Residual-based iterations for the generalized Lyapunov equation
For researchers in numerical linear algebra and control theory, this work provides theoretical extensions and new iterative methods for solving generalized Lyapunov equations, though the improvements are incremental.
This paper extends the theoretical justification of the alternating linear scheme (ALS) from stable Lyapunov equations to stable generalized Lyapunov equations, establishes connections to H2-optimality in bilinear control systems, and proposes a residual-based generalized rational-Krylov-type subspace method.
This paper treats iterative solution methods to the generalized Lyapunov equation. Specifically it expands the existing theoretical justification for the alternating linear scheme (ALS) from the stable Lyapunov equation to the stable generalized Lyapunov equation. Moreover, connections between the energy-norm minimization in ALS and the theory to H2-optimality of an associated bilinear control system are established. It is also shown that a certain ALS-based iteration can be seen 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. Lastly a residual-based generalized rational-Krylov-type subspace is proposed for the generalized Lyapunov equation.