Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. (English) Zbl 1066.93028

Xu, Gen-Qi; Guo, Bao-Zhu

SIAM J. Control Optimization 42, No. 3, 966-984 (2003).

The problem of stabilizing elastic vibrations has been with us since the early 1960s, and much effort has been invested in finding the proper setting. The authors set their investigation of a vibrating string, or of coupled strings, in the familiar topological setting of a Hilbert space \({\mathbf H}= L^2(0,T)\), where stabilization is needed for the vector system \(dx/dt= Ax(t)\). Here \(A\) is the generator of a \(C_0\) semigroup. \(A\) is not self-adjoint and therefore its eigenvectors do not form an orthogonal basis of the Hilbert space \({\mathbf H}\), but a Riesz basis. However, one is not necessarily restricted to the eigenvectors of \(A\), because of Bari’s theorem stating that if \(\{\phi_i\}\) form a basis of \({\mathbf H}\), then, given any infinite and linearly independent system \(\{\psi_i\}\) such that \(\Sigma_i|\phi_i -\psi_i|<\infty\), \(\{\psi_i\}\) also form a Riesz basis. A theorem of Levin and Golovin dating to 1961 asserts that the set of exponentials \(\exp\{i\lambda_nt\}\), which they call sine-generating functions, forms a Riesz basis. Here \(\lambda_n\) are separated zeros of the corresponding sine functions, and lie in a strip parallel to the imaginary axis.
An important 1967 article of D. L. Russell [J. Math. Anal. Appl. 18, 542–560 (1967; Zbl 0158.10201)] started much of this activity involving non-harmonic Fourier series in control theory. The authors apply this theory to the control of vibrating coupled elastic strings, with the state equation reduced to the vector form \(dY/dt= AY(t)\), with simple boundary conditions at the ends, and with the control acting at the joint between them. The authors assert that the development of their theory would be similar for other popular choices of end conditions.

Reviewer: Vadim Komkov (Florida)

MSC:

93C20	Control/observation systems governed by partial differential equations
93D15	Stabilization of systems by feedback
74K05	Strings
35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs

Keywords:

Riesz basis; control of elastic vibrations; vibrating string; coupled strings; sine-generating functions

Citations:

Zbl 0158.10201