The paper considers linear distributed-parameter systems described by the diffusion equation. A functional-analysis approach is used to develop a technique for obtaining explicit solutions to the class of optimum-control problems that are reducible to the abstract minimum-norm problem. To illustrate the technique, two specific problems are solved. In both cases, the initial state of the system is specified. In the first problem, it is required to find the spatially distributed control which drives the system to a specified terminal state at a specified terminal time, while minimising a generalised quadratic-performance index. In the second problem, the spatially distributed control is required to bring the system at a specified terminal time to a state which is closest to a specified state, while satisfying an inequality energy constraint.