In Part I, the problem of heating a thin plate or material travelling through a furnace, in which the system is described by first order linear partial differential equations, is introduced as an example of optimal control theory in distributed parameter systems. The variational technique in a fixed domain is used to obtain the necessary conditions for optimality. Many cases of the problem with the state equation described by first order linear partial differential equations are discussed, in which the control function enters into the state equation in different positions. The problems are analysed and solved by making use of characteristic curves. In Part II, we have studied the variation of a functional defined on a variable domain, and we apply it to the problem of finding the optimum shape of the domain in which some performance criterion has an extremum. The problem in which the state equation is Laplace's equation defined on the variable domain of an annular shape with given boundary conditions is discussed and completely solved for the case when the inner boundary of the domain is only a small departure from a circle. We also introduce the method of logarithmic potential of a single layer to solve the boundary value problem of Laplace's equation with mixed boundary conditions and two simple examples are solved by using this method which leads to coupled integral equations.