Mathematical Sciences: Spectral Operators Generated by Damped Hyperbolic Equations

抽象舒博夫 本文的主要目的是研究Hilbert空间中一类非自伴算子的谱分析。 这些算子是三维非齐次非球对称系数波动方程所支配的系统的动力学生成元，其中含有一阶阻尼项。 本提案基于PI的12项工作（由NSF Grant DMS-92 12037和两项德克萨斯州高级研究计划赠款：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，并证明了两类非自伴算子的根向量的Riesz基性质：在有限区间两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在有限区间两端具有耗散边界条件的三维阻尼波动方程的动力学生成元。球上具有耗散边界条件的常球对称系数。这个项目的第一个目标是使下一个显着更困难的步骤：扩展的频谱结果的三维阻尼波方程的非球对称系数和耗散边界条件的球。 该分析的计划是基于在上述作品和PI的早期作品的三维薛定谔运营商与非球对称的潜力的共振和共振态的方法相结合。 第二个目标是将所得结果应用于线性分布参数系统的控制与镇定问题。 现代数学分析的一个重要领域与所谓的分布参数系统的控制有关，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，其元件包括膜、壳等。控制的目的是通过向系统施加外部控制力来实现系统的期望行为。机械臂的操纵）和航空航天工程（例如，飞机的稳定性）。 近年来，在广泛的数学文献中考虑了上述控制问题，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，只有所需的控制功能的存在被严格证明，留下了一个悬而未决的问题，直接和有效的数值计算的控制。 已知一种为控制的计算提供显式算法的方法。 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，显着限制其适用范围的实际问题。 也就是说，该方法涉及现代分析的一个复杂领域--“非自伴算子的谱理论”。“自伴算子”的谱理论是数学的经典章节，也是量子物理学的主要工具。 由于该领域的困难，该方法的应用直到最近才限于一维结构（如弦或杆）。 此外，由于内部摩擦的能量耗散的现象没有考虑在这些模型。 这个项目的主要目标是将结果扩展到最实际的三维系统的内部摩擦。

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.