Control and Stabilization of Elastic Systems: Anisotropic Elasticity and Other Coupled Systems

9803547Horn在这个建议中解决的问题包括，首先，各向异性弹性系统的一致稳定性。 最近的研究结果表明，这样的系统是一致稳定的，如果域是一个球。 如果系统是各向同性的，PI已经获得了更一般的区域的一致稳定性。 因此，目标是获得两者的最佳结果：各向异性系统在更一般域上的一致稳定性。证明中涉及的技术包括通过使用微局部分析获得的尖锐痕迹规律性估计。 特别令人感兴趣的是需要唯一的各向异性系统的延续。 其次，即使假设系统是各向同性的，精确可控性的结果也是有问题的。虽然存在一些结果，但在大多数情况下，证明无法超越专业观众或几何限制非常严格。 一种方法的基础上使用Carleman型估计，允许高阶耦合项被认为是狄利克雷或诺依曼边界条件。 最后，进一步的应用程序中看到的情况下，圆柱壳的尖锐的轨迹规律性的结果。 一个特别感兴趣的问题是稳定性特性相对于壳的厚度是否是鲁棒的。精确可控性，即，将系统驱动到期望状态的能力是工程中出现的重要问题。可能发生的随机或不受控制的振动可能导致严重后果。 从塔科马海峡大桥倒塌的目击者到一个被手机信号失真困扰的商人，几乎每个人都经历过缺乏控制的影响。 如果一个系统是一致稳定的，任何不希望的振动都可以通过适当的输入选择来衰减，例如，通过沿物体边界的直接力或扭矩沿着移动。 本项目的研究集中在控制的理论方面，当一个弹性体被假定为由一种以上的材料组成或不均匀。

9803547HornQuestions 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.