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- W2022129315 abstract "This paper considers the effect of linear feedback control on the eigenvalues of a system of small-amplitude waves on a fluid surface. When such waves are controlled by a vibrating wall, the system has the form w = A 0 w + 〈f, w〉b, where the evolution operator A 0 is unbounded and skew-adjoint with compact resolvent on a Hilbert space H, the fixed vector b∊H reflects the design of the controlling wall (and is subject to a mild smoothness condition) and f∊H is to be chosen as a control function. The kth eigenvalue of A 0 is λ k = ik 1/2 + O(k−3/2), k = ±1, ±2,[tdot] with corresponding eigenvector φ k . If 〈 f , φ k 〉 is denoted by bk , then the main result of this paper is: let {η k } be any sequence of complex numbers for which (η k 7minus; λ k )/bk is square summable. Then there exists a unique control f∊H for which {η k } is precisely the set of eigenvalues of the closed-loop operator A 0( ·) + 〈f, · 〉b, and the corresponding eigenfunctions form a basis for H. For skew-adjoint systems of the same form where λ k = ikr + O(k 2−r ) for 1/2 < r < 1, the above result remains true. A weaker result is given for the case where 0 < r < 1/2." @default.
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- W2022129315 date "1987-03-01" @default.
- W2022129315 modified "2023-09-25" @default.
- W2022129315 title "Water waves: eigenvalue placement by linear feedback" @default.
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- W2022129315 doi "https://doi.org/10.1080/00207178708933789" @default.
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