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组最优控制和反问题的研究进行了研究和总结。在Lions的变分框架下，我们将非线性偏微分方程组表示为Hilbert空间中的一阶和二阶非线性发展方程，并在该框架下进行了研究。首先，在Ha博士的帮助下，我们建立了方程最优控制和辨识问题的一般理论。一般理论适用于理论问题。在此理论和方法的基础上，我们研究了更重要的非线性方程，如反应扩散方程、Hopfield型神经网络方程、Sine-Gordon方程、非线性梁方程、Klein-Gordon方程和非线性粘弹性方程。这些方程具有很强的非线性结构，需要特殊的、恰当的…分析更多的是解决方案。为了解决我们的问题，我们必须获得对解决方案的微妙和适当的估计。事实上，索波列夫嵌入定理在分析解的结构时是需要的。在有问题意识的情况下，我们在Ha，Vanualailai，ElGamal和Wang等人的帮助下，成功地得到了上述方程的解。其他研究者的研究成果如下。研究员Nambu研究了线性抛物型系统的输出镇定问题。调查人员Tabata提出并研究了人口流动的数学模型。研究人员Naito研究了半线性热方程自相似解的结构。研究人员Miyakawa研究了整个区域上的Navier-Stokes方程的渐近轮廓。小岛研究员研究了热传导的电磁场和速度场的反问题。所有调查人员的结果都发表在下面给出的期刊上。较少

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