Interface Control for Systems of Strongly Coupled Partial Differential Equations

University Of Memphis, Memphis TN

Investigators

Abstract

Interactive systems are ubiquitous in technological applications and critical for modern society. An important class of such coupled systems have components with differing dynamic properties, whose coupling occurs at the interface of the different media in which each component evolves. The role of a control action is to force the system to behave in a desired, pre-assigned way: suppressing flutter, suppressing turbulence, hitting a target, etc. One eloquent illustration is flutter in aero-elasticity, a phenomenon that may occur when a structure is subject to a surrounding gas or fluid flow. It results in a periodic-like instability that can be fatal for structural stability due to fatigue of the material, such as the collapse of the Tacoma Narrows Bridge in 1940 during strong winds. In such flow-structure interactions, the control action of flutter suppression can be used to avoid structural failure. Another illustration of control is the suppression of turbulence arising in a fluid as part of fluid-structure interaction. Examples include body fluids flowing within arterial walls or veins and motion of a solid vehicle immersed in a fluid, be it an aircraft flying in the air, a ship or a submarine moving in water, etc. Another area of interest is non-linear acoustics, in particular high intensity ultrasound, which has innumerable uses in medical and industrial technology: lithotripsy, thermotherapy, ultrasonic cleansing, detection of cracks, concealed weapon detection, etc. This project aims to extend and deepen the mathematical underpinnings of control systems for these and other important applications. The research project is focused on the study of control-theoretic issues for systems of strongly coupled partial differential equations (PDE), where the (active, passive) control action is exercised in the transmission conditions at the interface between two media. One example is a hyperbolic-like/hyperbolic interaction: a von Karman plate (displacement of an aircraft wing) sitting in the horizontal plane is immersed in, and interacts with, a 3D-gas flow which occupies the upper-half space and moves over the plate. The flow is mathematically modeled by a modified wave equation in terms of the flow potential, allowing various types of boundary conditions. Couplings occurs in each equation through the trace of the other variable. The normalized constant speed of the passing flow determines regimes: subsonic, transonic and supersonic. Mathematical studies of this system include: (i) well-posedness of various models under various types of boundary conditions; (ii) stability and attractors for the structure, to include control techniques for flutter suppression at both subsonic and supersonic regimes. Another example is the parabolic-hyperbolic interaction, which couples a fluid-gas equation inside the vessel with the full vectorial Karman (shell) dynamical equation, describing the external shell. Here again, mathematical studies of the system include (i) well-posedness and (ii) stabilization by means of stabilizing controls acting on the plate to transmit dissipation on the unstable fluid. In many acoustic applications, only the external boundary is accessible, not the interior. Control theoretic methods can also be applied to the acoustic equation, a third order (in time) PDE with either Dirichlet or Neumann boundary controls. Specific topics of study include: (i) optimal regularity from the boundary to the interior and its trace (ii) exact boundary controllability, (iii) boundary stabilization; (iv) optimal control and min-max game theory with quadratic cost functional. The research involves a diverse set of tools including non-linear functional analysis, micro-local analysis, and Riemannian geometry methods.