SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2143-2170, January 2020.
The proportional-integral (PI) boundary stabilization of nonlinear hyperbolic systems of balance laws is investigated for the $H^2$-norm, in which the control and output measurements are all located at the boundaries. The boundary conditions of the system are subject to unknown constant disturbances. The induced closed-loop system is proven to be locally exponentially stable with respect to the steady states. To this end, a set of matrix inequalities is given by constructing a new Lyapunov function as a weighted $H^2$-norm of the classical Cauchy solution and the integral of boundary output. Furthermore, the traffic flow dynamics of a freeway section are modeled with the Aw--Rascle--Zhang model. To stabilize the oscillations of traffic demand, a local PI boundary feedback controller is designed with integration of on-ramp metering and variable speed limit control. The exponential convergence of the nonlinear traffic flow dynamics in an $H^2$ sense is achieved and validated with simulations.

中文翻译：

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平衡律拟线性双曲方程组的局部比例-积分边界反馈镇定

SIAM控制与优化杂志，第58卷，第4期，第2143-2170页，2020年1月。
针对$ H ^ 2 $范数，研究了非线性双曲线平衡律系统的比例积分（PI）边界稳定，其中控制和输出测量值均位于边界处。系统的边界条件受到未知的恒定干扰。事实证明，感应式闭环系统相对于稳态是局部指数稳定的。为此，通过构造新的Lyapunov函数作为经典Cauchy解的加权$ H ^ 2 $-范数和边界输出的积分，给出了一组矩阵不等式。此外，使用Aw-Rascle-Zhang模型对高速公路路段的交通流动力学进行建模。为了稳定交通需求的波动，设计了本地PI边界反馈控制器，集成了匝道计量和变速限制控制。在$ H ^ 2 $的意义上实现了非线性交通流动力学的指数收敛，并通过仿真进行了验证。