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Abstract
Stochastic control models are a class of mathematical models that have applications invarious fields. They are widely used in applications like autopilot, robotics, and financial
economics due to their ability to handle uncertainty in decision-making processes. The naive
stochastic control models estimate noise distributions directly from the dataset, which makes
it vulnerable to both erroneous data points and over-fitting. While distributionally robust
optimization techniques have been developed in recent years to tackle one-stage problems,
due to the complexities involved in multiple stages, it is not sufficiently developed in control
theory. The focus of this thesis is to develop distributionally robust control with statistical
methods.
In this thesis, we will cover four topics relating to distributionally robust control. We will
begin by discussing the framework of distributionally robust control, bridging the gap between
classical risk-averse control and Wasserstein control. Then, we dive into risk-averse control
and generalize the classical risk-averse linear quadratic Gaussian control to the mixture of
Gaussian scenarios. We will discuss scenarios with and without uncertainty on components’
probability. While the latter can be solved with a closed-form solution, the former is more
complex due to the curse of dimensionality. We propose a relaxation of the former and prove
a minimax theorem for it. Following that, we introduce group lasso into distributionally
robust control as a tool for outlier robustness. We provide a controller that is selectively
robust on high-influential points. It can also handle scenarios where erroneous data points are
present in the dataset. We also discuss the most general nonlinear non-Gaussian risk-averse
control, which may not be solvable but tractable with a sequential approximation.