Low rank approximations of infinite-dimensional Lyapunov equations and applications

Lyapunov equation. We analyze the approximation properties of solutions of abstract Lyapunov equations in the setting of a scale of Hilbert spaces associated to an unbounded diagonalizable operator which satisfies the Kato’s square root theorem. We call an (unbounded) operator A diagonalizable if there exists a bounded operator Q, with a bounded inverse, such that the (unbounded) operator Q−1AQ is a normal operator with a compact resolvent. We assume that A generates an exponentially stable analytic semigroup and we construct—using sinc-quadrature techniques—a rank (2k + 1) × Ran(B) approximation X2k to the operator X. We also give conditions under which an estimate √√√√ ∞ ∑ i=2k+2 σ2 i (X) ≤ ‖X −X2k‖HS ≤ O(exp −π √ ) , (1) holds. Here ‖X‖HS = √ tr(X∗X) denotes the Hilbert-Schmidt norm of X and we note that our technique allows us to give the constant in the O(·) notation explicitly in terms of the weighted norms of A, B and Q and an estimate on the spectrum of A. In the case of a (more strongly) unbounded control operator B, e.g. an operator only bounded in a weighted Hilbert space, we obtain the same type of convergence estimates in an associated weighted norm on (the subspace of) the space of compact operators. Based on our convergence estimates we also discuss ramifications of this analysis for the design of adaptive finite element methods including the analysis of the influence of linear algebra approximations on the overall process. In this talk we will also discuss possible interpretations of the solutions of Lyapunov equations in the (negative order) Sobolev spaces of mixed derivatives and relate weighted operator norms of the solution of Lyapunov equation with the total energy of the system.