified by numerical simulation experiments.

Keywords： ARZ traffic flow model; Time-delay boundary control; ISS-Lyapunov function; Input-to-state stability

Gao Caixia, Zhao Dongxia. The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances[J]. Acta Mathematica Scientia, 2024, 44(4): 960-977

## 1 引言

ARZ非平衡交通流模型能够较为准确地刻画城市快速路网车辆的运行, 在黎曼坐标变换和线性化处理后呈现为二阶双曲偏微分方程组的形式[1-3]. 近年来, 入口匝道控制和可变限速控制成为城市快速路网车辆交通流控制的两种有效方法[4], 将其应用于 AR/ARZ 非平衡交通流模型, 取得了一系列研究成果[5,6]. 对于线性/拟线性双曲系统柯西问题解的适定性, 文献 [7,定理 A.1,B.1] 采用 Lumer-Philipps 定理、不动点定理和能量估计等数学方法进行了证明. 在稳定性分析方面, 文献 [8] 研究了一维 $n\times n$ 线性双曲平衡律系统在比例-积分 (PI) 边界反馈控制下的稳定性, 结合 Lyapunov 函数和线性矩阵不等式 (LMI) 方法, 建立了系统指数稳定所满足的矩阵不等式条件. 进而, 将所得结论应用于 ARZ 交通流模型在入口匝道和可变限速反馈控制下的镇定问题中, 证明了反馈参数满足某种耗散条件时, 交通流状态以指数速率收敛到稳态. 为了处理由于不确定的交通需求导致进入上游路段的交通流量的振荡, 文献 [9] 考虑了闭环系统的输入-状态稳定性 (ISS). 文献 [10,11] 采用 Backstepping 方法设计完全状态反馈以抑制交通状况中的走走停停现象.

### 2.1 模型建立

$$$\begin{cases}\partial_t \rho+\partial_x (\rho v)=0,\\\partial_t v+(v-\rho p'(\rho))\partial_x v=\dfrac{V(\rho)-v}{\tau_0},\end{cases}$$$

$$$p(\rho)=v_f-V(\rho),$$$
$$$V(\rho)=v_f(1-(\dfrac{\rho}{\rho_m})^\gamma),$$$

$$$\begin{cases}z=v,\\w=v+v_f(\dfrac{\rho}{\rho_m})^\gamma.\end{cases}$$$

$$$\begin{cases}\partial_t w+\lambda_1\partial_x w=\dfrac{v_f-w}{\tau_0},\\[3mm]\partial_t z+\lambda_2\partial_x z=\dfrac{v_f-w}{\tau_0},\end{cases}$$$

$$$\lambda_1=z>0,\;\lambda_2=(1+\gamma)z-\gamma w,$$$

$\lambda_2$ 的正负分别代表交通流处于自由模态和拥堵模态[15].

$$$\tilde{p}_{in}(t)+k_p^r\tilde{\rho}(L,t)+k_d^r\tilde{\rho}(L,t-\tau)=\tilde{\rho}(0,t)v^*I+\rho^*Ik_p^v\tilde{v}(L,t)+\rho^*Ik_d^v\tilde{v}(L,t-\tau).$$$

\begin{align*} \tilde{w}(0,t)=\ &\tilde{v}(0,t)+\alpha\tilde{\rho}(0,t)\\=\ &k_p^v\tilde{v}(L,t)+k_d^v\tilde{v}(L,t-\tau)+\dfrac{\alpha k_p^r}{v^*I}\tilde{\rho}(L,t)+\dfrac{\alpha k_d^r}{v^*I}\tilde{\rho}(L,t-\tau)\\&-\dfrac{\alpha \rho^*k_p^v}{v^*}\tilde{v}(L,t)-\dfrac{\alpha \rho^*k_d^v}{v^*}\tilde{v}(L,t-\tau)+\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I}\\=\ &(k_p^v-\dfrac{\alpha \rho^*k_p^v}{v^*}-\dfrac{k_p^r}{v^*I})\tilde{z}(L,t)+(k_d^v-\dfrac{\alpha \rho^*k_d^v}{v^*}-\dfrac{ k_d^r}{v^*I})\tilde{z}(L,t-\tau)\\&+\dfrac{k_p^r}{v^*I}\tilde{w}(L,t)+\dfrac{ k_d^r}{v^*I}\tilde{w}(L,t-\tau)+\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I}.\end{align*}

