Inverse/observability estimates for Schrodinger equations with variable coefficients
Triggiani, R; Yao, PF
AbstractWe 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.
KeywordSchrodinger equation inverse/observability estimates exact controllability Riemannian manifold Carleman estimates
WOS Research AreaAutomation & Control Systems ; Computer Science
WOS SubjectAutomation & Control Systems ; Computer Science, Cybernetics
WOS IDWOS:000087853800013
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Document Type期刊论文
Corresponding AuthorTriggiani, R
Affiliation1.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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