Computational Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

We study parametric optimal control problems governed by a system of timedependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute optimal control functions and so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization is discussed that transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP) which can be solved efficiently by SQP methods. Second order sufficient conditions can be checked numerically, and we propose to apply an NLP-based approach for robust computation of sensitivity differentials of optimal solutions with respect to the perturbation parameters. The advertised numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation. This example demonstrates the general ability of the algorithm to efficiently and robustly calculate an accurate numerical solution.