$$$\tilde{z}(0,t)=k_p^v\tilde{z}(L,t)+k_d^v\tilde{z}(L,t-\tau).$$$

$$$\begin{cases}\tilde{w}(0,t)=k_1\tilde{w}(L,t)+k_2\tilde{z}(L,t)+k_3\tilde{w}(L,t-\tau)+k_4\tilde{z}(L,t-\tau)+\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I},\\\tilde{z}(0,t)=k_5\tilde{z}(L,t)+k_6\tilde{z}(L,t-\tau),\end{cases}$$$

\begin{aligned}&k_1=\dfrac{k_p^r}{v^*I},\;k_2=k_p^v-\dfrac{\alpha \rho^*k_p^v}{v^*}-\dfrac{k_p^r}{v^*I},\;k_3=\dfrac{k_d^r}{v^*I},\\&k_4=k_d^v-\dfrac{\alpha \rho^*k_d^v}{v^*}-\dfrac{k_d^r}{v^*I},\;k_5=k_p^v,\;k_6=k_d^v.\end{aligned}

### 2.3 PDE-PDE 闭环系统的重建

$$$\begin{cases}\partial_t u(x,t)+\dfrac{L}{\tau}\partial_x u(x,t)=0,\\[3mm]u(x,0)=\tilde{w}(L,-\dfrac{\tau}{L}x)\end{cases}$$$

$$$u(x,t)=\tilde{w}(L,t-\dfrac{\tau}{L}x).$$$

$$$\begin{cases} \partial_t g(x,t)+\dfrac{L}{\tau}\partial_x g(x,t)=0,\\[3mm] g(x,0)=\tilde{z}(L,-\dfrac{\tau}{L}x) \end{cases}$$$

$$$g(x,t)=\tilde{z}(L,t-\dfrac{\tau}{L}x).$$$

$$$\begin{cases}\partial_t \tilde{w}(x,t)+\lambda_1^*\partial_x \tilde{w}(x,t)=-c\tilde{w}(x,t)+c(v_f-w^*),\\\partial_t \tilde{z}(x,t)+\lambda_2^*\partial_x \tilde{z}(x,t)=-c\tilde{w}(x,t)+c(v_f-w^*),\\[2mm]\partial_t u(x,t)+\dfrac{L}{\tau}\partial_x u(x,t)=0,\\[3mm]\partial_t g(x,t)+\dfrac{L}{\tau}\partial_x g(x,t)=0,\\[2mm]\tilde{w}(0,t)=k_1\tilde{w}(L,t)+k_2\tilde{z}(L,t)+k_3u(L,t)+k_4g(L,t)+\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I},\\\tilde{z}(0,t)=k_5\tilde{z}(L,t)+k_6g(L,t),\\u(0,t)=\tilde{w}(L,t),\\g(0,t)=\tilde{z}(L,t).\end{cases}$$$

### 3 系统 (2.26) 的适定性

$$$\mathcal{H}=\big(L^2(0,L)\big)^4,$$$

$$$\langle X_1,X_2\rangle=\int_0^L\big[m_1(x)\overline{m_2(x)}+n_1(x)\overline{n_2(x)}\big]{\rm d}x+\tau\int_0^L\big[r_1(x)\overline{r_2(x)} +s_1(x)\overline{s_2(x)}\big]{\rm d}x,$$$

$$$\begin{cases}\dot{X}(t)=\mathcal{A}X(t)+\mathcal{B}_1(v_f-w^*)+\mathcal{B}_2\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I},\\X(0)=X_0,\\\end{cases}$$$

$$$\mathcal{B}_1=\begin{pmatrix} c \\ c \\ 0 \\ 0\end{pmatrix},\quad \mathcal{B}_2=\begin{pmatrix} \delta(x) \\ 0 \\ 0 \\ 0\end{pmatrix},$$$

$\delta(\cdot)$ 为狄拉克函数.

