Time-optimal control of a linear diffusion process

The paper describes how to solve some of the problems encountered in studying the time-optimal control of a distributed-parameter system. The problems are illustrated by considering the very simple example of a system, whose behaviour is described by the one-dimensional heat-conduction or diffusion equation, in which it is desired to make the temperature zero at all points in the slab in minimum time, using a bounded control input. In order to obtain numerical results, it is necessary to represent the continuous temperature distribution by a finite number of variables. Three methods of doing this are described: the subdivision method, the Fourier-series method, and the parabolic method. Comparative numerical results are given, and the relative merits of the three methods are discussed. A recently published treatment of the finite-control-time problem is also discussed.