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Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

  • Pierluigi Colli
  • Andrea SignoriEmail author
  • Jürgen Sprekels
Article
  • 49 Downloads

Abstract

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

Keywords

Distributed optimal control Tumor growth Cancer treatment Phase field system Evolution equations Chemotaxis Adjoint system Necessary optimality conditions 

Mathematics Subject Classification

35K55 35Q92 49J20 92C50 

Notes

Acknowledgements

The authors are very grateful to the anonymous referee for the careful reading of the manuscript and for some useful suggestions. The research of Pierluigi Colli is supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati”, University of Pavia. In addition, PC gratefully acknowledges some other support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia, Italy.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano–BicoccaMilanItaly
  3. 3.Department of MathematicsHumboldt-Univesität zu BerlinBerlinGermany
  4. 4.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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