### 3.2 算子 $\mathcal{A}$ 的性质

$$$\left |\begin{array}{ccccc}1-k_1{\rm e}^\frac{-c}{\lambda_1^*}-k_2\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big) & -k_2 & -k_3 & -k_4\\[3mm]-k_5\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^\frac{-c}{\lambda_1^*}-1\big) & 1-k_5 & 0 & -k_6\\[2mm]{\rm e}^\frac{-c}{\lambda_1^*} & 0 & -1 & 0\\[1mm]-\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big) & -1 & 0 & 1\end{array}\right|\neq0.$$$

$\mathcal{A}X=X_1,\;X=(m,n,r,s)\in D(\mathcal{A}),$

$$$\mathcal{A}\left( \begin{array}{cccc} m \\ n \\ r \\ s \\ \end{array}\right)=\left( \begin{array}{cccc} -\lambda_1^*m'-cm \\ -\lambda_2^*n'-cm \\[2mm] -\frac{1}{\tau}r' \\[3mm] -\frac{1}{\tau}s' \\ \end{array}\right)=\left( \begin{array}{c} m_1 \\ n_1 \\ r_1 \\ s_1 \\ \end{array}\right).$$$

$$$\begin{cases} -\lambda_1^*m'-cm-m_1=0,\\ -\lambda_2^*n'-cm-n_1=0,\\[2mm] -\frac{1}{\tau}r'=r_1, \;-\frac{1}{\tau}s'=s_1,\\[2mm] m(0)=k_1m(1)+k_2n(1)+k_3r(1)+k_4s(1),\\ n(0)=k_5n(1)+k_6s(1),\; r(0)=m(1),\; s(0)=n(1).\end{cases}$$$

\begin{align*}&m(x)={\rm e}^{\frac{-c}{\lambda_1^*}x}\bigg[\int_{0}^{x} \frac{m_1(\sigma)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}\sigma}{\rm d}\sigma+m(0)\bigg],\end{align*}
\begin{align*}n(x)=n(0)-\displaystyle\int_{0}^{x}\frac{1}{\lambda_2^*}(n_1(\sigma)+cm(\sigma)){\rm d}\sigma,\end{align*}
\begin{align*}r(x)=r(0)-\tau\displaystyle\int_{0}^{x} r_1(\sigma){\rm d}\sigma,\end{align*}
\begin{align*}s(x)=s(0)-\tau\displaystyle\int_{0}^{x} s_1(\sigma){\rm d}\sigma.\end{align*}

\begin{align*} n(x)&=n(0)-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x}n_1(\sigma){\rm d}\sigma-\dfrac{c}{\lambda_2^*}\displaystyle\int_{0}^{x}\bigg[{\rm e}^{\frac{-c}{\lambda_1^*}\sigma}\bigg(\displaystyle\int_{0}^{\sigma}\frac{m_1(t)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}t}{\rm d}t+m(0)\bigg)\bigg]{\rm d}\sigma\\[4mm]&=n(0)-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x}n_1(\sigma){\rm d}\sigma-\dfrac{c}{\lambda_2^*}\displaystyle\int_{0}^{x}{\rm e}^{\frac{-c}{\lambda_1^*}\sigma} \displaystyle\int_{0}^{\sigma}\dfrac{m_1(t)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}t}{\rm d}t{\rm d}\sigma-\dfrac{c}{\lambda_2^*}\displaystyle\int_{0}^{x}{\rm e}^{\frac{-c}{\lambda_1^*}\sigma} m(0){\rm d}\sigma\\[4mm]&=n(0)-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x} n_1(\sigma){\rm d}\sigma-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x}m_1(t)\big({\rm e}^{\frac{c}{\lambda_1^*}(t-x)}-1\big){\rm d}t+\frac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}x}-1\big)m(0).\end{align*}

