# Control and filtering of multiscale systems

## Description

Model reduction is a major issue for control, optimization and simulation of large-scale systems. Common spatial decomposition methods such as Proper Orthogonal Decomposition or the Principal Component Analysis aim at identifying a subspace of “high-energy” modes onto which the dynamics are projected. These modes, however, may not be relevant for the dynamics. Moreover these methods tacitly assume that all degrees of freedom can be observed or measured.

An alternative procedure for stable input-output systems is balanced model reduction. Unlike the aforementioned approaches balancing accounts for incomplete observability. It consists in finding a coordinate transformation such that modes which are least sensitive to an external perturbation (controllability) also give the least output (observability) and therefore can be neglected. In this project we study balanced model reduction in the context of singularly perturbation theory for both deterministic and stochastic control systems, with a particular focus on the backward stability of the reduced models.

## Funding

The project is funded through the Einstein Center for Mathematics Berlin (ECMarth).

## Selected Publications

- C. Hartmann, J. Latorre, G. Pavliotis and W. Zhang (2014), Optimal control of multiscale systems using reduced-order models, J. Computational Dynamics 1, 279-306
- C. Hartmann, A. Zueva and B. Schäfer-Bung (2013), Balanced model reduction of bilinear systems with applications to stochastic control, SIAM J. Control Optim. 51, 2356-2378
- C. Hartmann, V.-M. Vulcanov, Ch. Schütte (2010), Balanced truncation of linear second-order systems: A Hamiltonian approach, Multiscale Model. Simul. 8, No. 4, pp. 1348 – 1367.