2018-10-08

# Asymptotic stability of autonomous (P)DAEs with network structure

## Prof. Dr. Caren Tischendorf, Humboldt-Universität zu Berlin

24 Oct 2018, 13:00–14:30; Location: S4|10-314

We analyze autonomous differential algebraic equations (DAEs) with the following particular structure:

\begin{aligned} x'_1 & = \frac{\text{d}}{\text{d}t} f_1 (y_1) + g_1 (y_1) + r_1,\\ y'_2 & = \frac{\text{d}}{\text{d}t} f_2 (x_2) + g_2 (x_2) + r_2,\\ x_1 & = A_1^\top z,\\ x_2 & = A_2^\top z,\\ 0 & = A_1 y_1 + A_2 y_2, \end{aligned}

where ($$A_1$$, $$A_2$$) is an incidence matrix of full row rank. First we show that a transient modeling of the flow of different kind of networks (circuits, water network, gas network, blood circuits) leads to DAEs with this structure. The functions $$f_i$$ and $$g_i$$ represent the element models and their spatial discretizations (in case of PDE models). The matrices Ai describe the network topology.

We present sufficient criteria on the element model functions $$f_i$$ and $$g_i$$ for the asymptotic stability of DAEs with this structure. It includes a characterization of the eigenvalue structure for the related generalized eigenvalue problem. Furthermore, we discuss the correlation to port Hamiltonian modeling of networks.

Category: CE Seminar