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Plenary Speakers
Mean Field Games and the Control of Large Scale Systems: An Overview
Peter Caines
McGill University
http://www.cim.mcgill.ca/~peterc/
Tuesday, July 12, 09:00-10:00, Willey 175
Abstract: Mean Field Game (MFG) theory provides tractable strategies for the decentralized control of large scale systems. The power of the formulation arises from the relative tractability of its infinite population McKean-Vlasov (MV) Hamilton-Jacobi-Bellman equations and the associated MV-Fokker-Planck-Kolmogorov equations, where these are linked by the distribution of the state of a generic agent, otherwise known as the system's mean field. The resulting decentralized feedback controls yield approximate Nash equilibria and depend only upon an agent’s state and the mean field. Applications of MFG theory are being investigated within engineering, finance, economics and social dynamics, while theoretical developments include existence and uniqueness theory for solutions to the MFG equations, major-minor (MM) agent systems containing asymptotically non-negligible agents, non-linear estimation theory for MM-MFG systems, and the comparison of centralized (optimal) control and MFG control performance.
Presentation Slides
Rapid stabilization
Jean-Michel Coron
Pierre and Marie Curie University
http://www.ann.jussieu.fr/coron/
Wednesday, July 13, 09:00-10:00, Willey 175
Abstract: We start by presenting some old results on the "rapid" (including in finite time) stabilization of control systems modeled by means of ordinary differential equations. We show the interest and the limitation of the damping method for the stabilization of control systems. We then describe various methods to construct feedback laws leading to rapid stabilization. These methods will be applied to some control systems modeled by means of either nonlinear differential equations or partial differential equations (Korteweg-de Vries, hyperbolic, Schroedinger and heat equations).
The “power network” of genetic circuits: Hidden interactions and their mitigation through distributed feedback control
Domitilla Del Vecchio
Massachusetts Institute of Technology
Thursday, July 14, 09:00-10:00, Willey 175
Abstract: The past several years have witnessed a substantial increase in the size of synthetic genetic circuits that are inserted in living cells for biotechnology applications ranging from energy, to environment, to medicine. These circuits are “powered” by cellular resources, which are in limited amounts. As the circuit size increases, resource limitations cannot be neglected any longer and in fact they can even lead to dramatic failures, just like blackouts in the power grid. In this talk, I will illustrate that because nodes in a network compete for limited resources, hidden interactions appear that dramatically change the expected network’s behavior. I will present a systematic modeling framework that captures hidden interactions in resource-limited networks and provides simple graphical rules to draw emergent interaction graphs. With the aim of modularly designing synthetic genetic circuits in which the effects of hidden interactions is mitigated, I will illustrate a distributed feedback control scheme that makes each node robust to fluctuations of available resources. I will present initial experimental results from bacterial cells, which validate some of the theoretical predictions. These results improve the predictive power of current mathematical models of gene networks and present a first step toward the engineering of genetic circuits that are robust to unavoidable resource fluctuations. I will finish the talk by highlighting a number of research questions that remain open due to the challenging mathematical structure of these systems.
Presentation Slides
Optimal transportation between unequal dimensions
Robert J. McCann
University of Toronto
http://www.math.toronto.edu/mccann/
Friday, July 15, 09:00-10:00, Willey 175
Abstract: Over the last few decades, the theory of optimal transportation has blossomed into a powerful tool for exploring applications both within and outside mathematics. Its impact is felt in such far flung areas as geometry, analysis, dynamics, partial differential equations, economics, machine learning, weather prediction, and computer vision. The basic problem is to transport one probability density onto other, while minimizing a given cost c(x,y) per unit transported. In the vast majority of applications, the probability densities live on spaces with the same (finite) dimension. After briefly surveying a few highlights from this theory, we focus our attention on what can be said when the densities instead live on spaces with two different (yet finite) dimensions. Although the answer can still be characterized as the solution to a fully nonlinear differential equation, it now becomes badly nonlocal in general. Remarkably however, one can identify conditions under which the equation becomes local, elliptic, and amenable to further analysis.
Presentation Slides
Adventures in imaging
Peter Olver University of Minnesota
http://www.math.umn.edu/~olver/
Friday, July 15, 14:00-15:00, Willey 175
Abstract: I will survey developments in the application of invariants of various types, including invariant histograms and differential invariant signatures, for object recognition and symmetry detection in digital images. Recent applications, including automated jigsaw puzzle assembly and cancer detection, will be presented.
