Approximation of non-linear parabolic boundary control problems by the Fourier method - convergence of optimal control

Abstract

Motivating the consideration of more general optimal control problems a boundary control problem for the one-dimensional heat equation with non-linear boundary condition is approximated by the Fourier method. The corresponding infinite series defining the state-function (obtained by the Fourier method) is terminated after n terms, and the set of admissible controls is replaced by a suitable approximation, too. It is shown that the sequence of “approximating” controls is strongly convergent in L 2 to a certain locally optimal control for n → ∞, provided that second order sufficient optimality conditions are fulfilled at this control and some additional assumptions on the non-linearities of the problem are satisfied. The theory is worked out for a general class of control problems governed by non-linear integral equations, which covers boundary control as a particular case.