On a Finiteness Result for the Number of Critical Points of the H2 Approximation Criterion

The long-standing open problem about whether the number of critical points in the H2 SISO real model order reduction problem is finite is answered in the positive in the case the transfer function of the to-be-reduced model has distinct poles (ie. only has poles of algebraic multiplicity one). This has important implications for various search methods for finding critical points or local minima of the criterion function for this model reduction problem. In fact more is shown namely that in a particular parametrization the number of complex solutions of the first order conditions of the H2 real model order reduction problem is finite and lies in a bounded set of which the bound can be determined from information about the to-be-reduced model. This implies that the H2 model order reduction problem can be solved in principle by constructive algebra methods (such as Groebner basis methods) in case the to-be-reduced model has distinct poles. It simplifies the methods for co-order three reduction as described in a companion paper.