Problems on Partial Differential Equations
Project/Area Number 
61460003

Research Category 
GrantinAid for General Scientific Research (B)

Allocation Type  Singleyear Grants 
Research Field 
解析学

Research Institution  Osaka University 
Principal Investigator 
TANABE Hiraki Professor,Faculty of Science,Osaka University, 理学部, 教授 (70028083)

CoInvestigator(Kenkyūbuntansha) 
NAKAO Shintaro Associate Professor,Faculty of Science,Osaka University, 理学部, 助教授 (90030783)
KOMATSU Gen Associate Professor,Faculty of Science,Osaka University, 理学部, 助教授 (60108446)
IKAWA Mitsuru Professor,Faculty of Science,Osaka University, 理学部, 教授 (80028191)
WATANABE Takeshi Professor,Faculty of Science,Osaka University, 理学部, 教授 (50028081)
IKEDA Nobuyuki Professor,Faculty of Science,Osaka University, 理学部, 教授 (00028078)

Project Period (FY) 
1986 – 1987

Project Status 
Completed (Fiscal Year 1987)

Budget Amount *help 
¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 1987: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1986: ¥2,600,000 (Direct Cost: ¥2,600,000)

Keywords  time delay / parabolic equation / evolution operator / controllability / observability semigroup / decay of wave motions / poles of scattering matrix / 散乱行列の極 / ハミルトン作用素 / ヴォルテラ型積分微分方程式 / 初期境界値問題 / 比重付楕円型評価 / 漸近行動 / 漸近展開 / 解の減衰 
Research Abstract 
The evolution operator for parabolic evolution equations with time delay was successfully constructed.As for the solvability of that kind of equations various results are know under mild smsoothness hypothesis on the coefficient of the delay term.If the coefficient is Holder continuous,it was found that the evolution operator can be constructed.A remarkable difference from the case without time delay is that a singularity of the time derivative of the evolution operator appear at each integral multiple of the delay interval.With the aid of this evolution operator a considerable part of the theoty of control for equations with only bounded operators in delay terms which has been developped by S.Nakagiri of Kobe University can be extended to the case where operators in delay terms are unbouned.For that purpose the basic space is enlarged so that the solution is expressed by a semigroup S(t).However,the adjoint operatorS*(t)appears and the results of Nakagiri cannot be extended directly.H
… More
ence,assuming that the basic space is a Hilbert space and the main operator is associated with a srongly elliptic sesquilinear form and considering the equation also in the space of negative norm as occasion demands, it has been found that a fairly large part of Nakagiri's resutls can be extended. Namely,the first structural operator F which connects the semigroup S_T(t)associated with the equation adjoint to the original one and S(t) is defined and through the second structural operator defined by means of the above mentioned evolution operator the relationsFS(t)=S*_t(t)F,S*(t)F*=F*S_t(t)were established.In this manner the equivalence of the controllability of the original equation and the observability /f the adjoint equation was proved.However,in order to develop this type of theoty there are problems on whether the spectra of the infinitesimal generator of the semigroup S(t)are discrete or not and whether the generalized eigenfunctions are complete or not.In some special case there is an affirmative answer to this problem; however,in the general case it will be a future subject. In addition some new results were established by other investigators on the decay of wave motions and the poles of scattering matrix,some approxiamation theory for stochastic differential equations,and essential selfadjointness of quantum Hamiltonians. Less

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