4,332 research outputs found
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
Koopman operator-based model reduction for switched-system control of PDEs
We present a new framework for optimal and feedback control of PDEs using
Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a
linear but infinite-dimensional operator which describes the dynamics of
observables. A numerical approximation of the Koopman operator therefore yields
a linear system for the observation of an autonomous dynamical system. In our
approach, by introducing a finite number of constant controls, the dynamic
control system is transformed into a set of autonomous systems and the
corresponding optimal control problem into a switching time optimization
problem. This allows us to replace each of these systems by a K-ROM which can
be solved orders of magnitude faster. By this approach, a nonlinear
infinite-dimensional control problem is transformed into a low-dimensional
linear problem. In situations where the Koopman operator can be computed
exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach
yields optimal control inputs. Furthermore, a recent convergence result for
EDMD suggests that the approach can be applied to more complex dynamics as
well. To illustrate the results, we consider the 1D Burgers equation and the 2D
Navier--Stokes equations. The numerical experiments show remarkable performance
concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
Aggregation and Control of Populations of Thermostatically Controlled Loads by Formal Abstractions
This work discusses a two-step procedure, based on formal abstractions, to
generate a finite-space stochastic dynamical model as an aggregation of the
continuous temperature dynamics of a homogeneous population of Thermostatically
Controlled Loads (TCL). The temperature of a single TCL is described by a
stochastic difference equation and the TCL status (ON, OFF) by a deterministic
switching mechanism. The procedure is formal as it allows the exact
quantification of the error introduced by the abstraction -- as such it builds
and improves on a known, earlier approximation technique in the literature.
Further, the contribution discusses the extension to the case of a
heterogeneous population of TCL by means of two approaches resulting in the
notion of approximate abstractions. It moreover investigates the problem of
global (population-level) regulation and load balancing for the case of TCL
that are dependent on a control input. The procedure is tested on a case study
and benchmarked against the mentioned alternative approach in the literature.Comment: 40 pages, 21 figures; the paper generalizes the result of conference
publication: S. Esmaeil Zadeh Soudjani and A. Abate, "Aggregation of
Thermostatically Controlled Loads by Formal Abstractions," Proceedings of the
European Control Conference 2013, pp. 4232-4237. version 2: added references
for section
Self-Evaluation Applied Mathematics 2003-2008 University of Twente
This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008
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