Some Analytical Solutions to the Problems of Optimization of the Variable Heat Conductivity Coefficient
摘要
We give new statements and solutions to the problems ofoptimization of a variable heat conductivity coefficient for an inhomogeneous pipe and a flat wall undermixed boundary conditions. The cost functionals and constraints are either the average ormaximal temperature while the constraints are either the condition of constancy of the integral heatconductivity coefficient or a priori information about the change of the heat conductivity coefficient ina given range. To solve the problems for a pipe, we use the two optimization methods: the variational approachbasing on the conjugate functions and an extended Lagrange functionalas well as the Pontryagin’s maximum principle. To solve the optimization problem for a flat wall under the assumption ofweak material inhomogeneity, we apply the method of expansion in a small physical parameter. Thefourth problem under consideration is the optimization of the variable heat conductivity coefficient of an inhomogeneous flat wall underboundary conditions of the first kind. We find a solution to this singular optimization problemamong broken extremals.Considering particular examples, we compare the values of minimized functionalsfor bodies with a constant heat conductivity coefficient and the optimal variable coefficient.We also estimate the gain of using optimization.