Research of Optimal Control and Inverse Problems for Nonlinear Evolution Equations

根据研究计划，首席研究员Nakagiri对非线性发展方程的最优控制和反问题的研究进行了研究和总结。他证明了一阶和二阶非线性发展方程解映射关于强迫函数和初值的Gateaux和Frechet可微性。在此基础上，利用Lions的变分方法，我们研究了更具物理意义的非线性方程，如分布神经网络方程、Sine-Gordon方程、Klein-Gordon方程、粘弹性方程、强粘性振动膜方程等的非凸成本最优控制问题和反问题。这些方程具有很强的非线性结构，需要对其解进行特殊的、恰当的分析。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。在有问题的意识下，我们成功地获得了…在Ha，Vanualailai，Hwang和Wang等人的帮助下，得到了上述方程问题的更多结果。此外，我们还研究了抽象的非线性Volterra积分-微分方程解的最优控制和反问题，并将其应用于长记忆粘弹性方程。其他研究者的研究成果如下。调查员塔巴塔研究了人口流动和流行病学的数学模型。研究人员Naito研究了非线性偏微分方程解的自相似结构，Ishii研究了运动平均曲率算法的收敛问题。小岛研究员通过对反问题的分析，提出并研究了数据挖掘的新方法，并将其应用于无损检测的估计问题。八木研究员从无限维动力系统的角度研究了数学物理和生态学中出现的模型方程。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 summarized the research on optimal control and inverse problems for nonlinear evolution equations. He has proved the Gateaux and Frechet differentiability of solution mappings for nonlinear first and second-order evolution equations with respect to forcing functions and initial values. Based on the results and the variational method due to Lions, we have investigated nonconvex cost optimal control problems and inverse problems for more physically important nonlinear equations such as distributed neural network equations, sine-Gordon equations, Klein-Gordon equations, viscoelastic equations, and equations of vibrating membrane with strong viscosity. These equations have own hard nonlinear structures and required the special and proper analysis of solutions. In order to solve our problems we have to obtain the delicate and proper estimates of solutions. Under the conscious of problems, we have succeeded in obtaini … More ng the results of problems for the above equations with the help of Drs Ha, Vanualailai, Hwang, and Wang. In addition, we have studied the optimal control and inverse problems for abstract nonlinear Volterra integro-differential equations with applications to viscoelastic equations with long memory. The researches of other investigators are as follows. The investigator Tabata studied the mathematical models in population movements and epidemiology. The investigator Naito studied the structure of self-similar solutions for nonlinear PDE's, and the investigator Ishii studied the convergence of algorithm for motion by mean curvature. The investigator Kojima proposed and studied the new method of data mining by the analysis of inverse problems, and applied it to the estimation problem for nondestructive testing. The investigator Yagi studied model equations appearing in mathematical physics and ecology from the stand point of infinite dimensional dynamical systems. The results of all investigators were published in the journals given below. Less