Duality in the optimal control of distributed parameter systems governed by hyperbolic equations (Q1572815)

The authors model a hyperbolic distributed parameter control system as a minimization problem \(\mathcal P\) for a system described by a second order differential equation in a Hilbert space. There are control constraints, but no state constraints or target condition. They define a (maximization) dual problem \({\mathcal D}\) and show that \(\max K \leq \min J,\) where \(J\) (resp. \(K)\) is the objective functional of \({\mathcal P}\) (resp. of \(\mathcal D).\) Under additional conditions, they are able to prove that \(\max K = \min J.\)