A new numerical method for the boundary optimal control problems of the heat conduction equation

A new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate-gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct di>erentiation method or the adjoint variable method. The direct di>erentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very e?cient, the adjoint variable does not have su?cient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this di?culty with the adjoint variable method by introducing a partial di>erential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems e?ciently with su?cient accuracy