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

根据研究计划，在变分方法的框架下，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唯一性。对于非线性系统，我们研究了最优控制问题，并针对不同类型的观测建立了转置系统的必要最优性条件。这些条件是非线性系统的新条件。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。其次，针对耦合正弦-戈登方程，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。通过内域观测研究了抛物线型和双曲型线性分布参数系统的空间变化参数可辨识性。这是一类逆问题，他建立了可辨识性的几个充分必要条件。在李亚普诺夫博士的帮助下，利用李亚普诺夫函数解决了运动物体的寻径问题。哈和瓦努阿莱。其他调查的结果如下。研究者Nambu建立了一类具有反馈边界条件的椭圆型微分算子的分数阶幂域的刻划。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。研究非线性椭圆型分布参数系统，建立了径向正解存在和不存在的新条件。所有研究者的研究结果发表在以下期刊上。

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.