Residual-based iterations for the generalized Lyapunov equation

• This paper treats iterative solution methods for the generalized Lyapunov equation. Specifically, a residual-based generalized rational-Krylov-type subspace is proposed. Furthermore, the existing theoretical justification for the alternating linear scheme (ALS) is extended from the stable Lyapunov equation to the stable generalized Lyapunov equation. Further insights are gained by connecting the energy-norm minimization in ALS to the theory of H2-optimality of an associated bilinear control system. Moreover it is shown that the ALS-based iteration can be understood as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm.

Subject headings

NATURVETENSKAP  -- Matematik -- Annan matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Other Mathematics (hsv//eng)

Keyword

Alternating linear scheme

Bilinear control systems

Generalized Lyapunov equation

H2-optimal model reduction

Matrix equations

Projection methods

Rational Krylov

Publication and Content Type

ref (subject category)

art (subject category)