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.
