Optimum shape problems in distributed parameter control theory.

The work is concerned with optimum shape problems in the distributed parameter area and it consists of four parts. In Part I we consider first the basic variational theory due to Gelfand and Fomin emphasising the importance of the transversality condition in optimum shape situations; also in Part I we discuss an application of the basic theory in a particular problem where the state equations (the constraints) are hyperbolic in character. In Part II we consider a heat transfer problem between two streams of different temperatures, moving parallel to one another and with constant speeds, the aim being to choose the inlet conditions of one stream in order to achieve desired outlet conditions for the other stream. Two different aspects of the heat transfer problem are considered. In Part III we consider a hydrodynamic problem using shallow water theory in which we seek the optimum shape of a harbour boundary in order to redistribute the liquid energy in some desired way. Here one-dimensional and two-dimensional aspects of the problem are discussed, in the former fairly precise results are achieved, and in the latter the solution of the problem is shown to depend on the solution of coupled integral equations. In Part IV we consider the problem of optimum shape of an axially symmetric elastic body (subject to the classical equations of elasticity) in order to minimise the axial moment of inertia or the weight of the body. An approximate method for finding the optimum shape is presented though considerable work remains to be done in this problem.