Optimal Decisions with Limited Information

Abstract: This thesis considers static and dynamic team decision problems in both stochastic and deterministic settings. The team problem is a cooperative game, where a number of players make up a team that tries to optimize a given cost induced by the uncertainty of nature. The uncertainty is modeled as either stochastic, which gives the stochastic team problem, or modelled as deterministic where the team tries to optimize the worst case scenario. Both the stochastic and deterministic static team problems are stated and solved in a linear quadratic setting. It is shown that linear decisions are optimal in both the stochastic and deterministic framework. The dynamic team problem is formulated using well known results from graph theory. The dynamic interconnection structure is described by a graph. It appears natural to use a graph theoretical formulation to examine how a decision by a member of the team affects the rest of the members. Conditions for tractability of the dynamic team problem are given in terms of the graph structure. Tractability of a new class of information constrained team problems is shown, which extends existing results. For the presented tractable classes, necessary and sufficient conditions for stabilizability are given. The state feedback $mathcal{H}_2$ and $mathcal{H}_{infty}$ dynamic team problems are solved using a novel approach. The new approach is based on the crucial idea of disturbance feedback, which is used to separate the controller effect from the measured output, to eliminate the controller's dual role. Finally, a generalized stochastic linear quadratic control problem is considered. A broad class of team problems can be modeled by imposing quadratic constraints of correlation type. Also, power constraints on the control signals are very common. This motivates the development of a generalized control theory for both the finite and infinite horizon case, where power constraints are imposed. It is shown that the solution can be found using finite dimensional convex optimization.