The present research project aims at investigating the properties of control, optimization, stability and long-time behavior of interactive structures, which are mathematically modeled by inhomogeneous systems of strongly coupled partial differential equations with an interface. Both the single partial differential equations describing each a constitutive component of the overall structure as well as the coupling between them, may be linear or non-linear, dispersive, and oscillatory.
Investigation will be carried out initially on four canonical motivating classes of interactive structures. They are intended to serve as benchmark cases for the general topic of physically significant interactive and non-trivially coupled Partial Differential Equations. They are: (1) the noise reduction problem in a structural acoustic chamber, by use of 'smart material/structure' technology; (2) flutter control of a 2-dimensional wing immersed in a subsonic or supersonic 3-dimensional gas flow in aeroelasticity; (3) plasma heating and plasma confinement in the control of magnetohydrodynamics equations; (4) asymptotic suppression of turbulence in viscous incompressible fluid dynamics.
The best way to explain the significance of the research undertaken under this grant and its potential impact on the welfare of society at large is to provide two enlightening illustrations thereof. Illustration #1. The mathematics describing the interaction between airflow and a structure, canonically the wing of an aircraft. There are many things that inevitably we take for granted in our everyday life. One of them is boarding an aircraft and flying. Making this possible and safe is one of the many technological conquests of the human spirit, from the incipient attempts reported in the Greek mythology (Icarus) all the way to Leonardo da Vinci’s rational studies. Mountains of technical and scientific knowledge had to be accumulated, processed, and absorbed over the centuries to bring this endeavor to fruition. One key ingredient thereof is flutter suppression—flutter being an endemic instability of the wing structure of all aircraft which occurs at high enough air speed at any altitude, as a result of the interaction of the wing structure with the airflow. Flutter speed is the smallest speed at which the wing in airflow becomes unstable and motion becomes periodic. Avoiding flutter is rigidly regulated by the Federal Aviation Administration (FAA). One of the achievements of the present research under this NSF grant was to establish mathematically that the basic model coupling airflow and structure does not exhibit flutter if a certain dissipation mechanism is put in place on the structure. In other words, flutter may be eliminated by means of suitable dissipation in the structure. Illustration #2. The mathematics behind oil and gas recovery. In the relationship between causes and effects, most science proceeds through the direct way; that is, in analyzing the causes and determining their effects. There are a multitude of critical problems, however, where the effects are known, and the task is then to recover the causes that have determined them; this is an inverse problem. Oil and gas recovery, as well as a multitude of both civilian and military applications, is based on solving an inverse problem. In oil recovery on land or at sea, a wave is fired downward. The wave hits the bottom, is reflected, and the return wave is ultimately measured at the original surface. The measurement performed at the surface depends on the nature of the ground the wave has encountered and by which it has been reflected. The task of the scientist is then to determine the likelihood that the measurement reveals oil or gas underground before making the critical and expensive decision to drill. A mathematical version of the above scenario is to recover one or more unknown coefficients of an equation by means of an additional boundary measurement; this has been one of the successful mathematical investigations under the present NSF grant.
