Questions addressed in this proposal include, firstly, the uniform stability of an anisotropic system of elasticity. Recent results by Alabau and Komornik have shown that such a system is uniformly stable if the domain is a sphere. If the system is isotropic, the PI has obtained uniform stability for more general domains. Thus, the goal is to obtain the best of both: Uniform stability of an anisotropic system on a more general domain. Techniques involved in the proof include sharp traces regularity estimates, obtained through the use of microlocal analysis. Of particular interest is the need for unique continuation for the anisotropic system. Secondly, exact controllability results even under the assumption that the system is isotropic are problematic. While a few results exist, in most cases the proofs are not accessible beyond a specialized audience or the geometric restrictions are extremely stringent. An approach based on the use of Carleman type estimates, which allow higher order coupling terms to be considered, is considered for both Dirichlet or Neumann boundary conditions. Finally, further applications of sharp trace regularity results are seen in the case of cylindrical shells. A particular question of interest is whether the stability properties are robust with respect to the thickness of the shell.
Exact controllability, i.e., the ability to drive a system to a desired state, is an important question arising in engineering. Random or uncontrolled vibrations which may occur can lead to serious consequences. From witnesses to the collapse of the Tacoma Narrows Bridge to a businessman bothered by distortion on his cellular phone, almost everyone has experienced the effects of a lack of control. If a system is uniformly stable, any undesired vibrations can be damped out through an appropriate choice of input, e.g., through a direct force or torque along the boundary of the object. Investigations in this project focus on the theoretical aspects of control when an elastic body is assumed to be composed of more than one material or is nonuniform.
