Global Convergence of a Nonsmooth Newton Method for Control-State Constrained Optimal Control Problems

We investigate a nonsmooth Newton method for the numerical solution of optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton's method. We prove the global convergence and the locally superlinear convergence under certain regularity conditions. The globalized method is based on the minimization of the squared residual norm. Numerical examples for the Rayleigh problem conclude the article.