Indefinite and Singular Optimal (boundary) Control Problems for P.D.E's

小行星9705046 该项目的重点是边界稳定/最优控制问题， 有界域上的线性和非线性动力系统。 的 本研究的主要目的是：（1）建立有限时间不确定成本的抽象理论 问题（包括奇异问题）的抛物线和 双曲/Petrwoski偏微分方程（p.d.e.）的）与 非光滑观测算子 这一结果将是新的， 标准（正定）有限时间成本问题。 另外我们 我想研究某些微分方程可解的条件 一类非线性最优控制的Riccati方程 问题 (2)发展一个极大极小Riccati理论的不确定成本问题，后 首先发展标准的黎卡提理论（即，在没有 系统中存在的干扰）。 (3)验证极小极大理论所需的各种假设（我们 以前开发的）用于各种壳模型。 正在考虑的抽象问题产生于 机械和结构设计中问题的数学建模。 在 特别是，重点是大型振动结构，如卫星碟， 飞机，天线等。必要时，这些运动的建模 结构导致偏微分方程系统。 方式将是 研究在其中衰减（即，控制）不必要的振动 这些结构。此外，该项目还寻求解决控制 在存在外部干扰的情况下振动结构（例如，外部 飞机机翼上的力）。该项目的目标是帮助填补一个大的 大型柔性结构优化问题的文献空白。

9705046 McMillan This project focuses on issues of boundary stabilization/optimal control of several linear and nonlinear dynamical systems on a bounded domain. The main goals of this study are the following: (1) To develop an abstract theory for the finite horizon indefinite cost problems (including singular problems) for parabolic and hyperbolic/Petrwoski partial differential equations (p.d.e.'s) with nonsmoothing observation operators. This result will be new even for standard (positive definite) finite horizon cost problems. In addition, we would like to investigate conditions for solvability of certain differential Riccati equations which are related to certain nonlinear optimal control problems. (2) Develop a minimax Riccati theory for indefinite cost problems, after first developing the standard Riccati theory (i.e., the case where there are no disturbances present in the system). (3) Verify various assumptions required by the minimax theory (which we have developed previously) for various shell models. The abstract problems that are being considered arise out of the mathematical modeling of problems in mechanics and structural design. In particular, the focus is on large vibrating structures such as satellite dishes, aircraft, antennae, etc. By necessity, the modeling of the movement of these structures results in systems of partial differential equations. Ways will be investigated in which to damp out (i.e., control) unwanted vibrations of these structures. In addition, the project seeks to address the control of vibrating structures in the presence of outside disturbances (e.g., external forces on aircraft wings). The project's goal is to contribute in filling a large gap in the literature on optimization problems for large flexible structures.