An algebraic method for boundary control systems of parabolic type

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1.Stabilization of linear parabolic systems by means of boundary feedback is studied. The boundary condition is partially of the first kind and partially of the third kind. An entirely new and simple algebraic transform (isomorphism in LィイD12ィエD1-spaces) is introduced for the complicated boundary condition. By introducing a finite-dimensional dynamic compensator of general type, stabilization is achieved for more complicated control systems. An algebraic structure of the so called Silvester equation with unbounded linear operators as coefficients is also studied.2.Approximate null controllability problem is studied for a class of linear second order parabolic systems in Hilbert spaces by means of the so called HUM method. A numerical approximation of solutions to the sine-Gordon equation is also studied by means of FEM.3.An integro-partial differential equation is established as a mathematical model for studing logistic growth of human population with migration. It is proven that the Cauchy problem admits the unique global solution in time. The asymptotic behavior of the solutions is also investigated. The property of solutions is used for explaining about the recent change of labor dynamics in European Continent.4.A class of nonlinear elliptic partial differential equations is studied, where the appearing coefficients and the spatial domain has a kind of symmetry. Existence and uniqueness of solutions are proven via ODE approach. A generalization of the so called moving sphere method is also obtained.5.Stabilization problem is studied for linear parabolic control systems by means of HィイD1∞ィエD1-control method. Robustness of the stabilizing feedback control scheme is proven in some topology. Another stabilization problem is also studied, where the sensors and actuators are periodic functions in time. By generalizing the result of Brunovsky (JDE, 1969), the stabilization is achieved.