Water waves: Eigenvalue placement by linear feedback

Document Type

Article

Publication Date

1-1-1987

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 = A0w + ⟨f, w⟩b, where the evolution operator A0 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 A0 is λk = ik1/2 + O(k−3/2), k = ±1, ±2, ⃛ with corresponding eigenvector φk. If ⟨f, φk⟩ is denoted by bk, then the main result of this paper is: let ηk) be any sequence of complex numbers for which (ηk 7minus; λk)/bk is square summable. Then there exists a unique control f∈H for which ηk) is precisely the set of eigenvalues of the closed-loop operator A0(·) + ⟨f, · ⟩b, and the corresponding eigenfunctions form a basis for H. For skew-adjoint systems of the same form where λk = ikr + O(k2−r) for 1/2 < r < 1, the above result remains true. A weaker result is given for the case where 0 < r < 1/2. © 1987 Taylor & Francis Group, LLC.

Publication Title

International Journal of Control

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