## 93C20 Systems governed by partial differential equations

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Embedded in an interdisciplinary research project, the present work investigates the modeling, simulation, and optimization of specific processes for the production of functional nanoparticles. The examined material systems represent a selection of important core building blocks for future nanotechnologies. Depending on the application, a well-defined particle size distribution of the final product is required. The involved mechanisms (e.g. reaction, growth, ripening, and agglomeration) are described in the modeling by a hyperbolic partial integro-differential equation coupled to one or more ordinary differential equations. The successful validation against experimental data combined with the techniques of optimal control theory thereby allow for a targeted manipulation of the underlying processes. The developed methods establish in many cases for the first time a systematic approach for the production of tailor-made nanoparticles.

We study the stabilization of quasilinear hyperbolic PDE systems on networks. To do so we develop stabilizing boundary feedback controls. We study feedback controls without and with time-varying delays. To analyze the system evolution we introduce strict Lyapunov functions which are weighted and squared L2- and H1-norms of the solutions of the quasilinear systems. The exponential weights in the Lyapunov functions contain the system eigenvalues. The feedback controls with time delay are represented by delay terms in the Lyapunov functions. The boundary controls yield an exponential decay of the corresponding Lyapunov function on a given finite time interval. Furthermore, we prove the exponential decay of the L2- and H1-norms of the solutions and give exponential estimates for the C0- and C1-norms. We apply our results on the stabilization of the isothermal Euler equations with friction, a hyperbolic system of balance laws that models the gas flow through pipes. For this system we analyze stationary states and classical nonstationary solutions. We present methods to stabilize the isothermal Euler equations locally around a given stationary state both on a single pipe and on fan-shaped pipe networks with compressor stations.

We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.