Mathematical Sciences: Control of Singular Distributed Systems

对于全局不存在的动态系统，其解的最优控制将是本课题的研究重点。解存在的最长时间称为爆破时间。当动力学包含控制时，爆炸时间取决于控制。很自然地提出了最佳控制爆破时间的问题。本研究的主要方向是将爆破时间和相关爆破集（即爆破时间有限的集合）表征为问题初始条件的函数。其他令人感兴趣的问题包括消光时间的最优控制，即系统第一次达到零状态时的最优控制，这与经典的可控性问题有关，以及熄灭时间的最优控制，即导数第一次爆炸但函数保持有界。这个项目是关于控制奇异系统中无界行为的开始。热燃烧等现象的数学模型表现出这种被称为爆炸的行为。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，可能希望尽可能长时间地推迟爆炸的开始，以便允许，例如，在燃烧之前，温度尽可能多地升高。同样，某些生物和经济问题也可以用在有限时间内表现出爆炸的方程来描述。例如，在控制肿瘤生长的系统中，通过使用化疗或放射治疗作为控制，可以最大限度地延长爆炸时间。控制这种系统的消失时间相当于使肿瘤大小减小到零的时间最小化。

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.