Penalty Techniques for State Constrained Optimal Control Problems with the Wave Equation

• Authors: Martin Gugat
• Venue: SIAM Journal on Control and Optimization
• Year: 2009

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 $1/\sqrt{\gamma}$, 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 $1/\sqrt{\beta}$, 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 $L^\infty$-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.