In this paper, I describe the process and results of my study on the flight trajectory optimization of a continuously flying solar aircraft. Continuous flight is achieved by cyclic operation, where the trajectory is repeated indefinitely, typically every 24 hours. The word continuously is used in the theoretical sense, as continuous or perpetual flight is not achievable in practice due to degradation of batteries and aircraft components over time. The importance of flight trajectory optimization has been recognized in both general aviation and space applications. The prevalent class of algorithms for solving these problems are largely sequential in nature, where the differential equations that describe flight motion are solved in an inner loop while an outer loop performs the optimization of the control variables. These methods can be computationally expensive as they require repeated solution of the differential equations for each guess of the control variable in addition to calculation of gradients for the optimizer … In this research, I built upon a simultaneous solution method called orthogonal collocation on finite elements to develop a robust trajectory optimization system with an effective initialization strategy …

I am fascinated by problems that require a blend of computational topology, geometry, and number theory. I have also been studying fractals which interesting geometrical objects that have been used in diverse applications such as cryptography, seismology, network optimization, and even weather forecasting. However, despite the wide range of applications and interest in fractals, the general theory of these objects is still in its infancy. My work on this research project has developed some theorems and conjectures in the field of combinatorics and has begun to shed some light on some areas of fractals, one-cell automata and dynamical systems … This need for optimization has become increasingly more important in today’s society from the perspective of both resource management as well as leveraging new opportunities. In terms of resource allocation, combinatoric optimization is being used to improve the efficiency of scheduling transportation (the traveling salesperson problem) to allocating scarce resources (such as militaryequipment or food distribution), through improving internet network traffic throughput, latency, and infrastructure costs. But the field has broader impact than just efficient resource allocation as it can more help in more efficiently processing large amounts of data. Increasingly, we are producing more information that we can efficiently sort through and understand, whether it is the 100k plus tweets per minute of the Presidential debates, the information gathered about global warming, or the data mining of consumer information.