$$$\begin{cases}\big(1-k_1{\rm e}^\frac{-c}{\lambda_1^*}-k_2\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big)\big)m(0)-k_2n(0)-k_3r(0)-k_4s(0)=k_1{\rm e}^\frac{-c}{\lambda_1^*}\displaystyle\int_0^1\frac{m_1(\sigma)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}\sigma}{\rm d}\sigma\\[3mm]-\dfrac{k_2}{\lambda_2^*}\displaystyle\int_0^1n_1(\sigma){\rm d}\sigma-\dfrac{k_2}{\lambda_2^*}\displaystyle\int_0^1m_1(t)\big({\rm e}^{\frac{c}{\lambda_1^*}(t-1)}-1\big){\rm d}t-k_3\tau\displaystyle\int_{0}^{1}r_1(\sigma){\rm d}\sigma-k_4\tau\displaystyle\int_{0}^{1}s_1(\sigma){\rm d}\sigma,\\[3mm]-k_5\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^\frac{-c}{\lambda_1^*}-1\big)m(0)+\big(1-k_5\big)n(0)-k_6s(0)=-\dfrac{k_5}{\lambda_2^*}\displaystyle\int_{0}^{1}n_1(\sigma){\rm d}\sigma-k_6\tau\displaystyle\int_{0}^{1}s_1(\sigma){\rm d}\sigma\\[3mm]-\dfrac{k_5}{\lambda_2^*}\displaystyle\int_{0}^{1}m_1(t)\big({\rm e}^{\frac{c}{\lambda_1^*}(t-1)}-1\big){\rm d}t,\\[3mm]{\rm e}^{\frac{-c}{\lambda_1^*}}m(0)-r(0)=-{\rm e}^{\frac{-c}{\lambda_1^*}}\displaystyle\int_{0}^{1}\frac{m_1(\sigma)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}\sigma}{\rm d}\sigma,\\[3mm]-\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big)m(0)-n(0)+s(0)=-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{1}n_1(\sigma){\rm d}\sigma-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{1}m_1(t)\big({\rm e}^{\frac{c}{\lambda_1^*}(t-1)}-1\big){\rm d}t.\end{cases}$$$

$$$D=\left |\begin{array}{cccc}1-k_1{\rm e}^\frac{-c}{\lambda_1^*}-k_2\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big) & -k_2 & -k_3 & -k_4\\[3mm]-k_5\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^\frac{-c}{\lambda_1^*}-1\big) & 1-k_5 & 0 & -k_6\\[3mm]{\rm e}^\frac{-c}{\lambda_1^*} & 0 & -1 & 0\\[1mm]-\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}}-1\big) & -1 & 0 & 1\end{array}\right|\neq0$$$

$$$m(0)=\frac{D_1}{D},\;n(0)=\frac{D_2}{D},\;r(0)=\frac{D_3}{D},\\s(0)=\frac{D_4}{D},$$$

$$$\begin{cases}m(x)={\rm e}^{\frac{-c}{\lambda_1^*}x}\bigg[\displaystyle\int_{0}^{x} \frac{m_1(\sigma)}{-\lambda_1^*}{\rm e}^{\frac{c}{\lambda_1^*}\sigma}{\rm d}\sigma+\dfrac{D_1}{D}\bigg],\\[3mm]n(x)=\dfrac{D_2}{D}-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x} n_1(\sigma){\rm d}\sigma-\dfrac{1}{\lambda_2^*}\displaystyle\int_{0}^{x}m_1(t)\big({\rm e}^{\frac{c}{\lambda_1^*}(t-x)}-1\big){\rm d}t+\dfrac{\lambda_1^*}{\lambda_2^*}\big({\rm e}^{\frac{-c}{\lambda_1^*}x}-1\big)\dfrac{D_1}{D},\\[3mm]r(x)=\dfrac{D_3}{D}-\tau\displaystyle\int_{0}^{x} r_1(\sigma){\rm d}\sigma,\\[3mm]s(x)=\dfrac{D_4}{D}-\tau\displaystyle\int_{0}^{x} s_1(\sigma){\rm d}\sigma.\end{cases}$$$

$\begin{equation*}\mathcal{A}X=\lambda X,\;\;\forall\; X=(m,n,r,s)\in D(\mathcal{A}),\end{equation*}$

