In recent years, pursuit-based feedback control laws have helped realize complex spatio-temporal behaviors of robot collectives by utilizing relative information (e.g. optic flow) of the target with respect to the pursuer. For instance, these algorithms can enable a team of Unmanned Aerial Vehicles (UAVs) perform search, rescue and surveillance. However, such platforms are far from being completely autonomous and frequently require human intervention to reset the goals for the mission midstream, to be accomplished by choosing one from a pool of control laws. While this can ensure achievement of very specific goals over a short duration, such as reaching a search location and performing motions to cover an annular region around it, there is a need to autonomously generate high level goals especially in the face of adverse or unexpected events. This requires using sensory information gathered from the environment in which the agents operate to decide the next course of action. The broad aim of this thesis is to establish a mathematical framework to enable a collective of robotic agents, each with a finite set of actions to choose from, arrive at a cognitive decision that is justified by aggregated evidence. We motivate the use of models from evolutionary game theory, particularly the replicator dynamics, to model the evolution of the probabilities associated with choosing each action.

We take inspiration from neuroscience for realizing context-dependent decision making by means of a three-layer cognitive hierarchy operating at multiple timescales. We show how evolutionary game theory offers a natural framework to model this hierarchy. In particular, replicator dynamics associated to fitness maps capture the evolution of a finite number of population fractions or probabilities that grow depending on the fitness or reward obtained for each population type. In the present setting, we interpret the types as synonymous with strategies implemented by feedback laws and the decision of an autonomous agent as represented by probabilities over its strategies. This formulation can be used to realize a combination of available control laws that will enable the agent to achieve its goal. In the bottom layer are the dynamics of an agent which responds to external stimuli from the physical environment at a fast timescale by a combination of its feedback laws. In the intermediate layer is the replicator dynamics evolving in a comparatively slower timescale, in which the decision making that goes behind choosing the feedback law in the lower layer is updated using knowledge of the fitness of each strategy. In the top layer evolving at the slowest timescale, we consider replicator control systems specified by control laws that seek to realize context dependence (cognition) at the higher level.

The contributions of this thesis are in all three layers of the cognitive hierarchy, explained through a top-down approach. We first consider the top layer by extending the replicator dynamics to a replicator control system whose controls vary the fitness of strategies in a time-dependent manner. We show a Lie algebraic structure in the space of fitness maps. We exploit this mathematical structure in the dynamics to modulate the fitness so that an arbitrary final set of probabilities can be attained from an initial state. In the process, we determine the associated controllability conditions. In the intermediate layer, we highlight an optimizing property of the replicator dynamics by showing that it satisfies first order necessary conditions for optimality for an appropriate cost function. In the bottom layer, we consider the interpretations of mixed strategies in the agent's physical world. An instance of dyadic pursuit in which the pursuer aims to capture a target using the motion camouflage pursuit strategy while trading off the accuracy of sensory information for the speed of response to the stimuli is explored.

In the final part of this thesis, we consider a cognitive description of starling flocks by treating each flock as a single decision-making entity. We use observations made from several flocking events and formulate a data smoothing problem using the game-theoretic formulation in this thesis to understand the temporal evolution of fractions of the relative kinetic energy allocated to the different behavioral modes. We propose a function, the optimal cost arising out of the solution to an underlying optimal control problem, as a measure of cognitive effort involved in producing these behaviors. Lastly, we conclude with a discussion on ongoing work, some challenges and future research directions.