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Finite-dimensional approximation and error bounds for spectral systems with partially known eigenstructure. (English) Zbl 0821.93043
Simple regular spectral linear systems are studied. Such systems can be described by the following abstract differential equation in a separable Hilbert space \(Z\): \[ \dot z = Az + Bu;\quad y = Cz; \quad z(0) = z_ 0, \] where \(B : \mathbb{R} \to Z\) and \(C : Z \to \mathbb{R}\) are bounded linear operators, and \(A\) is a closed linear operator having dense domain, compact resolvent, and simple eigenvalues with real parts bounded above. It is assumed, moreover, that the normalized eigenfunctions \(\{\varphi_ i\} ^ \infty_{i = 1}\) of \(A\) form a basis in \(Z\), and the eigenfunctions \(\{\psi_ i\}^ \infty_{i = 1}\) of \(A^*\) can be normalized so that \(\langle \varphi_ 1, \psi_ j \rangle = \delta_{ij}\) and \(\sup_ i \| \psi_ i \| < \infty\). For certain classes of such systems, bounds are given for the truncation error, i.e., the difference between the transfer function of the system and its truncation after a finite number of terms. For a class of hyperbolic systems (proportionally damped heat equation) the infinity norm of the truncation error is given in terms of the first \((n + 1)\) eigenvalues of \(A\) and its first \(n\) eigenfunctions. Thus, only those eigenvalues and eigenfunctions need to be calculated to compute the bound. For a class of hyperbolic systems (wave equation systems with viscous damping) a frequency dependent bound for the truncation error is developed. The bound again is based on the first \((n + 1)\) eigenvalues of \(A\) and its first \(n\) eigenfunctions. An illustrative example is given involving a simple wave equation.

MSC:
93C25 Control/observation systems in abstract spaces
93B60 Eigenvalue problems
93C20 Control/observation systems governed by partial differential equations
93B28 Operator-theoretic methods
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