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

1.研究了线性抛物型系统在静态反馈下的(Nambu)镇定问题：当传感器和执行器都有溢出时，镇定问题非常困难。通过研究有限维子结构基本解的性质，并引入一个小参数γ&gt；0，只要能观测性和可控性都满足且γ很小，就可以实现镇定。结果表明，当子结构的维度n增加时，γ的内部奇异性增加。然而，当n[小于或等于]5时，这种奇性是可以消除的。这一结果发表在2005.2早稻田大学7楼研讨会上。(Nambu)研究了线性抛物型系统在动态反馈方案下的镇定：证明了众所周知的通过相关椭圆算子的分数次幂的解析方法不再适用于各种复杂的边界控制系统。作为…此外，还引入了一种工作在标准L^2-空间上的全新的代数变换。通过一个通用型有限维动态补偿器，建立并实现了具有更复杂边界条件的一般控制系统的统一镇定方案。3.在正则状态镇定问题中，传感器和执行器的数目至少是不稳定特征值的乘数的最大值。当反馈数小于所要求的数目时，失去了能观测性和可控性条件，证明了通过适当构造反馈方案，至少可以实现输出镇定。4.(Nakagiri)在非线性双曲系统的参数估计问题中，得到了估计解的特征刻画。得到了时滞泛函微分方程解与输入之间的唯一性条件。研究了二阶半线性发展方程的(Nakagiri)最优控制问题，如Sine-Gordon方程和Klein-Gordon方程。(Tabata)在数学经济学的宏观分析中，研究了两种基于自参考主体的模型，并对它们进行了比较。给出了标度极限存在和不存在的充分条件。在描述人口动态的基于智能体的模型中，如最近欧洲大陆劳动力动态的变化，证明了该模型渐近于稳态(概率密度收敛)。7.(Naito)研究了一类半线性热传导方程的自相似解的结构。利用上下解方法证明了相关非线性椭圆型问题最小正解的存在性。在具有一类亚临界非线性项的OD方程边值问题中，也证明了多重解的存在性。较少

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