WIAS Preprint No. 3165, (2025)

On non-autonomous parabolic equations with measure-valued right hand sides and applications to optimal control



Authors

  • Kunisch, Karl
  • Rehberg, Joachim

2020 Mathematics Subject Classification

  • 35B65 35K20 35B45 28A75 49J20

Keywords

  • Non-autonomous evolution equations, parabolic initial boundary value problems, maximal parabolic regularity measure-valued right hand sides, optimal control

DOI

10.20347/WIAS.PREPRINT.3165

Abstract

The main aim of this paper is to develop a theory for non-autonomous parabolic equations with time-dependent measures on the spatial domain appearing as right hand sides. Restricting these measures to ones which have their supports on 'curves' or 'surfaces' -- the latter understood in the sense of geometric measure theory -- we succeed in interpreting them as distributional objects from a (negatively indexed) Sobolev--Slobodetskii space with differentiability index close to minus one. For these indices a tailor suited parabolic theory is established, based on previous results. It is also demonstrated that the proposed frame work is well-suited for optimal control with controls acting on sub-manifolds.

Download Documents