This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This project centers on control of dynamical systems governed by nonlinear hyperbolic partial differential equations: (i) stabilization of electromagnetic fields via nonlinear feedbacks restricted to a subset of the boundary or the interior of the domain; (ii) boundary control of structure-acoustic interactions with the elastic component described by the Reissner-Mindlin plate model; (iii) stabilization of acoustic noise from structures vibrating under influence of electromagnetic fields; (iv) stability and attractors for wave equations with memory terms and nonlinear damping. One of the primary goals is to investigate, in the context of these systems, control and energy dissipation mechanisms that are restricted in some sense: Either geometrically to a portion of the physical domain, and/or in the "strength" of the feedback as in under- and over-damped systems. The work will also address equations with non-dissipative controls (e.g. sheer force feedbacks for Euler-Bernoulli beams and Kirchhoff plates) which, in some cases, are more suitable for implementation, but whose energy-damping effects are not apparent and can only be studied via specialized techniques. The project is aimed at establishing the conditions on geometry, initial data, and structure of the controls necessary to steer/stabilize the system or, at least, ensure certain properties of the global attractors.
This research is expected to constructively impact engineering design in control of distributed parameter systems (e.g. acoustic and mechanical vibrations, thermal effects, electro-magnetic fields). Maxwell equations and stabilization of electromagnetic radiation arise in antenna design, nonlinear optics, semiconductor-superconductor modeling. Structure-acoustic interaction problems show up in a variety of areas ranging from active noise control to design of smart materials. In particular, study of acoustic-magneto-elastic coupling helps understand how to minimize noise from the gradient coils in magnetic resonance imaging (MRI) devices. It is desirable to engineer controls that are minimally invasive and convenient for implementation, thus, prompting investigation of actuators and energy dampers that are restricted in space and their strength. The goal is to establish conditions under which such mechanisms provide sufficient control over the system, or to quantify their deficiencies when full effectiveness is unattainable. This work will also connect with applied numerical analysis both in research and in education: Supplementary projects for undergraduate and graduate course will be developed offering a lower-level introduction to numerical methods for studying partial differential equations; a longer-term objective will be to augment some aspects of the above research with numerical algorithms in order to facilitate practical applications of these theoretical results.
