Problem Formulation

Below is the mathematical formulation of the control problem. The function y denotes the temperature of the probe and u is the control. The point x=1 represents the inside of the probe and x=0 the outside. The function yd represents the desired temperature distribution inside the probe. The functions T (tau) and k (kappa) denote the specific heat capacity and the heat conduction, respectively, y0 is the initial temperature distribution, q is the source term, and g is a given scalar.