Measure theoretic optimization in information theory and stochastic control subject to uncertainty

In the first chapter, two general classes of optimization problems are discussed. The application of these problems in information theory and stochastic control contexts is the main goal of this thesis. In chapters two and three, applications in source coding are discussed, then in the last two chapters, the applications of these problems in control of uncertain continuous time systems are explored.; Chapter two is concerned with lossless source coding for a family of source distributions whose relative entropy with respect to a given nominal distribution is bounded above by some fixed number. First the minimax average length source coding problem subject to the relative entropy constraint is considered. The minimizing players are the codeword lengths, while the maximizing players are the uncertain source distributions. This leads to coding with respect to the exponential pay-off. Second the minimax redundancy problem subject to relative entropy constraints is considered. It is shown that this problem reduces to an extension of the exponential Huffman coding.; In the third chapter, the problem of rate distortion formulation for abstract sources is considered. We introduce a general framework for rate distortion theory in Polish spaces. Previous well known results for abstract alphabets are extended to the more general case in which marginal measures on the reproduction space may not be absolutely continuous with respect to the optimal marginal measure. Moreover, the question of existence of a solution to the implicit equation of optimal distribution is addressed by proving a fixed-point theorem.; In chapter four, the problem of stochastic control for uncertain systems is considered. Here we aim at establishing explicit connections between the Legendre-Fenchel transform, robustness of uncertain systems, and risk-seeking and risk-averse optimization. The connections are obtained by introducing new classes of optimal stochastic uncertain systems in which the uncertainty is described by a family of probability measures, which satisfy a certain fidelity criteria. With respect to this formulation, the minimization of the relative entropy over the set of uncertain measures, subject to the fidelity criteria is shown to be equivalent to the Legendre-Fenchel transform.; In the last chapter, another class of stochastic optimal uncertain control systems are considered, in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures, while the pay-off is a linear functional of the uncertain measure. The theory is developed in an abstract setting and then applied to nonlinear partially observable continuous-time uncertain controlled systems.