Monique Chyba - US grants

In the design of controllers for governed physical processes, a central problem is to minimize a given quantity (time, amount of fuel, etc.) and to find a systematic way to characterize optimal trajectories. Due to technical and computational difficulties, it is impossible in most cases to find the optimal trajectories for a given criterion. In addition, often the situation is complicated by the existence of more than one quantity to be minimized.

This research project investigates a new approach to such minimization problems with two costs to be minimized. We do not try to find an optimal synthesis for a given cost. Instead, we start with the set of all extremals for one cost, say time, as guaranteed by the maximal principle. Then, using higher order necessary conditions coming from the maximum principle, coupled with differential geometric techniques, we reduce the set of candidate extremals to optimality for this cost to some significantly smaller set of extremals. We then minimize the other cost, say energy, over this smaller set of candidate time minimizers. We focus our method to apply to the very important class of controlled mechanical systems. The Lie algebra generated by the vector fields describing such systems possesses commutativity properties that make our computations possible.

The major application of our theory is the control of robotic underwater vehicles, which is an area of intense current interest among oceanographers, geophysicists, et al. The strategy described above is based on promising results previously obtained on controlling the motion of such a vehicle. The methods under development will be incorporated into a working vessel through collaboration with the University of Hawaii College of Engineering Autonomous Systems Laboratory. The results of the project will provide enhanced research tools to scientists in a wide variety of disciplines.

Robotic underwater vehicles are intended to be versatile and reliable tools designed, for instance, to intelligently and independently detect and monitor oceanographic phenomena, to collect samples, or to work on minesweeping where human intervention would be too risky. Unmanned underwater vehicles can be sent on long duration missions, can take measurements over large distances and variations in depth, and are not dependent on human skill for realization of mission goals. This research project develops mathematical control theory for direct application to improvement of robotic underwater vehicle capabilities.

A very important class of control systems, even though they are non-generic, is the class of controlled mechanical systems. Examples of these systems include the planar rigid body with a single variable direction thruster, the snakeboard. Trajectory design problems for such systems are of particular interest in this project. The main application considered in this project is the control of a submerged rigid body. Clearly, this application is particularly well adapted to analysis, both due to the practical motivation coming from the recent trend to build autonomous underwater vehicles and for more mathematically oriented reasons. Indeed, an underwater vehicle can be modeled as a simple mechanical control system, with dissipative forces. A major goal is to establish a mathematical formulation of the switching time parametrization algorithm, developed in a previous work, based on the differential geometric properties of the system such as the notion of decoupling vector field. The key notions involved in this project are the ones of decoupling vector fields for invariant systems on a Lie group, and the notion of singular extremals coming from optimal control. Indeed, a recent observation concerns a possible relationship between singular extremals of order greater than 1 and decoupling vector fields. The goal is to develop a proper generalization for the notion of decoupling vector field for forced affine-connection control systems.

In the exploration of our world's oceans, there comes a time when human controlled vehicles are not sufficient. Whether in the context of a deep ocean survey or a long term monitoring project, autonomous underwater vehicles are the natural choice to take the burden. Like any other machine, we need the ability to accurately control the underwater vehicle to perform even the simplest of tasks. We propose to combine our theoretical methods with experiments to show the efficiency of our algorithms versus the classic model-free controllers developed in the autonomous underwater vehicle field. The impact of the proposed research will reach several communities: mathematics, engineering and oceanography. Another major concern of the project is the involvement of graduate and undegraduate students at every level of the research as well as the development of outreach educational programs for younger generations.

ABSTRACT FOR THE NSF PROPOSAL GK-12: SUPER-M
PI: MONIQUE CHYBA
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HAWAII

The School and University Partnership for Educational Renewal in Mathematics project (SUPER-M) will bring the knowledge and expertise of research mathematicians into K?12 classrooms, making an important contribution to improving school mathematics in Hawaii. The project will provide K?12 students with an enriching learning environment where mathematics is interesting and dynamic. The project will also contribute to the formation of a cadre of highly qualified teachers, bringing new mathematics expertise to schools throughout the State of Hawaii and helping to sustain the program. Over five years, 38 Fellows will be selected to partner with K?12 schools on Oahu, the Big Island and Maui. Fellows will take a semester long course on ?Issues in K?12 Mathematics Education,? learning about best practices in the design of professional development courses for teachers. Fellows will design and lead workshops for K?12 teachers arising from their areas of research. Upon completion of the course, Fellows will partner with a cooperating teacher, creating and leading mathematics activities for K?12 students. In this way, SUPER-M will provide K?12 students with a solid grounding in mathematics, increasing their opportunities to pursue careers in STEM disciplines. SUPER-M will serve under-represented populations by placing a special emphasis on recruiting Native Hawaiian and women Fellows. SUPER-M expects to profoundly impact the community at large through special events such as summer camps, public outreach events, and family math days.

