Research on inverse problems and boundary control problems for partial differential equations having transport and nonlocal terms

具有输运项和非局部项的偏微分方程的反问题和边界控制问题研究

• 批准号: 23540240
• 负责人: NAKAGIRI Shin-ichi
• 金额: $ 2.16万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23540240/
• 关键词: 関数解析; 逆問題; 移流拡散方程式; ボルテラ型微分積分方程式; 変形公式; スペクトル解析; 境界制御; 最適制御; 非局所方程式; 係数逆問題; 非局所境界条件; 連続半群

Inverse problems and boundary control problems for partial differential equations having transport and nonlocal terms are investigated by the method of deformation formulas. For the transport diffusion equations, the investigator has determined uniquely the transport term by boundary observaton, and for the associated control system having nonlocal terms, he has constructed a concrete boundary feedback control law which makes the system a desirable stable state. The key to solve the problems is to use the deformation formula which deforms the transport term and nonlocal terms simultaneously, and the investigator has solved the problem of existence and construction of such a deformation kernel function. In addition, the deformation formula is extended to a system of first order Volterra integro-differential equations. By using the kernel, he has solved the related inverse and boundary control problems for the Volterra system.

利用变形公式的方法研究了具有输运项和非局部项的偏微分方程的反问题和边界控制问题。 对于输运扩散方程，研究者通过边界观测唯一地确定了输运项，对于具有非局部项的相关控制系统，他构造了具体的边界反馈控制律，使系统达到理想的稳定状态。 解决问题的关键是使用同时对输运项和非局部项进行变形的变形公式，研究者解决了这种变形核函数的存在和构造问题。 此外，将变形公式推广到一阶Volterra积分微分方程组。 通过使用该内核，他解决了Volterra系统的相关逆控制和边界控制问题。

项目成果

Semigroup theory for nonlocal parabolic equations

非局部抛物型方程的半群理论

• 发表时间: 2011
• 作者: Shin

移流拡散方程式の逆問題

对流扩散方程反问题

• 发表时间: 2011
• 作者: 中桐 信一

移流拡散方程式系の逆問題

对流扩散方程组反问题

• 作者: 中桐信一

Deformation formulas and inverse problems for advection-diffusion equations

平流扩散方程的变形公式和反问题

• 发表时间: 2012
• 作者: S. Nakagiri

Deformation formula and solution semigroups for hyperbolic Volterra integro-differential systems

双曲Volterra积分微分系统的变形公式和解半群

• 发表时间: 2013
• 作者: S. Nakagiri