2017/09/07 by Pierluigi Colli, Colli, Pierluigi, Gianni Gilardi +3 · 3 citations
Computer Science · Materials Science · Mathematics · #35K25 #35K61 #49J20 #49K20 #76R05 #80A22 #82C26 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1709.02335
openalex publication_date 2017/09/07 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
In this paper, we investigate a distributed optimal control problem for a\nconvective viscous Cahn-Hilliard system with dynamic boundary conditions. Such\nsystems govern phase separation processes between two phases taking place in an\nincompressible fluid in a container and, at the same time, on the container\nboundary. The cost functional is of standard tracking type, while the control\nis exerted by the velocity of the fluid in the bulk. In this way, the coupling\nbetween the state (given by the associated order parameter and chemical\npotential) and control variables in the governing system of nonlinear partial\ndifferential equations is bilinear, which presents an additional difficulty for\nthe analysis. The nonlinearities in the bulk and surface free energies are of\nlogarithmic type, which entails that the thermodynamic forces driving the phase\nseparation process may become singular. We show existence for the optimal\ncontrol problem under investigation, prove the Fr 'echet differentiability of\nthe associated control-to-state mapping in suitable Banach spaces and derive\nthe first-order necessary optimality conditions in terms of a variational\ninequality and the associated adjoint system. Due to the strong nonlinear\ncouplings between state variables and control, the corresponding proofs require\na considerable analytical effort.\n