Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems
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Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems Julieta Bollati1 · Claudia M. Gariboldi2 · Domingo A. Tarzia1 Received: 28 June 2019 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract We consider a steady-state heat conduction problem in a multidimensional bounded domain for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion 1 of the boundary and a constant heat flux q in the remaining portion 2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary 1 with heat transfer coefficient α and external temperature b. We obtain explicitly, for a rectangular domain in R2 , an annulus in R2 and a spherical shell in R3 , 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 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 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. Keywords Elliptic variational equalities · Distributed and boundary optimal control problems · Mixed boundary conditions · Explicit solutions · Optimality conditions Mathematics Subject Classification 35C05 · 35J25 · 35J86 · 35R35 · 49J20 · 49K20
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Domingo A. Tarzia [email protected]
Extended author information available on the last page of the article
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J. Bollati et al.
1 Introduction The goal of this paper is to show the explicit solution for eight elliptic optimal control problems in two and three dimensional cases. We consider a bounded domain in Rn (n = 2, 3), whose regular boundary consist of the union of three disjoint portions 1 , 2 and 3 with meas(1 ) > 0, meas(2 ) > 0 and meas(3 ) ≥ 0. We present the following steady-state heat conduction problems S and Sα (for each parameter α > 0) respectively, with mixed boundary conditions u = g, in −u α = g
in
∂u ∂u = q, = 0, ∂n 2 ∂n 3 ∂u α ∂u α = α(u − b), − = q, = 0, 2 ∂n ∂n 3
u = b, 1
∂u α − ∂n 1
−
(1) (2)
where g is the internal energy in , b is the temperature on 1 for (1) and the temperature of the external neighborhood of 1 for (2), q is the heat flux on 2 and α > 0 is the heat transfer coeffi
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