Mathematical Sciences: Spectral Operators Generated by Damped Hyperbolic Equations

本文的主要目标是建立Hilbert空间中一类非自伴随算子的谱分析。这些算符是由空间非齐次非球对称系数包含一阶阻尼项的三维波动方程控制的系统的动力学产生器。本提案基于PI的12项工作（由NSF资助DMS-92 12037和两项德克萨斯高级研究计划资助：00364- 116,93 -95和0036-44- 124,95 -97）。在这些工作中，PI对谱进行了详细的渐近分析，并证明了两类非自伴随算子的根向量的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器和球面上具有耗散边界条件的非常球对称系数三维阻尼波方程的动力学生成器。该项目的第一个目标是使下一个步骤变得更加困难：将光谱结果扩展到具有非球对称系数和球上耗散边界条件的三暗淡阻尼波动方程。该分析计划是基于上述工作中开发的方法和PI早期关于具有非球对称势的三维薛定谔算子的共振和共振态的工作的结合。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。所谓分布参数系统的控制是现代数学分析的一个重要领域，它在工程问题中有着广泛的应用。这种系统的例子包括复杂的振动弹性结构，其元素包括膜、壳等。控制的目的是通过对系统施加外部控制力来达到系统的预期行为。这类问题在机器人技术（如机械臂的操纵）和航空航天工程（如飞机的稳定）中具有重要意义。近年来，大量的数学文献研究了上述控制问题，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，只严格证明了所需控制函数的存在，这就留下了直接和有效的控制数值计算的开放性问题。已知有一种方法为控制的计算提供显式算法。这就是光谱分解的方法。然而，这种方法的应用遇到了严重的数学困难，这大大限制了它对实际问题的适用范围。也就是说，该方法处理现代分析的一个复杂领域——“非自伴随算子的谱理论”。“自伴随算子”的谱理论是数学的经典篇章，也是量子物理的主要工具。由于该领域的困难，直到最近，该方法的应用仅限于一维结构（如弦或棒）。此外，这些模型没有考虑内摩擦引起的能量耗散现象。本项目的主要目标是将结果扩展到具有内摩擦的最实际重要的三维系统。

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.