Neumann boundary optimal control problems governed by parabolic variational equalities

C.M. BOLLO; C. M. GARIBOLDI; D. A. TARZIA

CONTROL AND CYBERNETICS

POLISH ACAD SCIENCES SYSTEMS RESEARCH INST

Año: 2021 vol. 50 p. 227 - 251

0324-8569

We consider a heat conduction problem S with mixedboundary conditions in an n-dimensional domain Ω with regularboundary and a family of problems Sα with also mixed boundaryconditions in Ω, where α > 0 is the heat transfer coefficient on theportion of the boundary Γ1. In relation to these state systems, weformulate Neumann boundary optimal control problems on the heatflux q which is definite on the complementary portion Γ2 of theboundary of Ω. We obtain existence and uniqueness of the optimalcontrols, the first order optimality conditions in terms of the adjointstate and the convergence of the optimal controls, the system stateand the adjoint state when the heat transfer coefficient α goes toinfinity. Furthermore, we formulate particular boundary optimalcontrol problems on a real parameter λ, in relation to the parabolicproblems S and Sα and to mixed elliptic problems P and Pα. Wefind a explicit form for the optimal controls, we prove monotonyproperties and we obtain convergence results when the parametertime goes to infinity.