Dear President Braatz, colleagues, students and friends.
I am very grateful and indeed humbled by being honored to receive the Richard E. Bellman Control Heritage Award for 2019 and to join the distinguished list of prior recipients. I wish to express my sincerest thanks to those who nominated me and supported my nomination and to the awards committee. I am deeply moved by the honor I receive today.
More as a rule than an exception, such an honor is not a credit to a single individual but rather the result of collective work and many collaborations over the years. This is particularly true in areas which are by nature interdisciplinary. And control theory, as such, is one of these. It offers an excellent example of synergy where purely theoretical questions, mathematical in nature, are prompted and stimulated by technological advances and engineering design.
I was attracted to mathematical control theory from my early days at the University of Warsaw, where I was privileged to join a distinct and (at that time) experimental program, called Studies in Applied Mathematics. This was an interdisciplinary initiative under the collaboration of a few home departments. After graduating with a Master Degree, I was fortunate to receive a doctoral fellowship which allowed me to complete my PhD in Applied Mathematics-Control Theory within 3 years, with a thesis on a problem of non-smooth optimization, which extended Milutin-Dubovitski's work and had applications to control systems with delays.
I am extremely grateful to my mentors of that time: Professors A. Wierzbicki, A. Manitius from Control Theory [the latter now chair at George Mason University], the late Professor S. Rolewicz and Professor K. Malanowski both from the Polish Academy of Sciences. They, along with other colleagues, gave me an opportunity to embrace a large spectrum of the field of control theory, to include functional analysis, abstract optimization, differential equations.
My further education took a critical turn at UCLA in Los Angeles, which I joined in 1978, at the invitation of the late Professor A.V. Balakrishnan, the 2001 recipient of the Bellman's award. Bal for all of us. Here, under his mentorship, I was offered the challenge to get involved in the mathematical area of boundary control theory for Distributed Parameter Systems, still at its infancy at that time, even from the viewpoint of Partial Differential Equations, with many basic mathematical problems still open. That was about the time when Richard Bellman's book on Dynamic Programming appeared, in 1977, rooted on Bellman's equation and the Optimality Principle. I always looked at Bellman as a problem-solving mathematician, and the mathematical theory of boundary control of DPS is in line with this philosophy.
Controlling or observing an evolution equation from a restricted set [such as the boundary of a multi-dimensional bounded domain where the controlled system evolves] is both a mathematical challenge and a technological necessity within the realm of practical and physically implementable control theory. Most often, the interior of the domain is not accessible to external manipulations. One first goal of the time within the DPS control community was to construct an appropriate control theory, inspired also by the late R. Kalman, the 1997 recipient of the Bellman's award. Main initial contributors were J.L. Lions, A. Bensoussan and their influential school in Paris, and A.V. Balakrishnan and his associates. But DPS come in a large variety. It requires that each distinct class (parabolic, hyperbolic, etc.) be studied on its own with properties and methods pertinent to it, which however fail for other classes. The systematic study of boundary control, which leads to distributional calculus for various distinct classes of physically significant DPS, became the first long-range object of my research. Both, the results and the methods are dynamics dependent. Finite or infinite speed of propagation becomes an essential feature in controllability theory. For instance, the wave equation is boundary exactly controllable on a sufficiently large time, while the heat equation is only null-controllable yet on an arbitrary short time. Existence, uniqueness and robustness of solutions to nonlinear dynamics were just the first questions asked but still open within the existing PDE culture.
Topics investigated over the years included: optimal control, Riccati and H-J-Bellman theory and their numerical implementation, appropriate controllability and stabilization notions, all in the framework of boundary control of partially observed systems. This research effort, which continues to this very day, was conducted with collaborators and PhD students. It started with my association with A.V. Balakrishnan at UCLA, J.L. Lions at College de France and R. Kalman during my 7 years at the University of Florida. And it continued during my subsequent 26 years at the University of Virginia, the home of MacShane, and now at the University of Memphis. In both cases with talented PhD students. Some of these occupy now distinguished positions in the US academia.
Once the control theory of single distinct DPS classes became mature, engineering applications motivated the need to move on toward the study of more complex DPS consisting of interactive structures where different types of dynamics coupled at an interface define a given control system. Propagation of control properties through the interface then plays a main role.
Thus, in its second phase, my research in DPS then evolved toward these coupled interactive systems of several PDEs. Applications include large flexible structures, structural acoustic interaction, fluid-structure interaction, attenuation of turbulence in fluid dynamics [Navier Stokes] and flutter suppression in nonlinear aero-elasticity. In the latter area, my collaboration with Earl Dowell [Duke Univ.] was most enlightening, and is a further proof of the interdisciplinary nature of the field. These problems, while deeply rooted in engineering control technology, were also benchmark models at the forefront of developing a PDE-based mathematical control theory, which accounts for the infinite dimensional nature of continuum mechanics and fluid dynamics.
In closing, I would like to acknowledge with gratitude my personal and professional interaction over the years with people such as the late David Russell [VPI], Walter Littmann [U of Minnesota], Giuseppe Da Prato [Scuola Normale, Pisa], Michel Delfour [Univ. of Montreal] and Sanjoy Mitter [MIT], the latter the 2007 recipient of the Bellman award. Their pioneering works paved the way to further developments along a road-map which I am proud to be a part of.
Special thanks to my long-time collaborator and husband Roberto Triggiani, to the late Igor Chueshov [both co-authors of major research monographs, two with Roberto in Cambridge University Press and one with Igor in Monograph Series of Springer], as well as to my former students, now collaborators and colleagues.
Many thanks also to funding agencies such as NSF, AFOSR, ARO and NASA for many years of generous support.
Philadelphia, July 11, 2019.