An interior point method for a parabolic optimal control problem with regularized pointwise state constraints

In this talk, we extend our investigations on interior point methods for elliptic state-constrained optimal control problems in [4] and [2] to the parabolic case. The main difficulty of the numerical analysis of interior point methods for such problems is the lack of regularity of Lagrange multipliers associated with the state constraints. Therefore, it is helpful to improve the properties of the multipliers have to be improved by suitable regularization techniques. To consider the interior point algorithm in function space, we suggested in [4], [2] a Lavrentiev type regularization. The Lavrentiev regularization of elliptic problems was introduced in [3]. This method ensures regular Lagrange multipliers and preserves, in some sense, the structure of a state-constrained control problem. Moreover, compared with a direct application of interior point methods to state-constrained problems, the regularization improves the performance of the algorithm, [2]. Here we prove the convergence of a conceptual primal interior point method in function space. We confine ourselves to a problem with linear equation and an objective functional with observation at the final time. This seems to be more challenging in the analysis than functionals of tracking type.