2007 Fiscal Year Final Research Report Summary
Research of "Algebraic methods on spectrums of boundary control systems and applications to nonlinear systems"
| Project/Area Number |
17540195
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kobe University |
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Principal Investigator |
NAMBU Takao Kobe University, Faculty of Engineering, professor (40156013)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAGIRI Shin-ichi Kobe University, Faculty of Engineering, professor (20031148)
NAITO Yuki Kobe University, Faculty of Engineering, associate professor (10231458)
TABATA Minoru Osaka Prefectural University, Faculty of Engineering, professor (70207215)
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| Project Period (FY) |
2005 – 2007
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| Keywords | static feedback control scheme / internal singularity / parabolic boundary control systems / boundary stabilization / Klein-Gordon equations / self-similar solutions / self-referential agent-based model / dynamic compensator / CDMA |
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| Research Abstract |
1. (Nambu) In feedback stabilization studies, a control scheme is designed so that the "state" of the system decays with a designated rate as t → ∞. Any linear functional of the state thus decays at least with the same decay rate. A specific feedback control scheme is constructed with the property such that some nontrivial linear functionals decays exactly faster than the state. In constructing the control scheme, a new problem of an arbitrary allocation of the eigenvulues which is subject to constraint appears, and the necessary and sufficient condition is obtained in terms of the observabililty assumptions. This is the essential extension of the celebrated theory of pole allocation by W. M. Wonham (1967).
2. (Nambu) Stabilization of linear parabolic systems in dynamic feedback scheme is studied: The system is characterized by the boundary conditions such that the boundary of the first kind is continuously connected with the boundary of the third kind. Instead of the method via the ass
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ociated fractional powers of the elliptic operators, it new and simple algebraic method is developed in the standard L^2-spaces to cope with these complicated boundaries. The construction of the finite dimensional stabilizing compensator is simpler and, new and it is also of algebraic nature.
3. (Nambu) Stabilization of linear parabolic systems in static feedback scheme is studied in the most difficult case where both the sensors and the actuators have spillovers. The key idea is the detailed property of the fundamental solution to the finite-dimensional substructure. By 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, the internal singularity in γ increases. It is shown, however, that the singularity can be removed, by the algebraic cancellation of the singularities of the related poles, if n ? 5.
4. (Nakagiri) Necessary conditions of optimal control problems are sought for the equation of motion of membrane with strong viscosity, corresponding to the distributive and terminal values observations. The Gateaux differentiability of solution mapping on control variables is proven.
5. (Nakagiri) A finite element method for numerical solutions of the distributed systems of Hopfield-type neural network equations is studied. When time keeps continuous and the spatial dimension is one, a semi-discrete algorithm for numerical solutions using quadratic interpolation functions is constructed, in which the Gauss-Legendre quadrature of numerical integrations of nonlinear term and the Runge-Kutta method for solving ordinary differential equation are efficiently used.
6. (Naito) Existence and non-existence of global solutions, and multiple solutions to a class of semilinear elliptic equations with singular forcing terms are studied in connection with compactness properties of solutions.
7. (Naito) The blow-up rate of solutions for the semilinear heat equation with critical power nonlinearity is studied. Via the backward self-similar solutions, the criteria of the blow-up rate is derived. The existence of the so-called type II blow-up solutions is shown for the Cauchy-Dirichlet problems on suitable shrinking domains.
8. (Tabata) In the nonlinear integro-partial differential equation describing human migration in a bounded domain, the so-called Kramers-Moyal expansion (an infinite series expansion) is rigorously studied, which gives the mathematical justification of this formal expansion.
9. (Tabata) Capacity problem in the DS/CDMA (the direct-sequence code-division multiple-access) communication system is studied vis the statistical approach to code acquisition. In the counting method for code acquisition, it is shown that the method can acquire the target user's code under any number of interference users that simultaneously transmit their signals. The capacity of the DS/CDMA communication system is set by the post-acquisition-based criterion rather than the acquisition-based capacity. Less
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