Superlinear Convergence of a Semismooth Newton Method for Some Optimization Problems with Applications to Control Theory

In this paper, we formulate a semismooth Newton method for an abstract optimization problem and prove its superlinear convergence by assuming that the no-gap second order sufficient optimality condition and the strict complementarity condition are fulfilled at the local minimizer. Many control problems fit this abstract formulation. In particular, we apply this abstract result to distributed control problems of a semilinear elliptic equation, to boundary bilinear control problems associated with a semilinear elliptic equation, and to distributed control of a semilinear parabolic equation.