Control Problems of Systems of Strongly Coupled Partial Differential Equations

本提案旨在研究由强耦合偏微分方程描述的、受边界和点控制作用的交互系统模型的数学方面以及控制和优化特性。这既是数学上最具挑战性的案例，也是物理上最相关的案例。所选择的系统足够复杂和现实，可以作为下文所述技术应用的基准问题。对于整个结构和/或其组成部分，需要研究的具体问题包括：(1)解的适定性和正则性；(2)壳等系统在振荡情况下的精确能控性；(3)结构解的一致稳定性以及在非线性情况下的渐近行为的相应研究；(4)不确定二次代价的最优控制理论和极小-最大对策理论，以及鲁棒镇定。虽然近年来这些问题对于单类动力学--扩散动力学和振荡动力学--已经相当成熟，但它们对具有扩散和/或振荡型分量的强耦合系统的研究还处于起步阶段。动力系统的控制、优化和稳定性问题长期以来在广泛的不同应用中扮演着公认的和关键的角色。近年来，新兴的智能材料和结构技术向科学研究的前沿提出了控制、优化和稳定“交互结构”的迫切需要，这些结构的部件的动力学行为由偏微分方程组控制。最近的民用和军用研究令人信服地证明，智能材料/结构将成为实验室现实。这些新的结构概念积极地抑制噪声和振动，抑制翼型后缘的颤振，抑制地面车辆、飞机和潜艇的空中或水上声学特征，并降低它们内部的应激噪声水平。目前民用和军事行业都非常感兴趣的一个现实问题是，通过粘接或嵌入弹性墙的智能材料，如压电片或记忆合金，减少或消除由壳体模拟的具有曲面墙壁的三维机舱中不需要的噪声场。当连接适当的电压(控制)时，弹性墙产生的弯矩预计会产生声波，以对抗不想要的噪音，从而消除它们。

9804056LasieckaThe present proposal aims at investigating mathematical aspects, as well as control and optimization properties, of models of interactive systems, which are described by strongly coupled partial differential equations, and which are subject to boundary and point control action. These are both the mathematically most challenging and the physically most relevant cases. The systems chosen are complex and realistic enough to serve as benchmark problems for the technological applications described below. Specific issues to be examined for the entire structure, and/or a component thereof, include: (1) well-posedness and regularity of solutions; (2) exact controllability in the oscillatory case for systems such as shells; (3) uniform stability of the structure solutions as well as, in the non-linear case, a corresponding study of the asymptotic behavior; (4) optimal control theory and min-max game theory with non-definite quadratic cost, as well as robust stabilization. While in recent years these problems have reached a considerable level of maturity for single classes of dynamics - diffusive and oscillatory - their study to strongly coupled systems with components of both diffusive and/or oscillatory type is just at its infancy.Control, optimization and stability problems for dynamical systems have long played a well-recognized and critical role on a broad range of diverse applications. In recent years, the emerging technology of smart materials and structures has brought forth to the front of scientific investigation the pressing need to control, optimize and stabilize "interactive structures", whose components' dynamical behavior is governed by partial differential equations. Recent civilian and military research has convincingly demonstrated smart materials/structures to be a laboratory reality. These new structural concepts actively damp noise and vibration, suppress flutter at the trailing edges of airfoils, suppress the airborne or water borne acoustic signatures of ground vehicles, aircraft, and submarines, and reduce the stressful noise levels within them. A realistic problem of keen, present interest to both the civilian and military industry is the problem of reducing, or eliminating, the unwanted noise field in a three dimensional cabin with curved walls, modeled by shells, by means of smart materials such as piezo-electric patches or memory alloys bonded to, or embedded in, an elastic wall. When wired with an appropriate voltage (control), the resulting bending moment generated by the elastic wall is expected to produce acoustic waves that counter the unwanted noise, thus leading to their elimination.