Some Analytical Solutions to the Problems of Optimization of the Variable Heat Conductivity Coefficient

We give new statements and solutions to the problems of optimization of a variable heat conductivity coefficient for an inhomogeneous pipe and a flat wall under mixed boundary conditions. The cost functionals and constraints are either the average or maximal temperature while the constraints are either the condition of constancy of the integral heat conductivity coefficient or a priori information about the change of the heat conductivity coefficient in a given range. To solve the problems for a pipe, we use the two optimization methods: the variational approach basing on the conjugate functions and an extended Lagrange functional as well as the Pontryagin’s maximum principle. To solve the optimization problem for a flat wall under the assumption of weak material inhomogeneity, we apply the method of expansion in a small physical parameter. The fourth problem under consideration is the optimization of the variable heat conductivity coefficient of an inhomogeneous flat wall under boundary conditions of the first kind. We find a solution to this singular optimization problem among broken extremals. Considering particular examples, we compare the values of minimized functionals for bodies with a constant heat conductivity coefficient and the optimal variable coefficient. We also estimate the gain of using optimization.