Project
EF-Dec-Supp-3
Moving Sensor and Actuator Design for Dynamical Systems
Project Heads
Tobias Breiten, Daniel Walter
Project Members
Sophie Gehricke
Project Duration
01.10.2025 – 30.09.2028
Located at
HU Berlin, TU Berlin
Description
Predicting a spatiotemporal process u from scarce measurements, the so-called state estimation problem, is a classical yet nevertheless extremely relevant task situated at the intersection of engineering sciences and mathematics. Examples span from crack monitoring in bridges via stationary sensor grids to predicting the track of airborne chemical agents or wildfires by deploying micro air vehicles. From a mathematical perspective, the described problem can often be modeled by a linear dynamical system, describing the evolution of u in a Hilbert space H of functions on a spatial domain, which we couple with a measurement model describing the sensors.
Naturally, the success of this estimation process will critically rely on the measurement setup. To emphasize, calculating the mean-squared error of the process u and estimated process no longer depends on the actual realization of the measurements but only on the number of employed sensors and their trajectories. As a consequence, an optimal sensor setup, i.e. one that minimizes the uncertainty of the estimate, can be determined a priori by solving a mathematical program.
This approach can be put into the statistical field of optimum experimental design. In the context of linear system theory, similar problems were considered and have remained a vivid and interdisciplinary research area until today. Regarding its practical realization, we note that the problem of finding an optimal sensor setup is both non-convex and combinatorial. For stationary sensors applying the celebrated continuous design approach alleviates these issues by searching for an optimal measurement design in the set of probability measures rather than optimizing individual sensors. In the present project we aim at extending this approach by proposing a continuous design approach for moving sensors which relies on dynamical optimal transport regularization and favors measurement setups consisting of finitely many sensors on continuous trajectories.
As a starting point for our investigation we consider parabolic systems driven by controls represented as finite sums of moving point sources. After formulating the abstract framework, we compare different regularization approaches: an (L^2)-based regularization leading to measurable trajectories, and an optimal transport-inspired regularization promoting (H^1) trajectories. To solve the resulting problems, we adapt a generalized conditional gradient method that exploits the geometric structure of the regularizers. This is complemented by a consistent discretization scheme relying on an FE-approximation of the underlying dynamical system as well as a semi-variational discretization of the measure space.
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