# Convergent Snapshot Algorithms for Infinite-Dimensional Lyapunov Equations

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#### Abstract

We consider two algorithms to approximate the solution Z of a class of stable operator Lyapunov equations of the form AZ + ZA* + BB* = 0. The algorithms utilize time snapshots of solutions of certain linear infinite-dimensional differential equations to construct the approximations. Matrix approximations of the operators a and B are not required and the algorithms are applicable as long as the rank of B is relatively small. The first algorithm produces an optimal low-rank approximate solution using proper orthogonal decomposition. The second algorithm approximates the product of the solution with a few vectors and can be implemented with a minimal amount of storage. Both algorithms are known for the matrix case. However, the extension of the algorithms to infinite dimensions appears to be new. We establish easily verifiable convergence theory and a priori error bounds for both algorithms and present numerical results for two model problems.

*This paper has been withdrawn.*