LAUSR

We consider a steady-state heat conduction problem in a multidimensional bounded domain $$\Omega $$ Ω for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion $$\Gamma _1$$ Γ 1 of the boundary and a constant heat flux q in the remaining portion $$\Gamma _2$$ Γ 2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary $$\Gamma _1$$ Γ 1 with heat transfer coefficient $$\alpha $$ α and external temperature b . We obtain explicitly, for a rectangular domain in $${\mathbb {R}}^{2}$$ R 2 , an annulus in $${\mathbb {R}}^{2}$$ R 2 and a spherical shell in $${\mathbb {R}}^{3}$$ 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 $$\Gamma _1$$ Γ 1 converge, when $$\alpha \rightarrow \infty $$ α → ∞ , to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on $$\Gamma _1$$ Γ 1 . Also, we analyze the order of convergence in each case, which turns out to be $$1/\alpha $$ 1 / α being new for these kind of elliptic optimal control problems.