ABSTRACT
In this section we provide three comparatively elementary examples to see how the considerations above apply to some control-theoretic questions. The first relates to a simplified version of a quite practical heat transfer problem and is treated with the use of rough approximations, essentially to see how such heuristic treatment can be used to obtain practical results. The second describes the problem of control to a specified terminal state-which would be a standard problem in the case of ordinary differential equations but which involves some new considerations in this distributed parameter setting. The final example is a “coefficient identification” problem: using interaction (input/output) at the boundary to determine the function q = q(x) in an equation of the form ut = uxx − qu, generalizing Equation 69.21.
We consider a slab of thickness a and diffusion coefficient D within which heat is generated at constant rate ψ. On the one side, this is insulated (vx = 0) and on the other, it is in contact with a stream of coolant (diffusion coefficient D′) moving in an adjacent duct with constant flow rate F in the y-direction. Thus, the slab occupies {(x, y) : 0 < x < a, 0 < y < L} and the duct occupies {(x, y) : a < x < a¯, 0 < y < L}with a, a¯ $ L and no dependence on z.
