Vorträge im SoSe 2024

In most cases, one shows the stability of a nonlinear system by the construction of a suitable Lyapunov function for it. However, if the system is a complex large-scale or potentially infinite network of nonlinear agents, direct analysis of the stability of such a system becomes very challenging. The problem becomes even more complex, if the number of agents, which are acting in the network is unknown. Nonlinear small-gain theory for finite networks of ODE systems has provided efficient methods to counteract these challenges and has become the basis for the powerful methods of nonlinear controller design. In this talk, we present the full generalization of the Lyapunov-based nonlinear small-gain theorem to the case of countably many interacting systems. We use Lyapunov methods combined with the analysis of nonlinear monotone systems on a positive cone of certain sequence spaces. If the gain operator is homogeneous and subadditive, we prove constructive results giving an explicit expression for the Lyapunov function of the infinite network. Though we focus on infinite ODE networks, we expect that the results can be extended to infinite networks consisting of infinite-dimensional components, and we have developed some machinery for this case as well.

Controllability of multidimensional (n-D) systems is defined in terms of patchability, over distant open sets, of arbitrary pairs of trajectories. For systems described by linear constant coefficient PDEs, there are algebraic tests to ascertain controllability. One of these tests, the Hautus test, dictates that controllability is equivalent to a certain algebraic variety having its dimension to be strictly less than n -1. When this particular algebraic variety is empty, the system exhibits a special feature of admitting an observable image representation. We argue that this phenomenon is only a special case of a gradation in controllability distinguished by the dimension of the said algebraic variety. In the talk we shall explore this idea of gradation of controllability with an emphasis on the special case of n=2.

Speaker bio: Debasattam did his Masters and PhD from the Indian Institute of Technology (IIT) Bombay, India in 2007 and 2012, respectively. After spending two years in IIT Guwahati as an Assistant Professor, he joined IIT Bombay, where he currently holds the position of an Associate Professor. At present, Debasattam is visiting the Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen as a guest researcher. Debasattam’s areas of interest include algebraic-geometric analysis of n-D systems, optimal control, multi-agent systems, switched systems and dissipativity theory.