$\begin{equation*}\mathcal{A}\left( \begin{array}{cccc} m \\ n \\ r \\ s \\ \end{array}\right)=\left( \begin{array}{cccc} -\lambda_1^*m'-cm \\ -\lambda_2^*n'-cm \\[2mm] -\frac{1}{\tau}r' \\[3mm] -\frac{1}{\tau}s' \\ \end{array}\right)=\left( \begin{array}{c} \lambda m \\ \lambda n \\ \lambda r \\ \lambda s \\ \end{array}\right).\end{equation*}$

$$$\Delta(\lambda)\doteq\left |\begin{array}{cccc}q_1 & -k_2{\rm e}^{-\frac{\lambda}{\lambda_2^*}} & -k_3{\rm e}^{-\tau\lambda} & -k_4{\rm e}^{-\tau\lambda}\\q_2 & k_5{\rm e}^{-\frac{\lambda}{\lambda_2^*}}-1 & 0 & k_6{\rm e}^{-\tau\lambda}\\q_3 & 0 & -1 & 0\\q_4 & {\rm e}^{-\frac{\lambda}{\lambda_2^*}} & 0 & -1\end{array}\right|=0$$$

$\begin{equation*}\begin{cases}q_1=1-k_1{\rm e}^{-\frac{\lambda+c}{\lambda_1^*}}+\dfrac{k_2c\lambda_1^*}{\lambda\lambda_1^*-(\lambda+c)\lambda_2^*}\big({\rm e}^{-\frac{\lambda+c}{\lambda_1^*}}-{\rm e}^{-\frac{\lambda}{\lambda_2^*}}\big),\\[3mm]q_2=-\dfrac{k_5c\lambda_1^*}{\lambda\lambda_1^*-(\lambda+c)\lambda_2^*}\big({\rm e}^{-\frac{\lambda+c}{\lambda_1^*}}-{\rm e}^{-\frac{\lambda}{\lambda_2^*}}\big),\;\;q_3={\rm e}^{-\frac{\lambda+c}{\lambda_1^*}},\\[3mm]q_4=\dfrac{-c\lambda_1^*}{\lambda\lambda_1^*-(\lambda+c)\lambda_2^*}\big({\rm e}^{-\frac{\lambda+c}{\lambda_1^*}}-{\rm e}^{-\frac{\lambda}{\lambda_2^*}}\big).\end{cases}\end{equation*}$

$\begin{equation*}\sigma(\mathcal{A})=\sigma_p(\mathcal{A})=\{\lambda\in C|\triangle(\lambda)=0\},\end{equation*}$

\begin{equation*}\begin{aligned}&m_\lambda(x)={\rm e}^{-\frac{\lambda+c}{\lambda_1^*}x},\;\;r_\lambda(x)=s_\lambda(x)={\rm e}^{-\tau\lambda x},\\&n_\lambda(x)={\rm e}^{-\frac{\lambda}{\lambda_2^*}x}-\dfrac{c\lambda_1^*}{\lambda\lambda_1^*-(\lambda+c)\lambda_2^*}{\rm e}^{-\frac{\lambda}{\lambda_2^*}x}\big({\rm e}^{\frac{\lambda\lambda_1^*-(\lambda+c)\lambda_2^*}{\lambda_1^*\lambda_2^*}x}-1\big).\end{aligned}\end{equation*}

### 3.3 算子 $\mathcal{A}$ 生成 $\mathcal{H}$ 上的 $C_0$ 压缩半群

\begin{align*} \langle X_1,X_2\rangle_1=&\int_0^1\big[\dfrac{p_1}{\lambda_1^*}{\rm e}^{-\frac{\mu x}{\lambda_1^*}}m_1(x)\overline{m_2(x)}+\dfrac{p_2}{\lambda_2^*}{\rm e}^{-\frac{\mu x}{\lambda_2^*}}n_1(x)\overline{n_2(x)}\big]{\rm d}x\\ &+\tau\int_0^1\big[{\rm e}^{-\tau x}p_3r_1(x)\overline{r_2(x)}+{\rm e}^{-\tau x}p_4s_1(x)\overline{s_2(x)}\big]{\rm d}x, \end{align*}

$$$\begin{cases}4p_1k_1^2-p_1{\rm e}^{\frac{-\mu}{\lambda_1^*}}+p_3\leq0,\\4p_1k_2^2-p_2{\rm e}^{\frac{-\mu}{\lambda_2^*}}+2p_2k_5^2+p_4\leq0,\\4p_1k_3^2-{\rm e}^{-\tau}p_3\leq0,\\4p_1k_4^2-{\rm e}^{-\tau}p_4+2p_2k_6^2\leq0,\\\mu(\mu+2c) p_1 \lambda_2^* {\rm e}^{-\mu(\frac{1}{\lambda_1^*}+\frac{1}{\lambda_2^*})}>c^2 p_2 \lambda_1^*,\end{cases}$$$

