Sharp discontinuous traveling waves in a hyperbolic Keller-Segel equation
This talk concerns a hyperbolic model of cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (the ''pressure'') which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem.
This is a joint work with Xiaoming Fu and Pierre Magal.
Power System Modeling and Optimization to Understand Critical Energy Pathways
We are witnessing a revolutionary transition of energy infrastructure around the world. As infrastructure systems transform, they will be required to deliver critical services reliably, resiliently, securely, affordably, and in an environmentally responsible fashion. Additionally, advances in computation and connectivity have enabled emerging opportunities and challenges for system integration to support these requirements of modern infrastructure. Infrastructure modernization is disrupting the status quo of markets, planning, and operational procedures. NREL strives to address these challenges and help evaluate and understand energy infrastructure transformation across a broad technology landscape through the development and analysis of leading-edge modeling tools, solution methods, and datasets. This presentation will showcase key NREL modeling and analysis activities focusing on NRELs open-access datasets and a new suite of open-source infrastructure system models.
We examine the problem of real-time optimization of networked systems and develop online algorithms that steer the system towards the optimal system trajectory. Zero-order algorithms are considered, wherein the primal step that involves the gradient of the objective function (and hence requires networked systems model) is replaced by its zero-order approximation with two function evaluations using a deterministic perturbation signal. The evaluations are performed using the measurements of the system output, hence giving rise to a feedback interconnection, with the optimization algorithm serving as a feedback controller. We provide insights on the stability and tracking properties of this interconnection. Finally, we apply this methodology to a real-time optimal power flow problem in power systems, for reference power tracking and voltage regulation.
