Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems

Abstract

We consider a steady-state heat conduction problem in a multidimensional bounded domain Ω for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Γ 1 of the boundary and a constant heat flux q in the remaining portion Γ 2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary Γ 1 with heat transfer coefficient α and external temperature b. We obtain explicitly, for a rectangular domain in R 2 , an annulus in R 2 and a spherical shell in R 3 , the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on Γ 1 converge, when α → ∞, to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on Γ 1 . Also, we analyze the order of convergence in each case, which turns out to be 1/α being new for these kind of elliptic optimal control problems.

BDF2 schemes for optimal parameter control problems governed by bilinear parabolic equations

“…In the past few decades, there has been an enormous research effort towards parameter identification problems involving elliptic and other evolution equations; see, for example, References 1‐8 and therein. In a typical mathematical model, linear distributed 9,10 or boundary controls 11,12 are usually used. These controls are also called additive controls due to arising in the model equations as additive terms.…”