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

点击翻译按钮获取中文摘要

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