Small penalty control of the end temperature in a long rod

The problem considered is that of maintaining the end temperature of a long rod near a prescribed level over a fixed time interval. Control is achieved via the heat flux at the near end, and it is optimal in the sense that it minimizes a given performance index of quadratic form. The performance index contains a penalty parameter associated with the magnitude of the control. Particular attention is given to the determination of the optimal control when the penalty parameter is small (i.e., cheap control). This gives rise to a singularly perturbed integral equation, which is solved asymptotically by a methodology which has recently been developed for a related class of problems.