**
Jonathan Epperlein**

Harold Frank Hall (HFH), Room 4164

This dissertation consists of three parts centered around the topic of spatially distributed systems.

The first part treats a specific spatially distributed system, the so-called Rijke tube, an experiment illustrating the unstable interplay of heat exchange and gas dynamics. The experiment is described and it is demonstrated how closed-loop system identification tools can be applied to obtain a transfer function model, before a spatially distributed model is developed and analyzed. The model in its most idealized form can be described in the frequency domain by a matrix of non-rational transfer functions, which facilitates analysis with classical methods such as the root locus.

The second part considers the following problem: for a given plant and cost function, could there be a finite-length periodic trajectory that achieves better performance than the optimal steady state? Termed optimal periodic control (OPC), this problem has received attention over several decades, however most available methods employ state-space based methods and hence scale very badly with plant dimension. Here, the problem is approached from a frequency-domain perspective, and methods whose complexity is independent of system dimension are developed by recasting the OPC problem for linear plants with certain memoryless polynomial nonlinearities as the problem of minimizing a polynomial.

Finally, the third part extends results for a special class within spatially distributed systems, that of spatially invariant systems, from systems defined on spaces of square-integrable functions to systems whose state space is an inner-product Sobolev space as they arise when considering systems of higher temporal order. It is shown how standard results on exponential stability, stabilizability and LQ control can be generalized by carefully keeping track of spatial frequency weighting functions related to the Sobolev inner products, and simple recipes for doing so are given.

**About Jonathan Epperlein:**

Jonathan Epperlein was awarded the Dipl.-Ing. degree in Engineering Cybernetics from Otto-von-Guericke Universität in Magdeburg, Germany in 2008, and in 2011 the M.Sc. in Electrical and Computer Engineering from UCSB, where he has been a PhD. candidate ever since. He has also spent time at the Hamilton Institute in Maynooth, Ireland, Yale University, and Harvard Medical School. His dissertation research was in modeling and control of thermoacoustic systems, spatially invariant distributed-parameter systems, and optimal periodic control of such systems.

**Hosted by: Professor Bassam Bamieh**