Reseach on Optimal Control and Inverse Problems for Nonlinear Partial Differential Equations

根据研究计划，首席研究员Nakagiri对非线性偏微分方程的最优控制和逆问题的研究进行了研究和总结。在狮子会引起的变分框架下，我们将非线性偏微分方程表述为Hilbert空间中的一阶和二阶非线性演化方程，并在此框架下进行研究。首先，在Ha博士的帮助下，我们构建了最优控制的一般理论和方程的辨识问题。一般理论适用于理论问题。在理论和方法的基础上，对反应扩散方程、hopfield型神经网络方程、正弦-戈登方程、非线性梁方程、Klein-Gordon方程和非线性粘弹性方程等具有重要物理意义的非线性方程进行了研究。这些方程本身具有较强的非线性结构，需要进行特殊的、适当的分析。为了解决我们的问题，我们必须对解决方案作出精确而适当的估计。实际上，在分析解的结构时需要用到Sobolev嵌入定理。在问题意识的帮助下，我们在dr . Ha, Vanualailai， Elgamal和Wang的帮助下，成功地得到了上述方程的问题结果。其他研究者的研究情况如下。研究者Nambu研究了线性抛物型系统的输出镇定问题。研究者Tabata提出并研究了人口流动的数学模型。研究者内藤研究了半线性热方程的自相似解的结构。研究者Miyakawa研究了Navier Stokes方程在整个域上的渐近轮廓。研究者Kojima研究了电磁场和换热速度场的反问题。所有研究者的研究结果发表在以下期刊上。少

According to the research plan, the head investigator Nakagiri studied and summerized the research on optimal control and inverse problems for nonlinear partial differential equations. In the variational framework due to Lions, we formulated the nonlinear partial differential equations as the first and second order nonlinear evolution equations in Hilbert space, and proceeded the research based on the framework. First, we have constructed the general theory of optimal control and identification problems for the equations with the help of Dr Ha. The general theory has applications to theoretical problems. Based on the theory and the method, we have investigated the more physically important nonlinear equations such as reaction diffusion equations, Hopfield-type neural network equations, sine-Gordon equations, nonlinear beam equations, Klein-Gordon equations and nonlinear viscoelastic equations. These equations have own hard nonlinear structures and required the special and proper analys … More is of solutions. In order to solve our problems we have to obtain the delicate and proper estimates of solutions. In fact, the Sobolev imbedding theorem is needed in analizing the structure of solutions. Under the conscious of problems, we have succeeded in obtaining the results of problems for the above equations with the help of Drs Ha, Vanualailai, Elgamal, and Wang. The researches of other investigators are as follows. The investigator Nambu studied the output stabilization problems for linear parabolic systems. The investigator Tabata pro-posed and investigated the mathematical model in population movements. The investigator Naito studied the structure of self-similar solutions for semilinear heat equations. The investigator Miyakawa investigated the asymptotic profile of Navier Stokes equations in the whole domain. The investigator Kojima studied the inverse problems for electro-magnetic field and velocity filed of heat transfer. The results of all investigators were published in the journals given below. Less