Semi-Plenary Speakers
Controllability of systems defined on graphs
Kanat Camlibel
University of Groningen
http://www.math.rug.nl/~kanat/
Tuesday, July 12, 14:00-15:00, Willey 175
Abstract: The study of networks of dynamical systems became one of the most popular themes within systems and control theory in the last two decades. This talk focuses on controllability of networks consisting of identical dynamical systems. For such networks, the overall dynamics determined by the (identical) dynamics of the individual systems as well as the graph capturing the network structure. First, we will focus on a particular system defined over graphs, namely diffusively coupled leader/follower systems. These systems admit models in which graph Laplacians play an important role. By studying certain partitions of the underlying graph, we will provide purely graph theoretical necessary as well as sufficient conditions for the controllability of diffusively coupled leader/follower multi-agent systems. After that, we shift our attention to more general graph related matrices than Laplacians and look at the problem of minimal leader selection, that is rendering the overall network controllable by choosing as few as possible number of leaders.
Presentation Slides
Reconstruction of interconnectedness in networks of dynamical systems based on passive observations
Murti Salapaka University of Minnesota
Abstract: Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. In this talk, a methodology for identifying the interrelatedness structure of dynamically related time series data based on passive observations will be presented. The framework will allow for the presence of loops in the connectivity structure of the network. The quality of the reconstruction will be quantified. Results on the how the sparsity of multivariate Wiener filter, the Granger filter and the causal Wiener filter depend on the network structure will be presented. Connections to graphical models with notions of independence posed by d-separation will be highlighted.
Presentation Slides
Resilient control of network flow dynamics
Giacomo Como
Lund University
http://www.control.lth.se/Staff/GiacomoComo.html
Thursday, July 14, 14:00-15:00, Willey 175
Abstract: In the control of large network systems efficiency, resilience, and scalability are key issues. We focus on distributed control of network flow dynamics, governed by routing and flow control policies within constraints imposed by the network structure and physics laws. Depending on the application (e.g., road transport or distribution networks), such policies are meant to represent local routing and scheduling controls, users’ behavior, or a combination of the two. The considered models include cascading failures mechanisms, whereby overloaded links become inactive and potentially induce the overload and failure of other nodes and links in the network. First, we focus on throughput and resilience properties of decentralized feedback policies that use local information only and require no global knowledge of the network. Then, we discuss cases in which optimal network flow control can be cast as a convex problem which is amenable to iterative distributed computation solutions. Finally, we deal with multi-scale flow dynamics and the use of incentive mechanisms to influence users' behaviors. Throughout, we illustrate how structural properties of the dynamics, such as monotonicity, contraction, and convexity, can be leveraged to obtain tractable models whose performance can be related to connectivity and other network properties.
Presentation Slides
Mathematics and swimming of aquatic organisms
Marius Tucsnak
University of Bordeaux
http://www.math.u-bordeaux1.fr/imb/fiche-personnelle?uid=mtucsnak
Thursday, July 14, 14:00-15:00, Willey 125
Abstract: Understanding the motion of aquatic organisms is a problem which fascinated scientists and philosophies for centuries. Our aim consists in showing that this modelling problem raises highly challenging questions for various fields of mathematics, such as partial differential equations, numerical analysis or control theory. We begin by emphasizing that, depending on the flow regime, the governing equations may exhibit a wide range of properties, leading to a rich mathematical structure. We next discuss wellposedness issues, where the major difficulty to be solved consists in tackling the presence of free boundaries. Finally we study the displacement of the solids (under the action of an exterior force or in a self-propelled manner) from a control theoretical perspective.
Presentation Slides
Linear Matrix Inequalities and Interpolation Problems for CP Maps
University of Auckland
Igor Klep
https://www.math.auckland.ac.nz/~igorklep/
Thursday, July 14, 14:00-15:00, Blegen 150
Abstract: Linear matrix inequalities (LMIs) are common in many areas: control systems, mathematical optimization, statistics, etc. The solution set of an LMI is called an LMI domain (or a spectrahedron) and is a convex subset of the euclidean space. LMIs often have unknowns which are scalars, but in many problems the unknowns enter naturally as matrices rather than scalars.
There is a class of maps naturally associated to LMI domains, the completely positive (cp) maps. A linear map is positive if it maps positive semidefinite (psd) matrices to psd matrices, and is completely positive (cp) if each of its ampliations is positive. (Completely) positive maps are ubiquitous in matrix theory and mathematical physics. For instance, trace-preserving cp maps, i.e., quantum channels, are fundamental objects in quantum information theory, while unital cp maps are central to understanding LMIs and their matrix solution sets.
In this talk we show some theory for cp maps recently developed for LMIs and analogous results for quantum channels, e.g. a tracial Hahn-Banach theorem and a quantum interpolation theorem.
The talk is based on joint work with Bill Helton and Scott McCullough.
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