WIAS Preprint No. 1599, (2011)

Boundary coefficient control --- A maximal parabolic regularity approach


  • Hömberg, Dietmar
    ORCID: 0000-0001-9460-5729
  • Krumbiegel, Klaus
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35K20 35B65 47F05 49J20 49K20


  • Parabolic equation, mixed boundary condition, maximal parabolic $L^p$-regularity, optimal control, sufficient optimality conditions




We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

Appeared in

  • Appl. Math. Optim., 67 (2013) pp. 3--31 under the title ``Optimal control of a parabolic equation with dynamic boundary condition"

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