Mathematical Sciences: Sequential Experimentation and Adaptive Control

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This research will study three problems. The first set is called bandit problems in sequential design and scheduling. A multi-armed bandit problem selects between k independent stochastic sequences at every stage n according to a specified rule. The "index rule" selects a sequence with maximum "Gittins index". These indices are not easily computable. The investigator will continue to develop simple approximations to the optimal stopping problems. The second set of problems uses the boundary crossing theory for sample means to find useful approximations to certain Bayes tests. The investigator has already found such approximations for general exponential families. Now he proposes to extend the results beyond the exponential families and to higher dimensions. The third set of problems deals with stochastic regression and adaptive control. The classical stochastic adaptive control is the optimal regulation of the output when it is designed as a linear combination of previous p outputs and q inputs. The output is regulated according to various criteria such as mimimizing the sum of squares etc.. The investigator already has several results in this area and he plans to pursue these problems further.