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

9705046麦克米兰这个项目主要研究有界域上几个线性和非线性动力系统的边界镇定/最优控制问题。本研究的主要目的如下：(1)建立抛物型和双曲型/Petrwoski偏微分方程具非光滑观测算子的有限时间不定成本问题(包括奇异问题)的抽象理论。即使对于标准的(正定的)有限期限成本问题，这一结果也是新的。此外，我们还研究了与某些非线性最优控制问题有关的某些Riccati微分方程解的条件。(2)首先发展了标准的Riccati理论(即系统中不存在扰动的情况)，然后发展了用于不确定成本问题的极小极大Riccati理论。(3)对各种壳模型验证了极大极小理论所要求的各种假设。正在考虑的抽象问题源于对力学和结构设计中的问题进行数学建模。特别是，重点是大型振动结构，如卫星天线、飞机、天线等。必要时，对这些结构的运动进行建模会产生偏微分方程组。将研究抑制(即控制)这些结构的不必要振动的方法。此外，该项目寻求解决在存在外部干扰(例如，飞机机翼上的外力)的情况下对振动结构的控制。该项目的目标是为填补大型柔性结构优化问题文献中的一大空白做出贡献。

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.