Controllability of Second-Order Equations in L2(Ω)
Joint Authors
Source
Mathematical Problems in Engineering
Issue
Vol. 2010, Issue 2010 (31 Dec. 2010), pp.1-11, 11 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2010-12-28
Country of Publication
Egypt
No. of Pages
11
Main Subjects
Abstract EN
We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space L2(Ω): ÿ+Ay=1ωu(t), t∈(0,τ], y(0)=y0, ẏ(0)=y1, where Ω is a domain in RN(N≥1), y0,y1∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and A:D(A)⊂L2(Ω)→L2(Ω) is an unbounded linear operator with the following spectral decomposition: Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k, with the eigenvalues λj given by the following formula: λj=j2mπ2m, j=1,2,3,… and m≥1 is a fixed integer number, multiplicity γj is equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors (eigenfunctions) of A.
Specifically, we prove the following statement: if for an open nonempty set ω⊂Ω the restrictions ϕj,kω=ϕj,k|ω of ϕj,k to ω are linearly independent functions on ω, then for all τ≥2/πm-1 the system is approximately controllable on [0,τ].
As an application, we prove the controllability of the 1D wave equation.
American Psychological Association (APA)
Leiva, Hugo& Merentes, Nelson. 2010. Controllability of Second-Order Equations in L2(Ω). Mathematical Problems in Engineering،Vol. 2010, no. 2010, pp.1-11.
https://search.emarefa.net/detail/BIM-449486
Modern Language Association (MLA)
Leiva, Hugo& Merentes, Nelson. Controllability of Second-Order Equations in L2(Ω). Mathematical Problems in Engineering No. 2010 (2010), pp.1-11.
https://search.emarefa.net/detail/BIM-449486
American Medical Association (AMA)
Leiva, Hugo& Merentes, Nelson. Controllability of Second-Order Equations in L2(Ω). Mathematical Problems in Engineering. 2010. Vol. 2010, no. 2010, pp.1-11.
https://search.emarefa.net/detail/BIM-449486
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-449486