2017/05/31 by Constantin Christof, Christian Clason, Christian Meyer +1
Computer Science · Engineering · Mathematics · #Adjoint equation #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computer science #Derivative (finance) #Differentiable function #Differential equation #Directional derivative #Elliptic curve #Elliptic partial differential equation #Fréchet derivative #Mathematical analysis #Mathematical optimization #Mathematics #Optimal control #Physics #Regularization (linguistics) #Stability and Controllability of Differential Equations #State variable #math.OC #msc:49J52 #msc:49K20 #msc:49M15
paper · pdf · doi:10.3934/mcrf.2018011
published as Mathematical Control and Related Fields 8 (2018), 247-276
arxiv created 2017/11/27 · openalex publication_date 2018/01/01 · arxiv updated 2018/01/29 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand sub-differential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.