Optimal control of the one-dimensional periodic wave equation (Q2365245)

The authors deals with maximum principles for an optimal control of the vibrating string (in the resonant case) with \(T\)-periodic conditions and a convex cost functional. \(T\) is a rational multiple of \(\pi\). A necessary and sufficient condition for the optimality of a pair \((y,u)\) (in \(L^\infty\)) is given. The proof is very delicate, due to the resonance phenomenon (the dimension of the kernel of the wave operator with \(T\)-periodic conditions is infinite).