Georgia Institute of Technology College of Sciences

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor. 

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

The talk is about controllability for uncertain linear systems. Our approach is 
based on the Controllability Function (CF) method proposed by V.I. Korobov in 
1979. The CF method is a development of the Lyapunov function method and the 
dynamic programming method. The CF includes both approaches at a certain values 
of parameters. The main advance of the CF method is finiteness of the time of motion 
(settling-time function). 
In the talk the feedback synthesis problem for a chain of integrators system 
with continuous bounded unknown perturbations is considered. This problem consist 
in constructing a control in explicit form which depends on phase coordinates and 
steers an arbitrary initial point from a neighborhood of the origin to the origin in a 
finite time (settling-time function). Besides the control is satisfies some preassigned 
constrains. The range of the unknown perturbations such that the control solving the 
synthesis problem for the system without the perturbations also solves the synthesis 
problem for the perturbed system are found. This study shows the relations between 
the range of perturbations and the bounds of the settling-time function.
In particular the feedback synthesis problem for the motion of a material 
point with allowance for friction is solved. 
Keywords: chain of integrators, finite-time stability, robust control, settling 
time estimation, uncertain systems, unknown bounded perturbation

Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.