KMS Of Academy of mathematics and systems sciences, CAS
Inverse/observability estimates for Schrodinger equations with variable coefficients | |
Triggiani, R; Yao, PF | |
1999 | |
Source Publication | CONTROL AND CYBERNETICS |
ISSN | 0324-8569 |
Volume | 28Issue:3Pages:627-664 |
Abstract | We consider a general Schrodinger equation defined on an open bounded domain Omega subset of R-n with variable coefficients in both the elliptic principal past and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H-1(Omega)-energy at time t = T, or at time t = 0, is dominated by the L-2(Sigma)-norms of the boundary traces partial derivative w/partial derivative v(A) and w(t), module an interior lower-order term. Once homogeneous B.C. are imposed, our results yield - under a uniqueness theorem, needed to absorb the lower order term - continuous observability estimates for both the Dirichlet and Neumann case, with an arbitrarily short observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neumann case. In contrast to existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in Omega subset of R-n to a problem on an appropriate Riemannian manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the variable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the Neumann case. |
Keyword | Schrodinger equation inverse/observability estimates exact controllability Riemannian manifold Carleman estimates |
Language | 英语 |
WOS Research Area | Automation & Control Systems ; Computer Science |
WOS Subject | Automation & Control Systems ; Computer Science, Cybernetics |
WOS ID | WOS:000087853800013 |
Publisher | POLISH ACAD SCIENCES SYSTEMS RESEARCH INST |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/14493 |
Collection | 中国科学院数学与系统科学研究院 |
Affiliation | 1.Univ Virginia, Charlottesville, VA 22903 USA 2.Acad Sinica, Inst Syst Sci, Beijing 100080, Peoples R China |
Recommended Citation GB/T 7714 | Triggiani, R,Yao, PF. Inverse/observability estimates for Schrodinger equations with variable coefficients[J]. CONTROL AND CYBERNETICS,1999,28(3):627-664. |
APA | Triggiani, R,&Yao, PF.(1999).Inverse/observability estimates for Schrodinger equations with variable coefficients.CONTROL AND CYBERNETICS,28(3),627-664. |
MLA | Triggiani, R,et al."Inverse/observability estimates for Schrodinger equations with variable coefficients".CONTROL AND CYBERNETICS 28.3(1999):627-664. |
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