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首页> 外文期刊>SIAM Journal on Control and Optimization >PSEUDO-BACKSTEPPING AND ITS APPLICATION TO THE CONTROL OF KORTEWEG-DE VRIES EQUATION FROM THE RIGHT ENDPOINT ON A FINITE DOMAIN
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PSEUDO-BACKSTEPPING AND ITS APPLICATION TO THE CONTROL OF KORTEWEG-DE VRIES EQUATION FROM THE RIGHT ENDPOINT ON A FINITE DOMAIN

机译：伪备份和应用于有限域对右端点的Korteeg-de Vries方程的应用

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摘要

In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear terms. At the end of the paper, we give numerical simulations to illustrate the main result.
机译：在本文中，我们设计了用于在域的右端点起作用的Korteweg-de Vries方程的Dirichlet-Neumann边界反馈控制器。允许域的长度是至关重要的。构建在域的右端点处采用的反向梗死控制器比其左端点对应更具挑战性。 BackStepping方法的标准应用失败，因为相应的内核模型变为过度规定。为了解决这个困难，我们介绍了伪逆向梗地方法，它使用满足所有所需边界条件的伪核。此外，可以通过其边界条件之一中的参数来控制伪内核的各种规范。我们证明，通过该假核构建的边界控制器仍然是指数稳定的，具有低指数衰减率的成本。我们表明单个Dirichlet控制器足以具有较慢衰减速率的指数稳定。我们还考虑在右侧的Dirichlet边界条件下的二阶反馈律。我们表明，如果主方程仅包括三阶项，则该方法有效，而当主方程涉及第一顺序和/或非线性术语，则相同的问题保持打开。在纸张结束时，我们给出了数字模拟，以说明主要结果。

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