Optimality, Synthesis and a Continuum Model for Collective Motion
Udit Halder, Ph.D.
University of Maryland
Special Seminar
Thursday, June 27, 2019
11:00 a.m. – 12:00 p.m.
301 CSL
Abstract:
It is of importance to study biological collectives and apply the wisdom so accrued to modern day engineering problems. 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. These problems arise from either a practical viewpoint or from attempts to describe biologically plausible feedback mechanisms. A striking result will be presented for a double agent scenario in which we have been able to show that under a particular constant bearing pursuit strategy, the problem shares certain features with the Kepler two body problem. 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 takes 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 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 specific example of an 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. Partial results along these lines for the case of a continuum of planar agents (SE(2) case) will be presented.
We will present another top-down and data-driven approach to analyze collective behavior. 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.
Biography: Udit Halder received M.S. and Ph.D. degrees in Electrical and Computer Engineering, both from the University of Maryland, USA in 2018 and 2019, respectively. Prior to joining UMD, he received his Bachelors in Electronics and Telecommunication Engineering from Jadavpur University, India in 2013. His research interests include geometric control theory, collective behavior, and robotics.
