Georgia Institute of Technology College of Sciences

In many applications, such as vibration of smart structures (piezoelectric, magnetorestrive, etc.), the physical quantity of interest depends both on the space an time. These systems are mostly modeled by partial differential equations (PDE), and the solutions of these systems evolve on infinite dimensional function spaces. For this reason, these systems are called infinite dimensional systems. Finding active controllers in order to influence the dynamics of these systems generate highly involved problems. The control theory for PDE governing the dynamics of smart structures is a mathematical description of such situations. Accurately modeling these structures play an important role to understanding not only the overall dynamics but the controllability and stabilizability issues. In the first part of the talk, the differences between the finite and infinite dimensional control theories are addressed. The major challenges tagged along in controlling coupled PDE are pointed out. The connection between the observability and controllability concepts for PDE are introduced by the duality argument (Hilbert's Uniqueness Method). Once this connection is established, the PDE models corresponding to the simple piezoelectric material structures are analyzed in the same context. Some modeling issues will be addressed. Major results are presented, and open problems are discussed. In the second part of the talk, a problem of actively constarined layer (ACL) structures is considered. Some of the major results are presesented. Open problems in this context are discussed. Some of this research presented in this talk are joint works with Prof. Scott Hansen (ISU) and Kirsten Morris (UW).

Collective movement is one of the most prevailing observations in nature. Yet, despite considerable progress, many of the theoretical principles underlying the emergence of large scale synchronization among moving individuals are still poorly understood. For example, a key question in the study of animal motion is how the details of locomotion, interaction between individuals and the environment contribute to the macroscopic dynamics of the hoard, flock or swarm. The talk will present some of the prevailing models for swarming and collective motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics regarding the build-up and maintenance of collective order in swarms. In particular, whether order and disorder correspond to different phases, requiring external environmental changes to induce a transition, or rather meta-stable states of the dynamics, suggesting that the emergence of order is kinetic. Different aspects of the phenomenon will be presented, from experiments with locusts to our own attempts towards a statistical physics of collective motion.

In this talk, I will introduce B-series, which are formal series indexed by trees, and briefly expose the two laws operating on them. The presentation of algebraic aspects will here be focused on applications to numerical analysis. I will then show how B-series can be used on two examples: modified vector fields techniques, which allow for the construction of arbitrarly high-order schemes, and averaging methods, which lie at the core of many numerical schemes highly-oscillatory evolution equations. Ultimately and if time permits, I will illustrate how these concepts lead to the accelerated simulation of the rigid body and the (nonlinear) Schrödinger equations. A significant part of the talk will remain expository and aimed at a general mathematical audience.

We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along an unknown path with subsonic velocity, and that data is collected over time on some detection surface. We explore the question of uniqueness for these problems, and show how to recover the times and locations of sources microlocally first, and then the smooth part of the source assuming that it is the same at each source location. In case the sources (now all different) are (roughly speaking) non-negative and of limited oscillation in space, and sufficiently separated in space-time, which is a model for microseismicity, we present an explicit reconstruction, requiring sufficient local energy decay. (Joint research with L. Oksanen and J. Tittelfitz)