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Topics in the Geometric Theory of Linear Systems (Interdisciplinary Mathematics Series 22): Robert Hermann

With this volume, I start a series aimed at developing new geometric methodology to be used in engineering and the physical sciences. This book deals with certain aspects of the theory of linear systems, which of course is the backbone of the applied subject.

While the engineers were developing the theory of linear systems in the 1960s, and early 70s, themathematicians were perfecting the beautiful theory of pseudodifferential and Fourier integral operators. My ultimate goal is to build a bridge between then using as one foundation, a geometric, Grassmann approach to the ``symbol'' developed by Clyde Martin and myself. In this volume, I do not even begin this job; an intermediate problem has been to incorporate the earlier work of Mikusinski into the work of contemporary control theory.

It is also clear that Sato's theory of hyperfunctions has many points of contact with linear systems and control theory, and will be very useful when someone does the (thankless!) job of modifying it to serve the needs of engineering problems. Sato himself is a mathematical physicist, and the theory of motivated by the goal of mechanising the way that physicists use distribution theory, e.g., in the Feynman integral calculus of quantum field theory. Convesely, I believe that in the Mikusinski theory, and its extensions into modern linear system theory, offers many interesting problems and insights into the ``geometric analysis.''

Another foundational topic I have and to think about - and my preliminary thoughts are in this volume - is the theory of scattering. Systems theoretically, one is dealing not with input-output relations in the usual form described in the elementary treatises, but with some more complicated sort of relation. The physicist of the 1950's who developed scattering theory in its modern form had some such version in his mind - for example, they talked about ``in states" and ``out states" - but the subject as it has developed since has diverged from these roots. Given my ``geometric" bias, I have developed in embryo form, a Grassmannian approach, in parallel with the Grassmannian approach to input-output linear systems developed by Martin and myself. I believe that geometric ways of thinking about what I have called scattering systems will be useful in wider areas than mathematical physics, e.g., the estimation problems of stochastic system theory.

My recent work has involved the pseudogroup and associated geometric structure ideas which I learned as a Princeton graduate student in the 1950's. In retrospect, this material was potentially applicable to a wide variety of situations originating in important scientific and engineering disciplines, but there was then no mechanism for combining these insights with the highly nontrivial and difficult, long term work necessary to use them effectively. My efforts to pursue my applied interests outside of the milieu of academic mathematics certainly received no encouragement or understanding from my peers of the system then in place! It seems that the mathematical work is now belatedly learning some of these lessons, but in the adverse circumstances of financial constraints and competition for a limited pool of talented people from thriving fields like computer science.

In 1953-1955 I studied with Ehresmann and Spencer. Ehresmann had a brilliant picture in his mind, which he could only partially communicate in print to his colleagues, of how the classical work (particularly that of Lie, Vessiot and Cartan) could be integrated into modern mathematics. Unfortunately, Ehresmann never wrote an authoritative version of his ideas as they might apply to differential geometry in the conventional sense. His mathematical intuition was mainly topological and algebraic. Spencer, cming from a background in analysis, saw most profoundly how to utilize these ideas (plus others from the theory of sheaves and the theory of elliptic differential operators) into a machinery which could carry out the Lie, Vessiot, and Cartan ideas (particularly in the context of "pseudogroups") in amodern framework.

I believe that It is the Ehresmann-Spencer material that is at the foundation of much of the successful advance of geometry into the applied world. Thus, it is unfortunate that no adequate exposition of it exists! Since my own career as an impure mathematician has placed me on the research frontiers in both control theory and physics, I have tried to steer a course which would keep in contact withboth disciplines. With the support Of Brian Doolin and George Meyer of the Ames Research left of NASA I have, since 1975, been able to concentrate my efforts on the long term mathematical ideas and techniques which might be useful in the control world, while also having the stimulus through Ames of contact with control problems as they are perceived by the working engineer.

In the Preface to Volume 21 I mentioned that I planned to channel more of my work to journal articles. In fact, this has absorbed most of my energy for the last four years, and a collection of papers I have published in the Journal of Mathematical Physics since 1980 should serve to indicate to the reader the character in which this, and future volumes, will go, pursuing directions which have been opened up in these papers.

Again, I would like to thark Kari Young for her typing and editorial work.

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