Optimality, Synthesis and a Continuum Model for Collective Motion

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It is of importance to study biological collectives and apply the wisdom so accrued to modern day engineering problems. In this dissertation we attempt to gain insight into collective behavior where the main contribution is twofold. First, a `bottom-up' approach is employed to study individual level control law synthesis and emergence thereby of collective behavior. Three different problems, involving single and multiple agents, are studied by both analytical and experimental means. These problems arise from either a practical viewpoint or from attempts at describing biologically plausible feedback mechanisms. One result obtained in this context for a double agent scenario is that under a particular constant bearing pursuit strategy, the problem exhibits certain features common with the Kepler two body problem. Laboratory demonstrations of the solutions to these problems are presented. It is to be noted that these types of individual level control problems can help understand and construct building blocks for group level behaviors.

The second approach is `top-down' in nature. It treats a collective as a whole and asks if its movement minimizes some kind of energy functional. A key goal of this work is to develop wave equations and their solutions for a natural class of optimal control problems with which one can analyze information transfer in flocks. Controllability arguments in infinite dimensional spaces give strong support to construct solutions for such optimal control problems. Since the optimal control problems are infinite dimensional in the state space and one cannot simply expect Pontryagin's Maximum Principle (PMP) to apply in such a setting, the work has required care and attention to functional analytic considerations. In this work, it is shown that under a certain assumption on finite co-dimensionality of a reachable set, PMP remains valid. This assumption is then shown to hold true for the case of a specific ensemble of agents, each with state space as the Heisenberg group H(3). Moreover, analysis of optimal controls demonstrates the existence of traveling wave solutions in that setting. Synchronization results are obtained in a high coupling limit where deviation from neighbors is too costly for every agent. The combination of approaches based on PMP and calculus of variations have been fruitful in developing a solid new understanding of wave phenomena in collectives. We provide partial results along these lines for the case of a continuum of planar agents (SE(2) case).

Finally, a different top-down and data-driven approach to analyze collective behavior is also put forward in this thesis. It is known that the total kinetic energy of a flock can be divided into several modes attributed to rigid-body translations, rotations, volume changes, etc. Flight recordings of multiple events of European starling flocks yield time-signals of these different energy modes. This approach then seeks an explanation of kinetic energy mode distributions (viewed as flock-scale decisions) by appealing to techniques from evolutionary game theory and optimal control theory. We propose the notion of cognitive cost that calculates a suitably defined action functional and measures the cost to an event, resulting from temporal variations of energy mode distributions.