Category: SCO

Title: Clustering-based model reduction of network systems containing cycle

Abstract: In the model reduction of networked systems, clustering-based methods are able to preserve the network structure and synchronisation properties. However, the selection of clustering that results in a reduced-order model with optimal approximation error remains an open problem. The method of [1] enables identifying two nodes/sub-systems that show similar behaviour and, therefore, choosing two nodes to be clustered and resulting in a good approximation, based on the analysis of the Gramians of the edge dynamics. This method works perfectly on networked systems with tree structures, but the extension to more general graph structures containing cycles is not straightforward. In this colloquium, I will present my ongoing research on solving the problem of finding the best-clustering of networks containing cycles using the concept of Gramians for semi-stable systems [2].

[1] Besselink, Bart, Henrik Sandberg, and Karl H. Johansson. “Clustering-based model reduction of networked passive systems.” IEEE Transactions on Automatic Control 61.10 (2015): 2958-2973.
[2] Cheng, Xiaodong, and Jacquelien MA Scherpen. “Novel Gramians for linear semistable systems.” Automatica 115 (2020): 108911.

Title: Nonlinear (switched) DAEs: normal forms, impulse-free jumps and stability

Abstract: In the first part of this talk, we deal with inconsistent initial value problems of nonlinear DAEs of the form $E(x)\dot x= F(x)$. We define  impulse-free jumps of  nonlinear DAEs as  parameterized curves with derivatives in the distribution $\ker E(x)$. Then with the help of a proposed nonlinear Weierstrass form, we study the existence and uniqueness of the impulse-free jumps. After that, a singular perturbed system approximation is proposed for  nonlinear DAEs; we show that  solutions of the perturbed system approximate both  impulse-free jumps  and  $\mathcal C^1$-solutions of nonlinear DAEs. In the second part of the talk, we extend the jump rule in the first part to the switched case, which generalizes the impulse-free condition of switched linear DAEs to the nonlinear case. Moreover, a novel notion called the jump-flow explicitation is used  to simply the common Lyapunov function condition for the stability analysis of switched nonlinear DAEs. Finally, we generalize the well-known commutativity condition of switched nonlinear ODEs to the DAEs case. We show that to guarantee the stability of nonlinear switched DAEs with all stable models, not only the commutativity of the flow vector fields but also some extra invariant conditions are needed.

Title: Analytic Properties of Heat Equation Solutions and Reachable Sets

Abstract: We consider heat equations on bounded Lipschitz domains Omega in R^d and show that solutions to the heat equation for positive times are analytically extendable to a subdomain of the complex plane containing Omega. Our analysis is based on the boundary layer potential method for the heat equation. In particular, our method gives an explanation for the shapes appearing in the literature in 1d, which is not so easy to explain using Fourier analysis alone. I will also discuss the converse theorem, namely that certain sets in the complex plane can be realized as solutions to the heat equation on the boundary of Omega when Omega is a ball. Boundary layer potential theory also gives an indication that this statement is more difficult if Omega is not a ball. This exciting new technique to analyze the question of reachable sets is joint work with Alexander Strohmaier.

The colloquium will take place online in Google Meet. You can email the organizer for a link to the meeting.

This talk has been cancelled.

This talk has been cancelled.