$$$\langle \mathcal{A}X,X\rangle_1\leq 0,\;\forall X \in D(\mathcal{A}).$$$

$$$\begin{array}{ll} {\rm (a)} \Big(\frac{\mu p_1}{2\lambda_1^*}+\frac{p_1 c}{\lambda_1^*}\Big){\rm e}^{-\frac{\mu}{\lambda_1^*}}>0,\\[3mm] {\rm (b)} \Big(\frac{\mu p_1}{2\lambda_1^*}+\frac{p_1 c}{\lambda_1^*}\Big){\rm e}^{-\frac{\mu}{\lambda_1^*}}\frac{\mu p_2}{2\lambda_2^*}{\rm e}^{-\frac{\mu}{\lambda_2^*}}-\frac{p_2^2 c^2}{4{\lambda_2^*}^2}{\rm e}^{\frac{-2\mu x}{\lambda_2^*}}>0.\end{array}$$$

$\begin{equation*}\mu(\mu+2c) p_1 \lambda_2^* {\rm e}^{-\mu(\frac{1}{\lambda_1^*}+\frac{1}{\lambda_2^*})}>c^2 p_2 \lambda_1^*.\end{equation*}$

\begin{align*} Re\langle\mathcal{A}X,X\rangle_1\leq&-\dfrac{1}{2}p_1{\rm e}^{-\frac{\mu}{\lambda_1^*}}m^2(1)+\dfrac{1}{2}p_1m^2(0)-\dfrac{1}{2}p_2{\rm e}^{-\frac{\mu}{\lambda_2^*}}n^2(1)+\dfrac{1}{2}p_2n^2(0)\\&-\dfrac{1}{2}{\rm e}^{-\tau}p_3r^2(1)+\dfrac{1}{2}p_3r^2(0)-\dfrac{1}{2}{\rm e}^{-\tau}p_4s^2(1)+\dfrac{1}{2}p_4s^2(0)\\&-\dfrac{\tau}{2}\displaystyle\int_{0}^{1}{\rm e}^{-\tau x}p_3r^2{\rm d}x-\dfrac{\tau}{2}\displaystyle\int_{0}^{1}{\rm e}^{-\tau x}p_4s^2{\rm d}x.\end{align*}

\begin{align*} Re\langle\mathcal{A}X,X\rangle_1\leq\ &\big(-\dfrac{1}{2}p_1{\rm e}^{-\frac{\mu}{\lambda_1^*}}+2p_1k_1^2+\dfrac{1}{2}p_3\big)m^2(1)+\big(2p_1k_2^2-\dfrac{1}{2}p_2{\rm e}^{-\frac{\mu}{\lambda_2^*}}+p_2k_5^2+\dfrac{1}{2}p_4\big)n^2(1)\\&+\big(2p_1k_3^2-\dfrac{1}{2}{\rm e}^{-\tau}p_3\big)r^2(1)+\big(2p_1k_4^2+p_2k_6^2-\dfrac{1}{2}{\rm e}^{-\tau}p_4\big)s^2(1)\\&-\dfrac{\tau}{2}\displaystyle\int_{0}^{1}{\rm e}^{-\tau x}p_3r^2{\rm d}x-\dfrac{\tau}{2}\displaystyle\int_{0}^{1}{\rm e}^{-\tau x}p_4s^2{\rm d}x.\end{align*}

$$${\rm Re}\langle\mathcal{A}X,X\rangle_1\leq 0.$$$

$$$\begin{cases}\dot{X}(t)=\mathcal{A}X(t)+\mathcal{B}_2\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I},\\X(0)=X_0.\\\end{cases}$$$

$$$\begin{cases}\dot{X}^*(t)=\mathcal{A}^*X^*(t),\;\; X^*(t)=(\tilde{w}^*,\tilde{z}^*,u^*,g^*)\in D(\mathcal{A}^*),\\y^*=\mathcal{B}_2^*X^*(t),\end{cases}$$$

