Mathematical Sciences: Control of Singular Distributed Systems

对于可能不是全局存在的动态系统的解的最优控制将是本项目的重点。解存在的最长时间称为爆破时间。当动力学包含控件时，放大时间取决于控件。自然会出现以最佳方式控制井喷时间的问题。本研究的主要方向是刻划爆破时间和相应的爆破集，即爆破时间有限的集作为问题初始条件的函数。其他感兴趣的问题包括灭绝时间的最优控制，即系统第一次达到零状态的时间，这与经典的可控性问题有关，以及熄灭时间的最优控制，即导数第一次爆破，但函数保持有界。本课题致力于控制广义系统中无界行为的发生。热燃烧等现象的数学模型显示了这种行为，即所谓的吹气。在这类问题中，模型问题的解在有限时间内变得无界。作为一个实际问题，可能希望尽可能地推迟爆裂的开始，以便允许，例如，在燃烧之前尽可能地提高温度。类似地，某些生物和经济问题可以用在有限时间内表现出爆破性的方程来描述。例如，在控制肿瘤生长的系统中，最大化放大时间将通过使用化疗或放射治疗作为对照来实现。控制这样一个系统的消退时间将相当于将肿瘤大小降至零的时间最小化。

Optimal control of solutions of dynamical systems which may not exist globally in time will be the focus of this project. The maximum time for which the solution exists is called the blowup time. When the dynamics contain a control the blowup time depends on the control. It is natural to pose the problem of optimally controlling the blowup time. The main direction of this research will be to characterize the blowup time and associated blowup set, that is, the set on which the blowup time is finite, as a function of the initial conditions of the problem. Additional problems of interest include optimal control of extinction time, that is, the first time the system reaches the zero state, which is related to classical questions of controllability, and optimal control of the quenching time, that is, the first time the derivative blows up but the function remains bounded. This project is concerned with controlling the onset of unbounded behavior in singular systems. Mathematical models of phenomena such as thermal combustion exhibit this behavior, which is known as blowup. In such problems the solution of the model problem becomes unbounded in finite time. As a practical matter, it may be desired to delay the onset of blowup for as long as possible in order to allow, for example, the temperature to increase as much as possible prior to combustion. Similarly, certain biological and economic problems can be described by equations which exhibit blowup in finite time. Maximizing the blowup time in a system governing the growth of tumors, for example, would be effected through the use of chemotherapeutic or radiation therapy as the control. Controlling the extinction time of such a system would amount to minimizing the time at which the tumor size decreases to zero.