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

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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.