Neumann Boundary Control of Hyperbolic Equations with Pointwise State Constraints

We consider optimal control problems for hyperbolic equations with controls in Neumann boundary conditions with pointwise constraints on the control and state functions. Focusing on the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the form of a pointwise Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.