ABSTRACT
We must begin by making an important distinction between the considerations appropriate to a great variety of practical problems, for example, controlled heat transfer or observed diffusion and those more theoretical considerations which arise when we wish to apply the essential ideas developed for the control theory of ordinary differential equations in the context of systems governed by partial differential equations-here, the linear heat equation
∂v
∂t = ∂
∂x2 + ∂
∂y2 + ∂
∂z2 for t > 0, x = (x, y, z) ∈ Ω ⊂ R3. (69.1)
Many of the former set of problems are ones of optimal design, rather than of dynamic control and many of the essential concerns are related to fluid flow in a heat exchanger or to phase changes (e.g., condensation) or to other issues which go well beyond the physical situations described by Equation 69.1. Some properties of Equation 69.1 are relevant for these problems and we shall mention these, but the essential concerns which dominate them are outside the scope of this chapter.
