Control Problems of Systems of Strongly Coupled Partial Differential Equations

9804056 Lasiecka本建议旨在调查数学方面，以及控制和优化性能，模型的相互作用系统，这是由强耦合偏微分方程，并受到边界和点控制行动。这些都是数学上最具挑战性和物理上最相关的情况。所选择的系统是复杂和现实的，足以作为下面描述的技术应用的基准问题。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）壳等系统振动情况下的精确可控性;（3）结构解的一致稳定性，以及在非线性情况下，对渐进行为的相应研究;（4）非定二次型最优控制理论和极小极大对策理论，以及鲁棒镇定。虽然近年来，这些问题已经达到了相当程度的成熟度的单类动力学-扩散和振荡-他们的研究，以强耦合系统的组件的扩散和/或振荡的类型是刚刚起步。近年来，随着智能材料与结构技术的兴起，对“交互作用结构”的控制、优化和稳定的迫切需要被提上了科学研究的前沿。“交互作用结构”的动力学行为由偏微分方程控制。最近的民用和军用研究令人信服地证明了智能材料/结构是实验室的现实。这些新的结构概念积极阻尼噪音和振动，抑制机翼后缘的颤振，抑制地面车辆，飞机和潜艇的空中或水上声学特征，并降低它们内部的压力噪音水平。民用和军用工业目前都很感兴趣的一个现实问题是减少或消除具有弯曲壁的三维舱室中不需要的噪声场的问题，该舱室由壳体模拟，通过智能材料如压电贴片或记忆合金粘结到或嵌入弹性壁中。当与适当的电压（控制）连接时，预期由弹性壁产生的所得弯曲力矩产生抵消不想要的噪声的声波，从而导致它们的消除。

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.