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This monograph applies the relative optimization approach to time
nonhomogeneous continuous-time and continuous-state dynamic
systems. The approach is intuitively clear and does not require
deep knowledge of the mathematics of partial differential
equations. The topics covered have the following distinguishing
features: long-run average with no under-selectivity, non-smooth
value functions with no viscosity solutions, diffusion processes
with degenerate points, multi-class optimization with state
classification, and optimization with no dynamic programming. The
book begins with an introduction to relative optimization,
including a comparison with the traditional approach of dynamic
programming. The text then studies the Markov process, focusing on
infinite-horizon optimization problems, and moves on to discuss
optimal control of diffusion processes with semi-smooth value
functions and degenerate points, and optimization of
multi-dimensional diffusion processes. The book concludes with a
brief overview of performance derivative-based optimization. Among
the more important novel considerations presented are: the
extension of the Hamilton-Jacobi-Bellman optimality condition from
smooth to semi-smooth value functions by derivation of explicit
optimality conditions at semi-smooth points and application of this
result to degenerate and reflected processes; proof of
semi-smoothness of the value function at degenerate points;
attention to the under-selectivity issue for the long-run average
and bias optimality; discussion of state classification for time
nonhomogeneous continuous processes and multi-class optimization;
and development of the multi-dimensional Tanaka formula for
semi-smooth functions and application of this formula to stochastic
control of multi-dimensional systems with degenerate points. The
book will be of interest to researchers and students in the field
of stochastic control and performance optimization alike.
Dynamic Systems (DEDS) are almost endless: military C31 Ilogistic
systems, the emergency ward of a metropolitan hospital, back
offices of large insurance and brokerage fums, service and spare
part operations of multinational fums . . . . the point is the
pervasive nature of such systems in the daily life of human beings.
Yet DEDS is a relatively new phenomenon in dynamic systems studies.
From the days of Galileo to Newton to quantum mechanics and
cosmology of the present, dynamic systems in nature are primarily
differential equations based and time driven. A large literature
and endless success stories have been built up on such Continuous
Variable Dynamic Systems (CVDS). It is, however, equally clear that
DEDS are fundamentally different from CVDS. They are event driven,
asynchronous, mostly man-made and only became significant during
the past generation. Increasingly, however, it can be argued that
in the modem world our lives are being impacted by and dependent
upon the efficient operations of such DEDS. Yet compared to the
successful paradigm of differential equations for CVDS the
mathematical modelling of DEDS is in its infancy. Nor are there as
many successful and established techniques for their analysis and
synthesis. The purpose of this series is to promote the study and
understanding of the modelling, analysis, control, and management
of DEDS. The idea of the series came from editing a special issue
of the Proceedings of IEEE on DEOS during 1988.
Performance optimization is vital in the design and operation of
modern engineering systems, including communications,
manufacturing, robotics, and logistics. Most engineering systems
are too complicated to model, or the system parameters cannot be
easily identified, so learning techniques have to be applied. This
book provides a unified framework based on a sensitivity point of
view. It also introduces new approaches and proposes new research
topics within this sensitivity-based framework. This new
perspective on a popular topic is presented by a well respected
expert in the field.
Performance optimization is vital in the design and operation of
modern engineering systems, including communications,
manufacturing, robotics, and logistics. Most engineering systems
are too complicated to model, or the system parameters cannot be
easily identified, so learning techniques have to be applied. This
book provides a unified framework based on a sensitivity point of
view. It also introduces new approaches and proposes new research
topics within this sensitivity-based framework. This new
perspective on a popular topic is presented by a well respected
expert in the field.
The theory of the operation of many modern man-made discrete event
systems such as manufacturing systems, computer and communications
networks largely belongs in the domain of queuing theory and
operations research. However, some recent research indicates that
the evolution of these man-made systems demonstrates dynamic
features that are similar to those of natural physical systems.
This monograph presents a multidisciplinary approach to the study
of discrete event systems, and is complementary to textbooks in
queuing and control systems theories.
Dynamic Systems (DEDS) are almost endless: military C31 Ilogistic
systems, the emergency ward of a metropolitan hospital, back
offices of large insurance and brokerage fums, service and spare
part operations of multinational fums . . . . the point is the
pervasive nature of such systems in the daily life of human beings.
Yet DEDS is a relatively new phenomenon in dynamic systems studies.
From the days of Galileo to Newton to quantum mechanics and
cosmology of the present, dynamic systems in nature are primarily
differential equations based and time driven. A large literature
and endless success stories have been built up on such Continuous
Variable Dynamic Systems (CVDS). It is, however, equally clear that
DEDS are fundamentally different from CVDS. They are event driven,
asynchronous, mostly man-made and only became significant during
the past generation. Increasingly, however, it can be argued that
in the modem world our lives are being impacted by and dependent
upon the efficient operations of such DEDS. Yet compared to the
successful paradigm of differential equations for CVDS the
mathematical modelling of DEDS is in its infancy. Nor are there as
many successful and established techniques for their analysis and
synthesis. The purpose of this series is to promote the study and
understanding of the modelling, analysis, control, and management
of DEDS. The idea of the series came from editing a special issue
of the Proceedings of IEEE on DEOS during 1988.
This Springer brief addresses the challenges encountered in the
study of the optimization of time-nonhomogeneous Markov chains. It
develops new insights and new methodologies for systems in which
concepts such as stationarity, ergodicity, periodicity and
connectivity do not apply. This brief introduces the novel concept
of confluencity and applies a relative optimization approach. It
develops a comprehensive theory for optimization of the long-run
average of time-nonhomogeneous Markov chains. The book shows that
confluencity is the most fundamental concept in optimization, and
that relative optimization is more suitable for treating the
systems under consideration than standard ideas of dynamic
programming. Using confluencity and relative optimization, the
author classifies states as confluent or branching and shows how
the under-selectivity issue of the long-run average can be easily
addressed, multi-class optimization implemented, and Nth biases and
Blackwell optimality conditions derived. These results are
presented in a book for the first time and so may enhance the
understanding of optimization and motivate new research ideas in
the area.
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