Applied Mathematics & Optimization

, Volume 79, Issue 3, pp 647–670 | Cite as

Sliding Mode Control for a Phase Field System Related to Tumor Growth

  • Pierluigi Colli
  • Gianni Gilardi
  • Gabriela Marinoschi
  • Elisabetta RoccaEmail author


In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn–Hilliard type equation for the phase variable with a reaction–diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn–Hilliard type equation and leads the phase onto a prescribed target \(\varphi ^*\) in finite time.


Sliding mode control Cahn–Hilliard system Reaction–diffusion equation Tumor growth Nonlinear boundary value problem State-feedback control law 

Mathematics Subject Classification

34H15 35K25 35K61 93B52 92C50 97M60 



This research activity has been performed in the framework of an Italian-Romanian three-year project on “Control and stabilization problems for phase field and biological systems” financed by the Italian CNR and the Romanian Academy. Moreover, the financial support of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese” is gratefully acknowledged by the authors. The present paper also benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations” for PC and GG, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for PC, GG and ER, and the UEFISCDI project PN-III-ID-PCE-2016-0011 for GM. Last but not least, the authors are grateful to the anonymous referee for the careful reading of the paper and for some useful suggestion.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Pierluigi Colli
    • 1
  • Gianni Gilardi
    • 1
  • Gabriela Marinoschi
    • 2
  • Elisabetta Rocca
    • 1
    Email author
  1. 1.Dipartimento di Matematica “F. Casorati”Università di Pavia and IMATI-C.N.R., PaviaPaviaItaly
  2. 2.“Gheorghe Mihoc-Caius Iacob”Institute of Mathematical Statistics and Applied Mathematics of the Romanian AcademyBucharestRomania

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