Control, Optimization and Transmission Problems for Nonlocal PDEs
Abstract
This main focus of the present research proposal is to make a significant contribution to the area of moving control, optimization and transmission problems of concrete models involving nonlocal Partial Differential Equations (PDEs) by addressing some key issues that remain poorly understood or unsolved. These problems appear systematically in real-life applications including but not limited to viscoelasticity, electrical signal propagation in cardiac tissue, phase filed models, fluid dynamics, diffusion of biological species, denoising algorithms, industrial processes like cooling/heating in steel and glass manufacturing, nonlocal image processing, anomalous transport and diffusion. Of concern in this research proposal are three specific models that have recently emerged as the main area of interest for several researchers all over the world, due to their importance in applications. More precisely we will be concerned with the observatibility of nonlocal PDEs with moving control, optimization of nonlocal parabolic problems and some nonlocal interface problems, in which we have a significant expertise. 1. The classical theory of controllability of PDEs consists to drive the solutions of a dynamical system to rest by means of the action of an applied control. Here we are mainly concerned with infinite-dimensional models from Continuum or Quantum Mechanics, modeled in terms of nonlocal PDEs and with a moving control. One important motivation for this kind of control is for example that the exact controllability of the wave equation with a pointwise control and Dirichlet boundary conditions fails if the point is a zero of some eigenfunction of the Dirichlet Laplacian, while it holds when the point is moving under some much more stable conditions. 2. Optimal control for local parabolic problems with state and gradient constraints have been most recently studied by using PDEs methods to show existence. Space-time discretizations, error estimates, optimality conditions and the validity of the obtained results by using some numerical methods, have been recently investigated. The corresponding problem for the nonlocal case remains unsolved and is one of the main objective of the present proposal. 3. The quasi-geostrophic system modeling large-scale atmospheric motion induced by the rotation of the Earth is given by a nonlocal transmission problem. The complete analysis of such problems is in its infancy. The main goal here is to contribute to fill this significant gap. We propose to study the long-term asymptotic behavior of such systems in terms of global attractors and ?-limit sets. The approach will be based on the rich and fruitful mathematical apparatus offered by PDEs, functional analysis, controllability, optimization and numerical analysis. However, the methods need to be substantially modified to deal with the nonlocal terms. This proposal shall also contribute to the training of graduate students. The considered problems take their origin in science and engineering and the treatment provided will avail the users with new ways to deal with issues encountered in material science, the aerospace industry and elsewhere. Well trained and highly qualified personnel in the Science, Technology, Engineering and Mathematics (STEM) disciplines are critical to the national defense mission.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 09, 2020
- Source ID
- W911NF2010115
Entities
People
- Mahamadi Warma
Organizations
- Army Contracting Command
- United States Army
- University of Puerto Rico