Summary: For \(p\neq 2\), only few results about analytic solutions of problems of optimal control of distributed parameter systems with minimal \(L^p\)-norm have been reported in the literature. In this paper we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. The aim is to steer the system from a position of rest to a constant terminal state in a given finite time. Also more general final configurations are considered.
The objective function that is to be minimized is the maximum of the \(L^p\)-norms of the control functions at both boundaries. It is shown that the analytic solution is, in fact, independent of the choice of the \(p\)-norm that is minimized. So the optimal controls solve a problem of multicriteria optimization, with the \(L^p\)-norms as objective functions.