Melnikov, Andrey (Ben-Gurion Univ.), Vinnikov, Victor (Ben Gurion Univ.)

Finite Dimensional Overdetermined 2D Systems Invariant in One Direction and Their Transfer Functions

Scheduled for presentation during the Mini-Symposium "Multidimensional Systems: Noncommutative and Overdetermined Systems II" (FrP05), Friday, July 28, 2006, 17:00−17:25, Room F

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 24, 2024

Abstract

The theory of two dimensional ($2D$) overdetermined time-invariant systems has been extensively developed for the last 20 years; it is closely connected to the theory of commuting operators.

Our goal will be a realization theorem for finite dimensional $2D$ systems, which are invariant in one of the variables (say $t_1$). Such an invariance allows us to perform a partial separation of variables and to define a transfer function, depending on the corresponding spectral parameter (say $lambda$), which will, additionally, depend on the second variable (say $t_2$).

In order to develop realization theorem for transfer functions of such systems we use a key feature that multiplication by $S(lambda, t_2)$ maps solutions of one ODE with spectral parameter $lambda$ (the input ODE) to solutions of another ODE with the same spectral parameter $lambda$ (the output ODE). Using it, we realize $S(lambda,t_2^0)$ for some value $t_2^0$ and then build a realization for the matrix function $S(lambda, t_2)$ according to the pole data at $t_2^0$.

The theory is interesting by itself, especially since it allows us to use frequency domain analysis in a time varying framework. It has also important connections with completely integrable nonlinear PDEs: the so called Lax equation appears naturally, and the passage from the input to the output ODE with a spectral parameter is analogous to the B" acklund transformation