PENALTY TECHNIQUES FOR STATE CONSTRAINED OPTIMAL CONTROL PROBLEMS WITH THE WAVE EQUATION

We consider a system that is governed by a PDE of hyperbolic type, namely, the wave equation, and is controlled by a Dirichlet boundary control. For the penalization of terminal constraints, we compare the convergence of three penalization techniques: a differentiable penalty method, an exact penalization, and a smoothed exact penalization. The error of the solution of the differentiably penalized problem is at most of the order 1/root gamma, where gamma is the penalty parameter. The exact penalization yields the exact solution if gamma is sufficiently large. If gamma is sufficiently large for the smoothed exact penalization, we obtain an error bound of the order 1/root beta, where beta is the additional smoothing parameter. This method yields differentiable objective functions that remain uniformly strongly convex, since gamma need not tend to infinity to obtain convergence. We also consider the penalization of distributed inequality state constraints that prescribe an upper bound for the L(infinity)-norm of the state. We prove the convergence with respect to the L(2)-norm for a penalization of the constraint violation measured in the L(1)-norm. We use this penalization technique to prove the existence of optimal controls for the inequality state constrained problem.