# Women in Control: New Trends in Infinite Dimensions (17w5123)

Arriving in Banff, Alberta Sunday, July 16 and departing Friday July 21, 2017

## Organizers

(Universidad Nacional Autonoma de Mexico)

(University of Waterloo)

(University of Memphis)

## Objectives

This workshop will have a very specific focus; control of partial differential equations (PDE's). The technical content of this workshop will be on several open issues in control of infinite-dimensional systems: controllability for nonlinear systems, estimation and related computational issues. These issues are strongly connected and rely upon developments in other areas of mathematics such as analysis, geometry, topology. A synergistic effect, cross-fertilization and interdisciplinary approach has an immense and beneficial effect on the development of control theory and in the sciences in general. The proposed workshop has several inter-connected objectives: to show young women interested in this field that they are not alone, introduce young female graduate students to potential future advisers and collaborators, as well as to increase the participation of women in research activities in control of PDE's in the long term. The second objective is to deepen worldwide networks of collaboration by inviting women distributed worldwide, particularly in Latin America, that will represent all levels of the career path: graduate students, postdoctoral fellows, women in mid-career and senior women. The participants will have the opportunity to interact through informal discussions and engage in scientific exchanges. Below, we expand on these objectives and how we intend to accomplish them.

Foster new research in control of infinite-dimensional systems. The panorama of applications of PDE's is huge, ranging from classical ones in engineering, such as the design of structures, to the more modern such as nanotechnology, biology, and medicine.

In general terms, existence results can be viewed as a confirmation of the correctness of the model. Uniqueness, regularity and stability results are related to the usefulness of the model. The control of systems governed by PDE's (observability, stabilization, exact controllability, optimal control, inverse problems and related identification theory) is a quickly growing area of mathematics.The complexity of systems (specifically, when nonlinear or delay terms appear and when dealing with vectorial structures) leads to many nontrivial difficulties and, consequently, many interesting questions remain open.

Our aim is gather scientists working in the general area of control of partial differential equations but representing different expertise and different points of view. This will promote cross-fertilization of ideas across artificial boundaries. The sharing of techniques and viewpoints should lead to new results in this important area. One area of interest is controllability and observability. For ordinary differential equations, most questions have been settled, starting the work of Kalman in the 1960's. Strong stability is the same as uniform stability and approximate controllability coincides with exact controllability. However, in the infinite-dimensional framework, topology enters the picture and basic concepts become subtle. It takes some thinking to understand why the heat equation is typically approximately controllable but not exactly controllable. The lack of well-posedness backward in time of the underpinning PDE is very relevant. On the other hand wave equation is often exactly controllable. Again, has to do with forward and backward propagation of waves in time. What happens when one deals with a control system which combines both parabolic and hyperbolic effects? Which characteristics dominate? And where to place controls in order to maximize a desired objective? These problems arise in interconnected systems such as modeling of fluid-flow structure interactions or acoustic structure interactions. and are of current interest. There are recognized computational difficulties in obtaining controllers for infinite-dimensional systems. These are particularly acute for problems involving weakly damped waves and vibrations. Finite-dimensional approximations of these systems often miss many key features, introduce spurious modes or do not reconstruct well (uniformly in the parameter of discretization) the underlying dynamics. Similar concerns apply to approximations of delay equations, which also evolve on infinite-dimensional space.

A frontier area of research is in control of nonlinear infinite-dimensional systems. The systems theory and the design approaches are in their infancy. The existing systems theory for infinite-dimensional systems is largely based on $L_2$-inputs and outputs. However, the theory for nonlinear finite-dimensional systems, on input-to-state stability, relies on $L_\infty$-inputs and outputs. One promising approach to the design of controllers for nonlinear infinite-dimensional systems is the extension of dissipative systems theory to infinite-dimensional systems. In this approach, the physics underlying many models is utilized to provide a formal mathematical statement of dissipation. This dissipation may be used in controller design to obtain closed loops that remain stable despite modelling errors and uncertainties.

Encourage more women to pursue academic careers, and help the success of those in academia. There are a number of senior women working in control of infinite-dimensional systems, most of whom are listed as participants. Thus, we are able to have a workshop for female mathematicians that is scientifically focused. The aim is to invite about half junior women, so that there will be a good mix of attendees from the graduate to full professor level. It is hoped that interactions with senior women will show the younger women, some of whom will be the only female in their group that they are not alone. It may also encourage more women to pursue academic careers. At the very least, attendees have invaluable networking opportunities with a number of current and future colleagues that will be beneficial in furthering their careers. The truly international representation of female scientists provides another benefit of transporting the best practices in a field that is often localized geographically.

Promote connections between Latin America and North America/Europe. There is increased scientific activity in Latin American in recent years, partly due to a concerted effort by a number of governments in that region to promote research and more broadly a high quality university system. However, academics in that region continue to be disadvantaged by various factors. One of the organizers of this workshop is from Mexico and a number of researchers from Latin America will be included amongst the attendees. The Mathematical Congress of the Americas will be held in Montreal, July 23 to 28, 2017. We hope to hold this workshop shortly before or after this congress.

Structure. The proposed theme is topical, dynamic and it perfectly fits within the current trend of worldwide scientific research. It is hoped that the talks, both formal and informal, at this meeting will inspire research along new lines and stimulate new collaborations. In order to achieve the overlapping goals listed above, the schedule will be a mix of talks and small informal discussion groups. The latter will aid participants in becoming acquainted with each other. We will also have at least one careers session, with such topics are grant applications, managing family commitments, and becoming involved in professional organizations such as editorial boards and SIAM.

This meeting is partially supported by NSF-HRD 1500481-AWM ADVANCE grant