Numerical Solution of Optimal Heating of Temperature Field in Uncertain Environment Modelled by the use of Boundary Control

‎In the present paper‎, ‎optimal heating of temperature field which is modelled as a boundary optimal control problem‎, ‎is investigated in the uncertain environments and then it is solved numerically‎. ‎In physical modelling‎, ‎a partial differential equation with stochastic input and stochastic parameter are applied as the constraint of the optimal control problem‎. ‎Controls are implemented as Dirichlet boundary conditions and representing the heating elements on the boundary of the field‎. ‎In numerical quantification‎, ‎stochastic input and parameter are approximated via Karhunen-Lo'eve expansion and inserted to the problem‎. ‎In fact‎, ‎for numerical discretization of the problem stochastic Galerkin method is applied to generalize polynomial chaos‎. ‎Numerical optimization is performed via gradient method‎. ‎The problem is fully implemented and in order to show the applicability of the method‎, ‎numerical examples are solved and numerical results are represented through figures.