Finite-dimensional approximation and error bounds for spectral systems with partially known eigenstructure.

*(English)*Zbl 0821.93043Simple 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.

Reviewer: L.Rodman (Williamsburg)