$\begin{equation*}\Vert(\tilde{w}(x,t),\tilde{z}(x,t))\Vert_{{L^2}(0,1)}\leq\beta(\Vert(\tilde{w}(0,t),\tilde{z}(0,t))\Vert_{{L^2}(0,1)})\\+\gamma(\sup\Vert(c(v_f-w^*),\dfrac{\alpha \tilde{p}_{in}(t)}{v^*I})\Vert_{{L^2}(0,1)}),\end{equation*}$

$$$\begin{cases}5p_1k_1^2-p_1{\rm e}^{\frac{-\mu}{\lambda_1^*}}+p_3\leq0,\;\;5p_1k_2^2-p_2{\rm e}^{\frac{-\mu}{\lambda_2^*}}+2p_2k_5^2+p_4\leq0,\\5p_1k_3^2-{\rm e}^{-\tau}p_3\leq0,\;\;5p_1k_4^2-{\rm e}^{-\tau}p_4+2p_2k_6^2\leq0,\\\mu+2c>\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{\mu}{\lambda_1^*}},\;\;(\mu {\rm e}^{\frac{-\mu}{\lambda_1^*}}+2c{\rm e}^{\frac{-\mu}{\lambda_1^*}}-\dfrac{p_1}{\lambda_1^*})(\mu {\rm e}^{\frac{-\mu}{\lambda_2^*}}-\dfrac{p_2}{\lambda_2^*})>\dfrac{p_2\lambda_1^*}{p_1\lambda_2^*}c^2,\end{cases}$$$

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

\begin{align*} &V_1(t)=\displaystyle\int_{0}^{1}\big(\dfrac{p_1}{\lambda_1^*}{\rm e}^{-\frac{\mu x}{\lambda_1^*}}\tilde{w}^2+\dfrac{p_2}{\lambda_2^*}{\rm e}^{-\frac{\mu x}{\lambda_2^*}}\tilde{z}^2\big){\rm d}x,\end{align*}
\begin{align*} V_2(t)=\tau \displaystyle\int_{0}^{1}{\rm e}^{-\tau x}(p_3u^2+p_4g^2){\rm d}x. \end{align*}

\begin{align*} \dot{V}_1(t)=&\displaystyle\int_{0}^{1}\big[\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{-\mu x}{\lambda_1^*}}2\tilde{w}\partial_t\tilde{w}+\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2\tilde{z}\partial_t\tilde{z}\big]{\rm d}x\\=&-p_1{\rm e}^{\frac{-\mu}{\lambda_1^*}}\tilde{w}^2(1,t)+p_1\tilde{w}^2(0,t)-p_2{\rm e}^{\frac{-\mu}{\lambda_2^*}}\tilde{z}^2(1,t)+p_2\tilde{z}^2(0,t)\\&-\dfrac{\mu}{\lambda_1^*}\displaystyle\int_{0}^{1}p_1{\rm e}^{\frac{-\mu x}{\lambda_1^*}}\tilde{w}^2{\rm d}x-\dfrac{\mu}{\lambda_2^*}\displaystyle\int_{0}^{1}p_2{\rm e}^{\frac{-\mu x}{\lambda_2^*}}\tilde{z}^2{\rm d}x-\displaystyle\int_{0}^{1}\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{-\mu x}{\lambda_1^*}}2c\tilde{w}^2{\rm d}x\\&-\displaystyle\int_{0}^{1}\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2c\tilde{w}\tilde{z}{\rm d}x+\displaystyle\int_{0}^{1}\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{-\mu x}{\lambda_1^*}}2\tilde{w}c(v_f-w^*){\rm d}x\\&+\displaystyle\int_{0}^{1}\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2\tilde{z}c(v_f-w^*){\rm d}x.\end{align*}

