Figures in Latex:
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\psfig{file=sn12a0.6-0.1.eps, width=2.5in, height=0.8in}&
\psfig{file=surface.eps, width=2.5in, height=0.8in}\\
{\small Fig.5: $p_1(0)=0.6, p_2(0)=0.1, \lambda=0.001$ }&
{\small Fig.6: $x_1,x_2=-0.3, t=0, \lambda=0.001$}
\end{tabular}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\epsfxsize 2.5 in
\epsfysize 1.2 in
\centerline{\mbox{\epsfbox{cell.eps}}}
\vskip -3 mm
\caption{ A typical cell. }
\label{fi0}
\end{center}
\end{figure}
packages to be included:
\usepackage{epsfig}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{float}
command for two sided printing:
statusdict begin true setduplexmode false settumble end
%
{ % start a procedure so PostScript `ignores' the dvips options.
h config.duplong
} pop % PostScript cleanup
dvips file.dvi -o file.ps -t letter
xfig -cbg white -cfg black -metric -pap letter
gcc filename.c -o filename -lm
gcc -O3 -ffast-math -fexpensive-optimizations -m486 file.c -o file -lm
dvips -o $(FILENAME).ps -t letter -Ppdf -G0 $(FILENAME).dvi
ps2pdf -sPAPERSIZE=letter $(FILENAME).ps
The second method
dvips -Ppdf G0 filename.dvi -o filename.ps
This thesis investigates control and optimization of distributed stochastic systems motivated by current wireless applications. In wireless communication systems, power control is important at the user level in order to minimize energy requirements and to maintain communication Quality of Service (QoS) in the face of user mobility and fading channel variability. Clever power allocation provides an efficient means to overcome in the uplink the so-called near-far effect, in which nearby users with higher received powers at the base station may overwhelm signal transmission of far away users with lower received powers, and to compensate for the random fluctuations of received power due to combined shadowing and possibly fast fading (multipath interference) effects. With the wireless uplink power control problem for dynamic lognormal shadow fading channels as an initial paradigm, a class of stochastic control problems is formulated which includes a fading channel model and a power adjustment model. For optimization of such a system, a cost function is proposed which reflects the QoS requirements of mobile users in wireless systems. For the resulting stochastic control problem, existence and uniqueness of the optimal control is established. By dynamic programming, a Hamilton-Jacobi-Bellman (HJB) equation is derived for the value function associated with the stochastic power control problem. However, due to the degenerate nature of the HJB equation, the value function cannot be interpreted as a classical solution, which hinders the solution of explicit control laws or even the reliance on numerical methods. In the next step, a perturbation technique is applied to the HJB equation and a suboptimal control law using a classical solution to the perturbed HJB equation is derived. Control computation via numerical methods becomes possible and indicates an interesting equalization phenomenon for the dynamic power adjustment under an i.i.d. channel dynamics assumption. Analysis of the suboptimal control reveals an interesting bang-bang control structure which indicates simple manipulation in power adjustment. However, in view of the partial differential equations involved, implementation for systems with more than two users appears elusive. The above stochastic power control problem suggests an investigation of a wider class of degenerate stochastic control problems which are characterized both by a weak coupling condition for the components of the involved diffusion process, and by a particular rapid growth condition in the cost function. We analyze viscosity solutions to the resulting HJB equations. We develop a localized semiconvex/semiconcave approximation technique to deal with the rapid growth condition. A maximum principle is established for the viscosity subsolution/supersolution of the HJB equation and it is used to prove uniqueness of the viscosity solution. The theoretical tools thus developed serve as a mathematical foundation for our stochastic power control problem. At this point, with the aim of constructing an analytically more tractable solution to the wireless power control problem, we consider a linear quadratic optimization approach in which the power attenuation is treated as a random parameter. In this setup, the value function is expressed as a quadratic form of the vector of individual user powers, and the optimal feedback control is proved to be affine in the power. Unfortunately, the resulting control law remains too formidable to compute in large systems. However, based on the obtained analytic solution, we are able to develop local polynomial approximations for the value function and seek approximate solutions to the HJB equation by an algebraic approach under small noise conditions. Suboptimal control laws are also constructed using the approximate solutions. %Computations are scalable in this case. Remarkably, here the scheme for approximation solutions can be combined with a single user based design to construct a localized control law for each user in systems with large populations. The single user based design substantially reduces the complexity of determining the power control law. It is of significant interest to consider the asymptotics of power optimization for large population systems. In such systems, it may be unrealistic to apply the standard stochastic optimal control approach due to the complexity of implementing the centralized control law. Suboptimal but distributed control laws may be more desirable. Before proceeding to investigate this challenging issue, we first consider a large-scale linear quadratic Gaussian (LQG) model for which the agents contained in the system interact with each other either via a global cost or via related individual costs. We study both the optimal control problem based on the global cost, and the LQG game based on individual costs. For the LQG game, we develop an aggregation technique based on examining individual and mass behaviour; highly localized control strategies for all agents are obtained and a so-called $\e\mbox{-Nash}$ equilibrium property for these strategies is proved. Finally, we evaluate the loss incurred by opting for the the distributed game theoretic solution, versus the centralized optimal control solution, as measured by the associated costs differential. %At the end, the main discrepancy %between the optimal control and the LQG game is effectively revealed by %a cost gap. For the large population power control problem, apart from the centralized stochastic control approach, we also consider optimization in a game theoretic context by generalizing the techniques in the large-scale LQG problem. The combination of the individual costs and state aggregation leads to decentralized power control.