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变分法的框架下，研究了非线性二阶演化方程描述的分布参数系统解的存在性和唯一性。对于非线性系统，我们研究了最优控制问题，并根据各种类型观测的转置系统建立了必要的最优性条件。这些条件对于非线性系统来说是新的。结果应用于正弦-戈登方程、克莱因-戈登方程、非线性阻尼梁方程等实际系统中。接下来，对于耦合正弦-Gordon方程，我们研究了基于有限元法的近似解的数值分析。结果，我们观察到数值解的混沌行为，这在很大程度上取决于方程中出现的物理参数。此外，首席研究员通过内部域观测研究了抛物线型和双曲型线性分布参数系统中的空间变化参数可辨识性。这是一种反问题，他建立了可辨识性的几个充要条件。他还在 Drs. 的帮助下，通过 Liapounof 函数解决了移动物体的查找路径问题。哈和瓦努阿莱莱。其他调查结果如下。研究人员 Nambu 建立了具有反馈边界条件的一类椭圆微分算子的分数幂域的表征。研究者 Tabata 利用最优条件的思想，提出并研究了流行病地理传播的模型方程。研究员内藤研究了非线性椭圆分布参数系统，并建立了正径向解存在和不存在的新条件。所有研究人员的结果均发表在以下期刊上。

项目成果

J.Vanualailai: "A solution to the two-dimensional findpath problem" Dynamics and Stability of Systems. 13. 373-401 (1998)

J.Vanualailai：“二维查找路径问题的解决方案”系统动力学和稳定性。

T.Nambu: "Characterization of the domain of fractional powers of a class of elliptic differential operators with feedback boundary conditions" J.Differential Equations. 136-2. 294-324 (1997)

T.Nambu：“具有反馈边界条件的一类椭圆微分算子的分数幂域的表征”J.Differential Equations。

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。

Y.Naito: "Entire solutions of the inequality div (A (|Du|) Du) >= f (u)" Math.Z.225. 167-175 (1997)

Y.Naito：“不等式 div (A (|Du|) Du) >= f (u) 的完整解”Math.Z.225。