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Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions
Author(s) -
D'Apice Ciro,
De Maio Umberto,
Kogut Peter I.
Publication year - 2020
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2660
Subject(s) - mathematics , optimal control , boundary value problem , neumann boundary condition , nonlinear system , mixed boundary condition , boundary (topology) , elliptic boundary value problem , mathematical analysis , dirichlet distribution , free boundary problem , maximum principle , mathematical optimization , physics , quantum mechanics
Summary The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.