Estimation and Control in Krein Spaces with Applications to Robust Control and Machine Learning for PDE Systems
Abstract
The investigators propose to conduct a research program in collaboration with mathematicians and scientists at the Interdisciplinary Center for Applied Mathematics (ICAM) at Virginia Tech led by Professor John Burns. The current collaborations began in the spring of 2021 when Burns served on the Doctoral Thesis committee of Santiago Gonzalez Zerbo. This thesis was directed by Dr. Alejandra Maestripieri and focused on applying Krein space methods to solve indefinite optimization problems with applications to (finite dimensional) signal processing. At this time Burns observed that these methods and ideas could potentially be used for systems governed by partial differential equations. In particular, they started to have periodic Zoom meetings to discuss the mathematical challenges and potential benefits of developing a mathematical framework and corresponding computational algorithms to address robust estimation, control and optimization problems for distributed parameter (DP) systems. In 2022 Dr. Maestripieri became a Visiting Research Scholar at ICAM and during this period she worked with Burns and his colleagues to review the literature and to determine the state of the art of Krein space methods and applications. The main conclusions from this study were- -Most of the existing literature focuses on least squares estimation problems in Minkowski spaces (i.e., finite dimensional Krein spaces) (76, 77, 130) and this prevents applications to infinite dimensional systems defined by partial differential equations. -There are significant practical benefits to employing Krein space methods to robust estimation and control of finite dimensional systems. The papers (3, 44, 45, 76, 77, 94, 130, 154, 155) demonstrated how Krein spaces provide a natural framework to address robust estimation, optimization and control problems with uncertain data. Moreover, this structure can be employed to construct efficient (square root) computational algorithms and to analyze the condition numbers of these algorithms (see (76, 77, 91, 92)). -Almost all the existing research is focused on problems defined by finite dimensional dynamical systems so that these results are not applicable to robust estimation and control of systems defined by partial differential equations (PDEs). In particular, the system operators are assumed to by defined by finite dimensional matrices. There are several mathematical and computational challenges that need to be addressed in order to extend the current theories so that the results can be applied PDE systems. One of the most obvious issues is that of dealing with infinite dimensional unbounded system operators. Even if one focuses on discrete time parameter systems (see (62, 61, 63, 101, 146, 149)), the problems remain challenging because the dynamics evolve in an infinite dimensional state space. They propose to investigate the development of a rigorous Krein space mathematical framework and corresponding computational algorithms that are applicable to robust estimation, control, optimization of PDE and general DP control systems. This work will be based on the initial results found in (43, 44, 45, 64, 65, 66, 151) and by extending the computational methods in (62, 61, 63, 76, 77, 94, 130, 154, 155) to construct practical numerical methods. In addition, we are proposing to use the results of the Argentina group to extend the RKHS methods in (24, 28, 118, 119, 123, 125, 126) to a Krein space setting. Finally, they note that there has been huge advances in practical real-time computing for estimation and control of systems governed by PDEs (e.g., see (59)) and these methods will be exploited and enhanced to allow for additional applications in machine learning.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 06, 2025
- Source ID
- FA95502410433
Entities
People
- Alejandra Maestripieri
Organizations
- Air Force Office of Scientific Research
- National Scientific and Technical Research Council
- United States Air Force