Thesis advisor: Renato Mirollo / There are countless occurrences of oscillating systems in nature. Climate cycles and planetary orbits are a few that humans experience daily. Man has also incorporated, to his benefit, oscillation into his craft; the grandfather clock, for example, can keep track of time with astounding accuracy using the period of a long pendulum. Such systems can range in complexity in a number of ways. The governing equation for a given oscillator could be as simple as a sine curve, or its motion could appear so erratic that oscillatory motion is undetectable to viewers. The number of oscillators in a system can also vary, and oscillators can be coupled; that is, oscillators can be affected by the motion of neighboring oscillators. It is this last case we wish to study. We will briefly look at the case of finitely many oscillators and then move to analyzing a model consisting of infinitely many identical oscillators. Synchrony is the simplest collective behavior. We will study a more complicated pattern called splay states in which oscillators are equally staggered in phase, i.e. phase locked such that the system will return to this pattern if it is disturbed by an arbitrarily small amount. Mathematically, this requires us to find attracting fixed points in the system. We will approximate the local behavior of our model by linearizing the system near its fixed points. We will then apply our findings to a few specific cases of such models including: uniform density, linear distribution, alpha-function pulses, and integrate-and-fire. / Thesis (BS) — Boston College, 2013. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: College Honors Program. / Discipline: Mathematics.

With the recent development of reliable autonomous technologies for small unmanned air vehicles (UAVs), the algorithms utilizing teams of these vehicles are becoming an increasingly important research area. Unfortunately, there is no unified framework into which all (or even most) cooperative control problems fall. Five factors that affect the development of cooperative control algorithms are objective coupling, communication, completeness, robustness, and efficiency. We classify cooperative control algorithms by these factors and then present three algorithms with application to target and perimeter surveillance and a method for decentralized algorithm design. The primary contributions of this research are the development and analysis of decentralized algorithms for perimeter and target surveillance. We pose the cooperative perimeter surveillance problem and offer a decentralized solution that accounts for perimeter growth (expanding or contracting) and insertion/deletion of team members. By identifying and sharing the critical coordination information and by exploiting the known communication topology, only a small communication range is required for accurate performance. Convergence of the algorithm to the optimal configuration is proven to occur in finite-time. Simulation and hardware results are presented that demonstrate the applicability of the solution. For single target surveillance, a team of UAVs angularly spaced (i.e. in the splay state configuration) provides the best coverage of the target in a wide variety of circumstances. We propose a decentralized algorithm to achieve the splay state configuration for a team of UAVs tracking a moving target and derive the allowable bounds on target velocity to generate a feasible solution as well as show that, near equilibrium, the overall system is exponentially stable. Monte Carlo simulations indicate that the surveillance algorithm is asymptotically stable for arbitrary initial conditions. We conclude with high fidelity simulation and actual flight tests to show the applicability of the splay state controller to unmanned air systems.