Advances in Differential Equations and Control Processes
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Abstract: We consider a
steady-state heat conduction problem with mixed boundary conditions for
the Poisson equation in a bounded multidimensional domain W depending on a positive parameter a which represents the heat transfer coefficient on a
portion of the boundary of W. We consider, for each a cost function and we formulate boundary optimal
control problems with restrictions over the heat flux q
on a complementary portion of the boundary of W. We obtain that the optimality conditions are given by
a complementary free boundary problem in in terms of the adjoint state. We
prove that the optimal control and its corresponding system state and adjoint state for each a are strongly convergent to
andin and respectively when We also prove that these limit
functions are respectively the optimal control, the system state and the adjoint
state corresponding to another boundary optimal control problem with
restrictions for the same Poisson equation with a different boundary condition
on the portion We use the elliptic variational
inequality theory in order to prove all the strong convergences. In this paper,
we generalize the convergence result obtained in Belgacem et al. [3] by
considering boundary optimal control problems with restrictions on the heat flux
q defined on and the parameter a (which goes to infinity) is defined on
Keywords and phrases: variational inequality, boundary optimal control with restrictions, mixed elliptic problem, adjoint state, steady-state Stefan problem, optimality condition, free boundary problems.
