几类色散波方程能稳问题的统一处理

The stabilizability of a control system is one of the key questions in the control community. Different in the structure from the parabolic and hyperbolic equations both of which have a longer history, dispersive wave equations have solid physical backgrounds and extremely wide applicability. Nowadays, most of the problems related to dispersive wave equations remain to be in the group of the hottest topics in the scientific frontier. The stabilization problem of dispersive wave equations has received extensive attentions, but it still lacks a unified treatment. This project is devoted to a unified treatment of the stabilization problem of the Korteweg-de Vries equation(-Burgers), the Benjamin-Bona-Mahony equation and the nonlinear Schrodinger equation where the control acts on part of the region and/or of the boundary of the region. An appropriate multiplier estimate (i.e., a weighted energy estimate), together with the assumption that the solution of the equation under consideration vanishes in a subregion, leads to an improvement of the regularity of the afore-mentioned solution; this new observation that the solution is indeed more regular, together with the unique continuation theory for partial differential equations developed in 1980s by Saut & Scheurer, leads to a qualitative unique continuation property (UCP, in short) for solutions to the equation under consideration; the qualitative UCP, together with another appropriate multiplier argument, leads to a quantitative UCP; the quantitative UCP, together with the semigroup property of solutions, leads to the desired decaying estimates on solutions to the equation under consideration.

控制系统能稳与否是控制论领域关注的核心问题之一。色散波方程结构上有别于拥有更悠久发展史的抛物、双曲方程，有深刻的物理背景和巨大的应用价值，大部分与其有关的科技问题的研究都是最活跃的前沿。色散波方程的能稳问题已获得广泛关注，但尚缺统一系统的处理，本项目旨在对Korteweg-de Vries(-Burgers)方程、Benjamin-Bona-Mahony方程和非线性Schrodinger方程的能稳问题（控制作用在区域内部或边界）作统一处理：适当的乘子估计（即加权能量估计）结合所考虑方程解在子区域消失的假设得到解的更高正则性，新发现的正则性结合由Saut和Scheurer于上世纪80年代创立的偏微分方程唯一延拓理论得到在研方程的定性唯一延拓性；新得到的定性唯一延拓性再结合适当的乘子估计得到热切企盼的定量唯一延拓性；新得到的定量唯一延拓性最后结合方程解的半群性质便得到方程解的衰减估计。

本课题旨在研究由若干色散波方程描述的无穷维控制系统的反馈镇定问题。受资助人为考虑Schrodinger方程的边界反馈镇定问题，受到前人研究成果的启发，引入了波方程从而形成了一类传动系统，受资助人借助Green函数，建立了该传动系统所描述的受控系统的无穷小生成元的豫解算子估计，进而建立了此传动系统能量的多项式衰减性。借助类似的方法，受资助人为板方程引入了波方程而构成传动系统，并先后证明了此类传动系统的渐近稳定性与对数衰减稳定性。受资助人还考虑了一类拟线性波方程所描述控制系统的轨线的长时间存在性与有界性问题（宽泛的稳定性）。受资助人通过选择适当的能量空间（以对付非线性性），借助紧性唯一性方法(Compactness-Uniqueness Method，CUM)，能证明一类Klein-Gordon-Schrodinger系统的能量多项式衰减。..在已有的研究成果中，建立豫解式估计的常用方法有Carleman估计、微局部分析、乘子估计等，而受资助人采用的是Green函数技术，此类方法对研究其他类型的传动系统有一定的启发意义。