Tolgahan Tasci | Searcharxiv

arXiv2026

Well-posedness and passivity for a class of bilinear control systems

We study an abstract class of bilinear infinite-dimensional systems that arises in various applications, including district heating systems and quantum control. First, we analyze the existence of mild and classical solutions for abstract bilinear systems with admissible input operators in Banach spaces and study continuous dependence on the data. When the underlying space is a Hilbert space, these results are used to show passivity for a suitably defined co-located output and for mild solutions. The results are applied to the bilinear Schrödinger equation, the Fokker--Planck equation, and a district heating or cooling pipe.

math.AP↗

arXiv2025

A second order approach to the Kato square root problem on open sets

We obtain the Kato square root property for coupled second-order elliptic systems in divergence form subject to mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^n$ under two simple geometric conditions: The Dirichlet boundary parts for the respective components are Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the remaining Neumann boundary parts. In contrast to earlier work, our proof is not based on the first-order approach due to Axelsson--Keith--McIntosh but uses a second-order approach in the spirit of the original solution to the Kato square root problem on Euclidean space. This way, the proof becomes substantially shorter and technically less demanding.

math.FA↗