Verification of optimality for optimal control problems

Many technical processes are described by partial differential equations. Naturally, it is important to optimize these processes. This leads to optimization problems in an infinite-dimensional setting. Such problems cannot be solved by hand in general, since they involve partial differential equations. Hence, it is necessary to derive finite- dimensional approximations by discretization, which can then be solved with the help of computers. If one has computed a solution of such a discretized problem, the question arises, whether this solution approximates a local or even a global optimum of the original problem. This project will deal with this issue. The main focus is on establishing methods that enable to answer the question about optimality.

The project investigated methods to assess the accuracy of approximations of solutions of optimal control problems subject to partial differential equations. Since partial differential equations cannot be solved explicitly, the partial differential equations are discretized, which transforms them into a system of nonlinear equations. This discretization is used to obtain a finite-dimensional optimization problem as approximation of the original optimal control problem. If a solution of this finite-dimensional problem is available, the question turns up about the reliability and accuracy of the result. In the project, we developed methods to answer this question with the help of computers for two classes of problems. First, we investigated residual-based a-posteriori error estimators for state-constrained and convex optimal control problems. Based on an estimate about the distance of the approximate solution to the feasible set of the continuous problem, we could derive upper error bounds for the discretization error. Moreover, the error estimator is convergent, that is, if the discrete quantities converge to the continuous solution, then the estimator tends to zero. In numerical experiments, we used this error estimator to guide adaptive mesh refinement. We demonstrated that this adaptive procedure lead to efficient discretization of the problem. Second, we analyzed problems with a finite-number of optimization parameters. Such problems appear in applications in parameter identification problems, where unknown parameters should be determined based on measurements. One way to solve such problems is to use a least-squares formulation. Here, we could succeed in answering the question on accuracy of approximations. Key ingredients in our method are precise estimates on the smallest eigenvalue of the second derivative of the Lagrangian as well as constant-free estimates of the residuals in the first-order optimality system. If these residuals are small compared to the smallest eigenvalue mentioned above then the numerically obtained approximation is close to a solution of the continuous problem.