Boundary stabilization of first-order hyperbolic equations with input delay
Boundary stabilization of first-order hyperbolic equations with input delay
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DOI：
10.1007/s13160-019-00346-6
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发表时间：
2019-02
影响因子：
0.9
通讯作者：
H. Sano;M. Wakaiki
中科院分区：
文献类型：
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作者：
H. Sano;M. Wakaiki
研究了一类具有输入时滞的一阶双曲型方程的边界稳定性问题。在我们以前的工作中，同样的问题的抛物型方程的输入延迟解决。对于双曲型方程，同样地，时滞元素被迁移方程代替，并采用反推方法。然而，与抛物型方程的情况不同，双曲型方程的系统算子的本征值和本征函数很难获得。因此，我们不能同时使用有限维控制理论来构造目标系统和控制器。本文利用非局部边界条件下双曲型方程的-半群，证明了所提出的控制器在Hilbert空间中可以用一种抽象形式表示，即所谓的预测器，并且在一定条件下该预测器是有意义的.进一步，对于逆积分变换，在同样的条件下，我们得到了一个关于其连续性的有趣结果。本文还提出了一种数值算法来求解控制器设计中出现的一个非标准双曲方程。
This paper is concerned with the boundary stabilization problem of first-order hyperbolic equations with input delay. In our previous work, the same problem for parabolic equations with input delay was addressed. For the hyperbolic equation, the element of time lag is similarly replaced by a transport equation and a backstepping method is employed. However, unlike in the case of the parabolic equation, it is difficult to obtain the eigenvalues and eigenfunctions of the system operator for the hyperbolic equation. Hence, we cannot construct the target system and the controller by using the finite-dimensional control theory together. In this paper, using-semigroups for hyperbolic equations with nonlocal boundary condition, we show that the proposed controller is expressed by an abstract form in a Hilbert space so-called a predictor, and that the predictor makes sense under a condition. Further, for the inverse integral transformation, we obtain an interesting result on its continuity under the same condition. A numerical algorithm is also proposed for solving a non-standard hyperbolic equation appearing in our controller design.