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