Topics of projects for prospective research students.
Convex Analytic Approach to certain controlled Markov chains and jump
processes.
This project is mainly theoretical. As is known, the Convex Analytic
approach, dual to the Dynamic Programming approach, is convenient for the
study of constrained problems. Currently, there are several particular
models for which this approach is still not developed. (For instance,
absorbing Markov Decision Processes with Borel spaces and jump processes
with unbounded intensities.) The objective is to obtain and investigate the
characteristic equation for occupation measures, find necessary and
sufficient conditions of optimality and describe sufficient classes of
policies in multicriteria problems.
Analysis of communication networks.
The project involves mathemtical analysis of several Active Queue Management models of TCP connection. A typical model is the continuous
time Queuing System with a varying input stream of jobs (customers).
Moreover, one can control the input intensity, and the goal is to find an optimal
control policy. Generally speaking, such models are "piece-wise
deterministic" processes, and optimal control of them is at the cutting
edge of modern control theory. Another project in this field involves "Call
Admission Control" where the Decision Maker must decide whether
to accept or reject the newly arrived customer (job).
Approximations of controlled birth-and-death processes.
Such processes (with discrete or continuous time) appear in Queuing Theory,
Reliability, Epidemiology and many other areas. In some cases, the
underlying model can be approximated by a diffusion, or by a deterministic
ordinary differential equation (e.g. so called "Fluid Model" of a queuing
system). The objective of the project is to evaluate the accuracy of such
approximations and to elaborate optimal control policies for them.
To succeed in the project and to enjoy your study you must have good
background in at least two areas out of the following:
1. functional analysis (metric spaces, Lebesgue integrals and
so on);
2. standard calculus (knowledge of differential equations, ordinary and
partial, is an advantage);
3. basic probability (knowledge of random processes is an advantage);
4. methods of optimisation (knowledge of methods for constrained problems is
an advantage);
5. computer programming (knowledge of mathematical software like MATLAB and MAPLE is an advantage).
Books to read:
1. D.Burghes, A.Graham. Control and Optimal Control Theories with
Applications. Horwood Publ. Limited, 2004, ISBN: 190427501X
2. H.M.Taylor, S.Karlin. An Introduction to Stochastic Modeling. Academic
Press, 1998, ISBN: 0126848874
3. M.L.Puterman. Markov Decision Processes. Wiley, 1994, ISBN: 0471619779
4. S.M.Ross. Applied Probability Models with Optimization Applications.
Dover Publ. 1992, ISBN: 0486673146
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