Research of "Algebraic and analytic methods in the spectrum for boundary control systems and numerical analysis"

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1.(Nambu) Stabilization of linear parabolic systems in static feedback scheme is studied : When both the sensors and the actuators have spillovers, the stabilization problem is extremely difficult. By investigating the property of the fundamental solution to the finite-dimensional substructure and introducing a small parameter γ > 0, the stabilization is achieved as long as both the observability and the controllability assumptions are fulfilled and γ is small. When the dimension n of the substructure increases, it is shown that the internal singularity in γ increases. However, this singularity can be removed if n 【less than or equal】 5. This result is reported in the "7th-floor seminar, Waseda University", 2005.2.(Nambu) Stabilization of linear parabolic systems in dynamic feedback scheme is studied : It is shown that the well known analytic approach via fractional powers of the associated elliptic operator is no more applied to a variety of complicated boundary control systems. As an … More alternative, an entirely new algebraic transform working in the standard L^2-spaces is introduced. Via a finite-dimensional dynamic compensator of general type, a unified stabilization scheme is developed and achieved for general control systems with more complicated boundary conditions.3.(Nambu) In regular state stabilization problems the necessary number of the sensors and the actuators is at least the maximum of the multiplicities for the unstable eigenvalues. When the number is smaller than the required one and thus the observability and the controllability conditions are lost, it is shown that at least the output stabilization can be achieved by suitably constructing the feedback scheme.4.(Nakagiri) In the problem of parameter estimation for nonlinear hyperbolic systems, a characterization of the estimated solutions is achieved. A uniqueness condition between solutions and inputs is obtained in retarded functional differential equations. A related numerical computation for the perturbed Klein-Gordon equation is studied.5.(Nakagiri) Optimal control problems are studied for semilinear evolution equations of 2nd order, such as sine-Gordon equations and Klein-Gordon equations.6.(Tabata) In the macro analysis of mathematical economy, two self-referential agent-based models are studied and compared with each other. Sufficient conditions are obtained for existence and non-existence of scaling limits. In the agent-based model describing the population dynamics such as the recent change of labor dynamics in European Continent, it is shown that the model asymptotically approaches to the steady state (probabilistic density convergence).7.(Naito) The structure of self-similar solutions to a class of semilinear heat conduction equations is studied. The existence of the minimum positive solutions is shown via the super solution-sub solution method in the related nonlinear elliptic problem. In boundary value problems of o. d. equations with a class of subcritical nonlinear terms, the existence of multiple solutions is also shown. Less