Control problems for systems of strongly coupled partial differential equations with variable coefficients.

近年来，智能材料和结构的新兴技术使科学研究迫切需要控制、优化和稳定动态“相互作用”结构，其组分的行为由偏微分方程（PDEs）控制。一个典型的说明国家利益的例子是声室（飞机或旋翼机的座舱或客舱等）的降噪问题。这将腔内不需要的声压（噪声场）的振荡行为与腔室柔性壁的弹性振动耦合在一起，可能通过夹层加强，并可能解释热弹性效应。成对的压电贴片粘在柔性壁上，一旦适当地布线，就会产生弹性力矩，从而减弱声室中的噪音。在数学上，声压用二阶标量双曲方程（波动方程）来建模，而柔性壁面用带或不带结构阻尼的类板Kirchhoff方程来建模。在第一种情况下，板具有抛物线行为，在第二种情况下具有双曲行为，因此导致双曲/抛物线耦合，或整体结构的双曲/双曲耦合。本项目的另外两个关键的、新颖的、显著的特征是：(i)首先，描述耦合结构的线性或非线性偏微分方程在空间中具有可变系数，当介质的性质依赖于点到点时总是如此；（ii）而且，柔性壁可以是弯曲的（而不是平的），因此可以用壳（而不是板）来建模。因此，在整体耦合结构的控制理论分析中，提出了微分几何方法来解决这两个严重的难题。目标是根据预先分配的最优性准则进行最优控制，并使耦合结构渐近稳定。该项目的方法包括首先建立一个数学理论，然后进行数值分析，以产生有效的可计算算法。联邦政府最近的研究令人信服地证明，智能材料/结构已成为实验室现实。1993年为国家科学基金会编写的题为“重建和加强国家基础设施：智能材料系统和结构的作用”的讲习班报告也证实了这一点。这些新的结构概念能有效地抑制噪音和振动，抑制翼型后缘的颤振，使固定翼和旋翼飞机都能实现主动扭转/弯曲；衰减或抑制水性特征（主动声学特征控制）等。将新颖的智能结构转变为新的、真正革命性的平台面临许多障碍。最重要的是设计优化：智能结构及其通信和控制系统。本项目的目的是在坚实的数学基础和分析的基础上对这一领域作出贡献。

In recent years, the emerging technology of smart materials and structures has brought to the fore of scientific investigation the pressing need to control, optimize, and stabilize dynamical 'interactive' structures, whose components' behavior is governed by partial differential equations (PDEs). A canonical illustrative example of keen national interest is the noise reduction problem in an acoustic chamber (aircraft's or rotorcraft's cockpit or cabin, etc). This couples the oscillatory behavior of the unwanted acoustic pressure (noise field) within the chamber with the elastic vibrations of a flexible wall of the chamber, possibly reinforced by sandwiched layers, and possibly accounting for thermo-elastic effects. Pairwise sets of piezo-electric patches bonded on the flexible wall, once suitably wired, develop an elastic moment that is meant to dampen out the noise in the acoustic chamber. Mathematically, the acoustic pressure is modeled by a second order scalar hyperbolic equation (wave equation), while the flexible wall is modeled by a plate-like Kirchhoff equation with or without structural damping. In the first case, the plate has a parabolic behavior, in the second a hyperbolic behavior, resulting therefore in either hyperbolic/parabolic coupling, or in hyperbolic/hyperbolic coupling of the overall structure. Two additional key, novel, distinguishing features of the present project are: (i) first, the linear or non-linear PDEs describing the coupled structure have variable coefficients in space, which is always the case when the properties of the medium depend from point to point; (ii) and, moreover, the flexible wall may be curved (rather than flat), and thus modeled by a shell (rather than a plate). Accordingly, differential geometric methods are then proposed in the control theoretic analysis of the overall coupled structure, to cope with these two serious difficulties. The goal is to optimally control - according to a pre-assigned optimality criterion - and asymptotically stabilize the coupled structure. The methodology of this project consists in first establishing a mathematical theory, to be followed next by a numerical analysis thereof, to yield effective and computable algorithms.Recent Federal research has convincingly demonstrated smart materials/structures to be a laboratory reality. This is also confirmed by a Workshop Report for the National Science Foundation entitled: "Rebuilding and enhancing the Nation's infrastructures: a role for intelligent material systems and structures", 1993. These new structural concepts actively damp noise and vibration, suppress flutter at trailing edges of airfoils and enable active twist/camber of both fixed wing and rotorcraft; attenuate or suppress water borne signatures (active acoustic signature control), etc. The transition of novel smartstructures into new, truly revolutionary platforms faces many obstacles. The most significant is design optimization: both of the smart structure and its communication and control systems. The present project aims at producing a contribution in this area, based on solid mathematical foundations and analysis.