Approximation algorithm for an infinite-dimensional operator equation \(XL-BX=C\)
The operator equation \(XL- BX= C\) for operators in a separable Hilbert space is studied. The operator \(L\) is assumed to be unbounded selfadjoint with a discrete spectrum tending to infinity (the results can be extended to a certain class of spectral operators \(L\) with compact resolvent). The equation arises in a stabilization scheme of general linear parabolic systems, and the operators \(B\) and \(C\) appear in the compensator. The operator \(C\) is finite-dimensional, while \(B\) is an unbounded operator with discrete spectrum whose eigenvectors form a Riesz basis. The condition \(\sigma(L)\cap \sigma(B)=\emptyset\) guarantees existence of a unique solution \(X\) which can be expressed in terms of the eigenvalues and eigenvectors of \(B\) and \(C\) and the resolvent of \(L\) (a result proved in an earlier paper by the author). A concrete algorithm for the approximation of the solution \(X\), with an estimate of the rate of convergence, is devised. The approximation is based on solving finite- dimensional equations of the same type.
scientific article; zbMATH DE number 700868
scientific article