\begin{align*} \dot{V}_1(t)\leq&-p_1{\rm e}^{\frac{-\mu}{\lambda_1^*}}\tilde{w}^2(1,t)+p_1\tilde{w}^2(0,t)-p_2{\rm e}^{\frac{-\mu}{\lambda_2^*}}\tilde{z}^2(1,t)+p_2\tilde{z}^2(0,t)\\&-\dfrac{\mu}{\lambda_1^*}\displaystyle\int_{0}^{1}p_1{\rm e}^{\frac{-\mu x}{\lambda_1^*}}\tilde{w}^2{\rm d}x-\dfrac{\mu}{\lambda_2^*}\displaystyle\int_{0}^{1}p_2{\rm e}^{\frac{-\mu x}{\lambda_2^*}}\tilde{z}^2{\rm d}x-\displaystyle\int_{0}^{1}\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{-\mu x}{\lambda_1^*}}2c\tilde{w}^2{\rm d}x\\&-\displaystyle\int_{0}^{1}\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2c\tilde{w}\tilde{z}{\rm d}x+\displaystyle\int_{0}^{1}\dfrac{p_1^2}{(\lambda_1^*)^2}{\rm e}^{\frac{-2\mu x}{\lambda_1^*}}\tilde{w}^2{\rm d}x+\displaystyle\int_{0}^{1}\dfrac{p_2^2}{(\lambda_2^*)^2}{\rm e}^{\frac{-2\mu x}{\lambda_2^*}}\tilde{z}^2{\rm d}x\\&+2c^2(v_f-w^*)^2.\end{align*}

\begin{align*} W(t)=&-\dfrac{\mu}{\lambda_1^*}\displaystyle\int_{0}^{1}p_1{\rm e}^{\frac{-\mu x}{\lambda_1^*}}\tilde{w}^2{\rm d}x-\dfrac{\mu}{\lambda_2^*}\displaystyle\int_{0}^{1}p_2{\rm e}^{\frac{-\mu x}{\lambda_2^*}}\tilde{z}^2{\rm d}x-\displaystyle\int_{0}^{1}\dfrac{p_1}{\lambda_1^*}{\rm e}^{\frac{-\mu x}{\lambda_1^*}}2c\tilde{w}^2{\rm d}x\\&-\displaystyle\int_{0}^{1}\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2c\tilde{w}\tilde{z}{\rm d}x+\displaystyle\int_{0}^{1}\dfrac{p_1^2}{(\lambda_1^*)^2}{\rm e}^{\frac{-2\mu x}{\lambda_1^*}}\tilde{w}^2{\rm d}x+\displaystyle\int_{0}^{1}\dfrac{p_2^2}{(\lambda_2^*)^2}{\rm e}^{\frac{-2\mu x}{\lambda_2^*}}\tilde{z}^2{\rm d}x\\\leq&-\displaystyle\int_{0}^{1}\big(\dfrac{\mu}{\lambda_1^*}{\rm e}^{\frac{-\mu }{\lambda_1^*}}p_1+{\rm e}^{\frac{-\mu }{\lambda_1^*}}\dfrac{p_1}{\lambda_1^*}2c-\dfrac{p_1^2}{(\lambda_1^*)^2}\big)\tilde{w}^2{\rm d}x-\displaystyle\int_{0}^{1}\dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}2c\tilde{w}\tilde{z}{\rm d}x\\&-\displaystyle\int_{0}^{1}\big(\dfrac{\mu}{\lambda_2^*}{\rm e}^{\frac{-\mu}{\lambda_2^*}}p_2-\dfrac{p_2^2}{(\lambda_2^*)^2}\big)\tilde{z}^2{\rm d}x\\=&-\displaystyle\int_{0}^{1}A^TPA{\rm d}x,\end{align*}

$\begin{equation*}A=\left( \begin{array}{c} \tilde{w} \\ \tilde{z} \end{array}\right),\;P=\left( \begin{array}{cc} \dfrac{\mu}{\lambda_1^*}{\rm e}^{\frac{-\mu}{\lambda_1^*}}p_1+{\rm e}^{\frac{-\mu}{\lambda_1^*}}\dfrac{p_1}{\lambda_1^*}2c-\dfrac{p_1^2}{(\lambda_1^*)^2} & \dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}c \\[3mm] \dfrac{p_2}{\lambda_2^*}{\rm e}^{\frac{-\mu x}{\lambda_2^*}}c & \dfrac{\mu}{\lambda_2^*}{\rm e}^{\frac{-\mu}{\lambda_2^*}}p_2-\dfrac{p_2^2}{(\lambda_2^*)^2} \end{array} \right).\end{equation*}$

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