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

本课题主要研究有界域上若干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)建立具有非光滑观测算子的抛物型和双曲型/Petrwoski偏微分方程的有限视界不定代价问题（包括奇异问题）的抽象理论。这个结果对于标准（正定）有限水平代价问题也是新的。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微分方程的可解性条件。(2)在首先发展标准Riccati理论（即系统中不存在干扰的情况）之后，为不确定成本问题发展极小极大Riccati理论。(3)验证极大极小理论（我们之前开发的）对各种壳模型所要求的各种假设。所考虑的抽象问题源于力学和结构设计问题的数学建模。特别是，重点是大型振动结构，如卫星天线，飞机，天线等。必须对这些结构的运动进行建模，得到偏微分方程组。将研究如何消除（即控制）这些结构不必要的振动的方法。此外，该项目还试图解决在外部干扰（例如，飞机机翼上的外力）存在时振动结构的控制问题。该项目的目标是填补文献中关于大型柔性结构优化问题的巨大空白。

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.