2016/06/17 by M. Hassan Farshbaf-Shaker, Farshbaf-Shaker, M. Hassan, Christian Heinemann +1
Computer Science · Engineering · Materials Science · #35D35 #35M33 #35Q74 #49J20 #49K20 #74A45 #74D10 #74F99 #74P99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Solidification and crystal growth phenomena #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1606.05555
openalex publication_date 2016/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Controlling the growth of material damage is an important engineering task\nwith plenty of real world applications. In this paper we approach this topic\nfrom the mathematical point of view by investigating an optimal boundary\ncontrol problem for a damage phase-field model for viscoelastic media. We\nconsider non-homogeneous Neumann data for the displacement field which describe\nexternal boundary forces and act as control variable. The underlying\nhyberbolic-parabolic PDE system for the state variables exhibit highly\nnonlinear terms which emerge in context with damage processes. The cost\nfunctional is of tracking type, and constraints for the control variable are\nprescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann:\nA phase field approach for optimal boundary control of damage processes in\ntwo-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015),\n2749--2793], where global-in-time well-posedness of strong solutions to the\nlower level problem and existence of optimal controls of the upper level\nproblem have been established, we show in this contribution differentiability\nof the control-to-state mapping, well-posedness of the linearization and\nexistence of solutions of the adjoint state system. Due to the highly nonlinear\nnature of the state system which has by our knowledge not been considered for\noptimal control problems in the literature, we present a very weak formulation\nand estimation techniques of the associated adjoint system. For mathematical\nreasons the analysis is restricted here to the two-dimensional case. We\nconclude our results with first-order necessary optimality conditions in terms\nof a variational inequality together with PDEs for the state and adjoint state\nsystem.\n