WIAS Preprint No. 2863, (2021)

Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Signori, Andrea
    ORCID: 0000-0001-7025-977X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2020 Mathematics Subject Classification

  • 35K55 35K51 49J20 49K20 49J50


  • Phase field model, thermal memory, well-posedness, optimal control, first-order necessary optimality conditions, adjoint system




A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fréchet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.

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