Topics in Optimal Control and Game Theory
We consider a number of “non-classical” problems in optimal control and game theory, with special attention to random perturbations, trade-offs due to conflicting objectives, and possible adversarial interactions of many selfish agents. First, we consider models of pedestrian flow, with the goal of guaranteeing that anisotropic interactions in the models were handled in an internally consistent manner. We focus on two classes of models, one based on inter-crowd interactions and another based on intra-crowd interactions. For both cases, we prove sufficient conditions under which the direction of motion is determined uniquely almost everywhere in the domain. For the inter-crowd models, we also prove sufficient conditions under which the Nash Equilibrium is guaranteed to be unique. Next, we investigate time-dependent Surveillance-Evasion Games. These are modeled as a semi-infinite zero-sum game between two players: an Observer who chooses a probability distribution over a pre-defined set of patrol trajectories, and an Evader who chooses a proba- bility distribution over all possible trajectories from a starting location to a given target set. We develop efficient numerical methods for finding the Nash Equilibrium of these games. We also formulate control-theoretic models of environmental crime in protected areas, such as wildlife poaching or illegal logging. We carefully model the optimal control problem faced by extractors aiming to illegally extract resources from a protected area such as a national park. The extractors are assumed to rationally choose their paths in and out of the protected area in order to maximize their expected payoffs, balancing the resources obtained from extraction with their travel time and risk of detection by authorities. Finally, we look into uncertainty quantification in piecewise-deterministic Markov pro- cesses (PDMPs), where the dynamics of the system switch between deterministic modes at random times. We develop efficient numerical methods for approximating the cumulative distribution function (CDF) of the exit time, and bounds on this CDF when the transition rates between modes are only partially known. For controlled PDMPs, we further develop efficient numerical methods for optimizing this CDF.
Game Theory; Numerical Analysis; Optimal Control Theory; Uncertainity Quantification
Vladimirsky, Alexander B.
Townsend, Alex John; Levine, Lionel
Ph. D., Mathematics
Doctor of Philosophy
Attribution 4.0 International
dissertation or thesis
Except where otherwise noted, this item's license is described as Attribution 4.0 International