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; 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，我们研究了由于狮子而导致的变异方法框架中非线性二阶进化方程所描述的分布式参数系统的解决方案的存在和唯一性。对于非线性系统，我们研究了最佳控制问题，并根据用于各种观察结果的转置系统建立了必要的最佳条件。这些条件是非线性系统的新条件。结果应用于实用系统，例如正弦 - 戈登方程，klein-gordon方程，非线性阻尼束方程等。接下来，对于耦合的正弦 - 戈登方程，我们研究了基于有限元方法的近似解决方案的数值分析。结果，我们观察到了数值解的混乱行为，这在很大程度上取决于方程中出现的物理参数。此外，负责人研究员通过内部域观测研究了抛物线和双曲线类型的线性分布式参数系统中的空间变化参数可识别性。这是一种反问题，他为可识别性建立了几个必要和足够的条件。他还通过DRS的帮助来解决通过LiaPounof功能移动对象的发现路径问题。 ha and vanualailai。其他研究结果如下。研究人员Nambu确定了具有反馈边界条件的一类椭圆形差分运算符的分数幂的域的表征。研究者塔巴塔（Tabata）通过使用最佳条件的概念提出并研究了流行病的地理扩散模型方程。研究者Naito研究了非线性椭圆分布的参数系统，并为阳性径向溶液的存在和不存在的新条件建立了新的条件。所有研究人员的结果发表在下面给出的期刊上。