Speaker
Description
A classical result due to Arov [1] states that a rational matrix function $F(z)$ (of size $k\times l$) that takes contractive values for $z\in {\mathbb D}={ z \in{\mathbb C} : |z|<1}$ admits a contractive finite-dimensional realization; i.e., there exists a contractive matrix $ \left[ \begin{array}{cc} A & B \cr C & D \end{array} \right]\in \mathbb C^{d+k,d+l} $ such that $F(z) = D + zC (I~-~zA)^{-1} B ,\quad z\in {\mathbb D}.$
Since then, several other realizations have been introduced. In this talk we restrict attention to the case where $\left[ \begin{array}{cc} A & B \cr C & D \end{array} \right]$ is a matrix rather than an infinite dimensional operator. In particular, we present the annulus realization [2], the half-plane realization extending [3], and a multihalfplane realization.
Finally, we discuss the practical implications of these results in connection with MOR and Padé approximation.
[1] D. Z. Arov. Passive linear steady-state dynamical systems. Sibirsk. Mat. Zh., 20(2):211–228, 457, 1979.
[2] R. Baran, P. Pikul, H. Woerdeman, M. Wojtylak. Contractive realization theory for the annulus and other intersections of discs on the Riemann sphere. Journal of Functional Analysis, 111346 (2026).
[3] K. Cherifi, H. Gernandt, D. Hinsen. The difference between port-Hamiltonian, passive and positive real descriptor systems. Mathematics of Control, Signals, and Systems, 36(2), 451-482 (2024).