Mathematical Sciences: Spectral Operators Generated by Damped Hyperbolic Equations

摘要Shubov这个建议的主要目的是发展Hilbert空间中一类非自伴算子的谱分析。这些算子是三维波动方程系统的动力学生成器，这些波动方程的空间非齐次非球对称系数包含一阶阻尼项。这项建议是基于派的12项工作(由美国国家科学基金会拨款DMS-92 12037和两个德克萨斯高级研究计划拨款：00364-116，93-95和0036-44-124，95-97支持)。在这些工作中，PI对谱进行了详细的渐近分析，并证明了两类关于非自伴算子的根向量的Riesz基性质：有限区间末端具有耗散边界条件的非齐次阻尼弦方程的动力学生成元和球面上具有非常定球对称系数的具有耗散边界条件的三维阻尼波方程的动力学生成元。这个项目的第一个目标是使下一个更加困难的步骤：将谱结果推广到具有非球对称系数和球面上耗散边界条件的三维阻尼波方程。这一分析计划是基于前述工作中发展的方法和PI关于具有非球对称势的三维薛定谔算符的共振态和共振态的早期工作的组合。第二个目标是利用谱分解方法将结果应用于线性分布参数系统的控制和镇定问题。现代数学分析的一个重要领域是所谓的分布参数系统的控制，它在工程问题中有着广泛的应用。这类系统的例子包括以膜、壳、S等为单元的复杂的振动弹性结构。控制的目标是通过施加外部控制力来实现系统的期望行为。这类问题在机器人学(例如，机器人手臂的操作)和航空航天工程(例如，飞机的稳定)中是最重要的。上述控制问题近年来在广泛的数学文献中得到了考虑，并在解决这些问题方面取得了相当大的进展。然而，在许多情况下，只有所需控制函数的存在被严格证明，这留下了关于控制的直接和有效的数值计算的悬而未决的问题。已知一种提供用于计算控制的显式算法的方法。这就是谱分解的方法。然而，这种方法的应用遇到了严重的数学困难，这大大限制了它在实际问题中的适用范围。也就是说，该方法涉及现代分析的一个复杂领域--“非自伴算符的谱理论”。“自伴算符”的谱理论是数学的经典章节，也是量子物理的主要工具。由于该领域的困难，直到最近，该方法的应用还局限于一维结构(如弦或杆)。此外，这些模型没有考虑内耗引起的耗能现象。这个项目的主要目标是将结果推广到最重要的具有内耗的三维系统。

Abstract Shubov The primary goal of this proposal is to develop the spectral analysis for a class of non-selfadjoint operators in a Hilbert space. These operators are the dynamics generators of the systems governed by three-dimensional wave equations with spatially non-homogeneous non-spherically symmetric coefficients containing first order damping terms. This proposal is based on twelve works by PI (supported by NSF Grant DMS-92 12037 and two Texas Advanced Research Program Grants: 00364-116, 93-95 and 0036-44-124, 95-97). In these works, the PI carried out a detailed asymptotic analysis of the spectrum and proved the Riesz basis property of the root vectors for two classes on non-selfadjoint operators: the dynamics generators for the equation of a non-homogeneous damped string with dissipative boundary conditions at the ends of a finite interval and the dynamics generators for three-dimensional damped wave equation with non-constant spherically symmetric coefficients with dissipative boundary conditions on a sphere. The first objective of this project is to make the next significantly more difficult step: to extend the spectral results to the three-dim damped wave equation with non-spherically symmetric coefficients and dissipative boundary conditions on a sphere. The plan of this analysis is based on a combination of the methods developed in the aforementioned works and the PI's earlier works on resonances and resonance states for three-dimensional Schrodinger operators with non spherically symmetric potentials. The second objective is to apply the results to the control and stabilization problems for linear distributed parameter systems using the spectral decomposition method. An important area of modern mathematical analysis, which has a wide range of applications to engineering problems, is related to control of the so-called distributed parameter systems. Examples of such systems include complicated vibrating elastic structures containing as their elements membranes, shell s, etc. The objective of control is to achieve the desired behavior of the system by applying an external control force to it. Problems of this type are of the primary importance in robotics (e.g., manipulation of a robot arm) and in aerospace engineering (e.g., stabilization of an aircraft). The aforementioned control problems were considered in recent years in extensive mathematical literature and a considerable progress towards their solution was made. In many cases, however, only the existence of the desired control functions was rigorously proved which left an open question of direct and efficient numerical computations of the controls. A method that provides explicit algorithms for the computation of controls is known. This is the method of the spectral decomposition. The application of this method, however, encounters serious mathematical difficulties which significantly restricts its area of applicability to practical problems. Namely, the method deals with a complicated area of modern analysis- "the spectral theory of non-selfadjoint operators." The spectral theory of "selfadjoint operators" is a classical chapter of mathematics and serves as a main tool in quantum physics. Because of the difficulties in the area, the applications of the method were until recently restricted to one- dimensional structures (like strings or rods). Moreover, the phenomenon of the energy dissipation due to the internal friction was not taken into account in these models. The main objective of this project is to extend the results to the most practically important three-dimensional systems with the internal friction.