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
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.
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