Theoretical and Numerical Research of Optimal Control and Inverse Problems for Nonlinear Elliptic and Hyperbolic Distributed Parameter Systems

非线性椭圆和双曲分布参数系统最优控制与反问题的理论与数值研究

• 批准号: 09640186
• 负责人: NAKAGIRI Shin-ichi
• 金额: $ 1.22万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640186/
• 关键词: optimal control; inverse problem; nonlinear distibuted systems; elliptic operator; identifiability; finite element method; semigroup; least square method; 最小2乗法; 分数ベキ

According to the research plan, we studied the existence and uniqueness of solutions for distributed parameter systems described by nonlinear second order evolution equations in the framework of variational method due to Lions. For the nonlinear systems we studied optimal control problems, and established necessary optimality conditions in terms of transposed systems for various types of observations. The conditions are new ones for nonlinear systems. The results were applied to practical systems such as sine-Gordon equation, Klein-Gordon equation, nonlinear damped beam equations and others. Next, for coupled sine-Gordon equations, we studied the numerical analysis of approximate solutions based on finite element method. As a result we observed the chaotic behavior of numerical solutions which depends heavily on physical parameters appearing in the equations. Further the head investigator studied the spatially-varying parameter identifiability in linear distributed parameter systems of parabolic and hyperbolic types by interior domain observations. This is a kind of inverse problems and he established several necessary and sufficient conditions for the identifiability. Also he solved the findpath problem of moving objects by means of Liapounof functions with the help of Drs. Ha and Vanualailai. The results of other investigatos are as follows. The investigator Nambu established the characterization of the domain of fractional powers of a class of elliptic differential operators with feedback boundary conditions. The investigator Tabata, by using the idea of optimality conditions, proposed and investigated the model equations for geographic spread of an epidemic. The investigator Naito studied nonlinear elliptic distributed parameter systems and established new conditions for the existence and nonexistence of positive radial solutions. The results of all investigators were published in the journals given below.

根据研究计划，我们在Lions的变分方法框架下研究了由非线性二阶发展方程描述的分布参数系统解的存在唯一性。对于非线性系统，我们研究了最优控制问题，并建立了关于转置系统的最优性必要条件。这些条件对于非线性系统是新的。将所得结果应用于sine-Gordon方程、Klein-Gordon方程、非线性阻尼梁方程等实际系统。其次，针对耦合sine-Gordon方程，研究了基于有限元方法的近似解的数值分析。结果，我们观察到的数值解的混沌行为，这在很大程度上取决于物理参数出现在方程。此外，首席研究员研究了空间变化的参数可识别性的线性分布参数系统的抛物线和双曲型的内部域的意见。这是一类反问题，他建立了几个充分必要条件的可识别性。此外，他解决了findpath问题的移动对象的手段Liapounof功能的帮助下，哈博士和Vanualailai。其他研究的结果如下。研究者Nambu建立了一类带反馈边界条件的椭圆型微分算子的分数幂域的特征。研究者Tabata利用最优性条件的思想，提出并研究了流行病地理传播的模型方程。内藤研究了非线性椭圆型分布参数系统，建立了正径向解存在和不存在的新条件。所有研究者的结果均发表在以下期刊上。

项目成果

J.Ha: "Optimal control of a linear damped second order evolution equation system in Hilber space" Mem.Grad.School Sci,& Technol.,Kobe Univ.15-A. 127-145 (1997)

J.Ha：“希尔伯空间中线性阻尼二阶演化方程组的最优控制”Mem.Grad.School Sci，

S.Nakagiri: "Regional identifiability of spatially-varying parameters in distributed parameter systems of parabolic type" Proceedings of the 11th IFAC Symposium on System Identification. Vol.1. 351-356 (1997)

S.Nakagiri：“抛物型分布参数系统中空间变化参数的区域可辨识性”第 11 届 IFAC 系统辨识研讨会论文集。

M.Tabata: "The spectrum of the linear transport operator with a potential term under the spatial oeriodicity condition" Rendiconti dell Sem. Mat. Univ. Padova. 97-1. 1-23 (1997)

M.Tabata：“空间偶数条件下具有潜在项的线性传输算子的谱”Rendiconti dell Sem。

N.Eshima: "The RC (M) association model and canonical correlation analysis" J.Japan Statistical Society. 27-1. 109-120 (1997)

N.Eshima：“RC（M）关联模型和典型相关分析”J.日本统计学会。

M.Tabata: "A model for the geographic spread of an epidemic which infects human beings" Mem.Grad.School Sci.& Technol., Kobe Univ.16-A. 167-188 (1998)

M.Tabata：“感染人类的​​流行病地理传播的模型”Mem.Grad.School Sci。