Recent experimental projects in quantum control using finite-dimensional systems as the control of spin systems in nuclear magnetic resonance and references therein) are motivating new theoretical studies in the case where the system interacts with its environment. The primary objective of the proposed research is to apply techniques of geometric optimal control theory to the control of the spin dynamics by magnetic fields in Nuclear Magnetic Resonance (NMR). Through interaction with a magnetic field, NMR involves the manipulation of nuclear spins. It has many potential applications extending from the determination of molecular structures (NMR spectroscopy) and quantum computing, where NMR remains one of the most promising road in the construction of a scalable quantum computer, to medical imagery (MRI). The control technology developed over the past 50 years allows the use of sophisticated control fields and permits the implementation of complex quantum algorithms such as the Deutsh-Jozsa and the Grover algorithms. NMR is therefore an ideal experimental testbed for quantum control. The proposed research will also impact the domain of quantum mechanics. First, solving the contrast imaging problem can potentially have a profound impact on how medical imaging is done in hospitals. Indeed, by designing magnetic fields to maximize the distance between the two spin we increase the image resolution and therefore improve its quality which improves patient care. Second, by using geometric techniques our approach will complement existing efficient numerical tools for pulse sequence optimization, such as the GRAPE (gradient ascent pulse engineering) by providing an understanding of the qualitative structures of the dynamics of the system. In particular, the physicists will gain insight about the control mechanism that lead to the optimal solutions.

Starting as a tool for characterization of organic molecules, the use of nuclear magnetic resonance (NMR) has spread to areas as diverse as pharmaceutics, medical diagnostics (medical resonance imaging) and structural biology. The principles of NMR have served as a paradigm for other physical methods that rely on interaction between radiation and matter. It is therefore not surprising that experiments in NMR also serve as good model problems in control of quantum systems. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. Until now, the analysis of nonlinear optimal control has been mostly concerned with the class of single-input systems. Via this application in quantum control, it is proposed to extend the analysis to the multidimensional case. The proposal research will also impact significantly the field of quantum mechanics. First, medical imagery can be drastically improved through the contrast problem which will directly impact patient care in hospitals. Second, by using geometric tools to develop optimal synthesis, we provide physicists with an additional understanding about the control mechanism of the optimal solutions which in turn will help developing improved numerical schemes. We also propose an initiative to address the underrepresentation of minorities, and particularly of Native Hawaiian women, in the fields of science, technology, engineering, and mathematics (STEM disciplines) at UH Manoa and nationwide. The goal is to address the dual barriers of gender and ethnicity that Native Hawaiian females face in these fields.

In the midst of the COVID-19 pandemic the state of Hawaii, being an archipelago, is in an exclusive position to carry out measures no other state could do ? it essentially sealed its borders to virtually all travel-related infections including inter-island ones by instituting a two-week quarantine of incoming air, water, and inter-island passengers, thus providing a critical data set that can help researchers understand the spread of the virus and the effectiveness of mitigation and isolation strategies. Hawaii also tracks the currently limited arrivals onto the various islands, and this collection of information will continue as mitigation levels change. This project will use the unique data from Hawaii to provide a predictive understanding of the virus through modeling of spread and mitigation effects, focusing on a critical gap in understanding variability of COVID-19 spread within different communities and a lack of dynamic modeling. Incorporation of data sets from a controlled environment will greatly enhance predictive understanding and enable mitigation approaches with better certainty based on real data. The project will use advanced computational techniques to make the models run efficiently and make them readily available to the public and decision makers involved in the COVID-19 response strategy.

Many current COVID-19 models only consider a totality of the population of any given state/county and do not take into account patterns of spatial activities or specificity of the region under consideration. This project will implement new dynamic and computationally-optimized models that incorporate compartmentalized populations to study variability in the spread of the disease as well as rapidly changing mitigation strategies. These elastic models are easily adaptable to different environments and employable locally and around the world, thus helping to minimize the negative effects of COVID-19 on public health at a global level. Use of discrete compartmentalized epidemiological models, as well as models based on spatio-temporal stochastic processes, can take into account different population communities distinguished through a variety of attributes that potentially affect the susceptibility of individuals to the disease. Such enhanced granularity will improve predictive capability of the models and provide better insights into the spread of COVID-19. The project will also engage students thus providing training for the future generation of researchers in data-driven sciences using a critical and urgent topic.

This project is jointly funded by CCF Division Software and Hardware Foundations Program and the Established Program to Stimulate Competitive Research (EPSCoR).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.