Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

In the present paper, optimal control problems for the well-known wave equation are investigated. The intention is to study the order of the numerical approximations for both the optimal state and the optimal control variables for problems with known analytical solutions. The numerical method chosen here is a full discretization method based on appropriate finite differences by which the PDE constrained optimal control problem is transformed into a nonlinear programming problem (NLP). Hence, we follow here the approach 'first discretize, then optimize', which allows us to make use not only of powerful methods for the solution of NLPs, but also to compute sensitivity differentials, a necessary tool for real-time control.