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

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Abstract

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.

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References

  1. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)

    Book  MATH  Google Scholar 

  2. Barbu, V., Colli, P., Gilardi, G., Marinoschi, G., Rocca, E.: Sliding mode control for a nonlinear phase-field system. SIAM J. Control Optim. 55, 2108–2133 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartolini, G., Fridman, L., Pisano, A., Usai, E. (eds.): Modern Sliding Mode Control Theory New Perspectives and Applications. Lecture Notes in Control and Information Sciences, vol. 375. Springer, New York (2008)

    MATH  Google Scholar 

  4. Bonetti, E., Colli, P., Tomassetti, G.: A non-smooth regularization of a forward-backward parabolic equation. Math. Models Methods Appl. Sci. 27, 641–661 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, vol. 5. Elsevier, Amsterdam (1973)

    MATH  Google Scholar 

  6. Brezzi, F., Gilardi, G.: Chapters 1–3. In: Kardestuncer, H., Norrie, D.H. (eds.) Finite Element Handbook. McGraw-Hill Book Co., New York (1987)

    Google Scholar 

  7. Byrne, H.M., Chaplain, M.A.J.: Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130, 151–181 (1995)

    Article  MATH  Google Scholar 

  8. Cheng, M.-B., Radisavljevic, V., Su, W.-C.: Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Autom. J. IFAC 47, 381–387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colli, P., Gilardi, G., Hilhorst, D.: On a Cahn–Hilliard type phase field model related to tumor growth. Discret. Contin. Dyn. Syst. 35, 2423–2442 (2015)

    Article  MATH  Google Scholar 

  10. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Vanishing viscosities and error estimate for a Cahn–Hilliard type phase-field system related to tumor growth. Nonlinear Anal. Real World Appl. 26, 93–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth. Discret. Contin. Dyn. Syst. Ser. S. 10, 37–54 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30, 2518–2546 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colturato, M.: Solvability of a class of phase field systems related to a sliding mode control problem. Appl. Math. 6, 623–650 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Colturato, M.: On a class of conserved phase field systems with a maximal monotone perturbation. Appl. Math. Optim. 4, 1–35 (2017)

    MathSciNet  Google Scholar 

  15. Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer. An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  16. Cristini, V., Li, X., Lowengrub, J.S., Wise, S.M.: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58, 723–763 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dai, M., Feireisl, E., Rocca, E., Schimperna, G., Schonbek, M.: Analysis of a diffuse interface model for multispecies tumor growth. Nonlinearity 30, 1639–1658 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. Taylor and Francis, London (1999)

    Google Scholar 

  19. Edwards, C., Colet, E., Fridman, L. (eds.): Advances in Variable Structure and Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol. 334. Springer, New York (2006)

    Google Scholar 

  20. Fridman, L., Moreno, J., Iriarte, R. (eds.): Sliding Modes After the First Decade of the 21st Century: State of the Art. Lecture Notes in Control and Information Sciences, vol. 412. Springer, New York (2011)

    Google Scholar 

  21. Frigeri, S., Grasselli, M., Rocca, E.: On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26, 215–243 (2015)

    Article  MATH  Google Scholar 

  22. Frigeri, S., Lam, K.F., Rocca, E.: On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, to appear. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds.) Solvability, Regularity, Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, Springer, Milan, pp. 1–28 (2017) (see also preprint arXiv:1703.03553)

  23. Garcke, H., Lam, K.F.: Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Math. 1, 318–360 (2016)

    Article  Google Scholar 

  24. Garcke, H., Lam, K.F.: Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis. Discret. Contin. Dyn. Syst. 37, 4277–4308 (2017)

    Article  MATH  Google Scholar 

  25. Garcke, H., Lam, K.F.: Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28, 284–316 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Garcke, H., Lam, K.F., Sitka, E., Styles, V.: A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26, 1095–1148 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Garcke, H., Lam, K.F., Nürnberg, R., Sitka, E.: A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis (2017), pp. 1–43. Preprint arXiv:1701.06656v1 [math.AP]

  28. Garcke, H., Lam, K.F., Rocca, E.: Optimal control of treatment time in a diffuse interface model for tumor growth. Appl. Math. Optim. (2017). doi:10.1007/s00245-017-9414-4

  29. Hawkins-Daarud, A., van der Zee, K.G., Oden, J.T.: Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Methods Biomed. Eng. 28, 3–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hilhorst, D., Kampmann, J., Nguyen, T.N., van der Zee, K.G.: Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25, 1011–1043 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Itkis, U.: Control Systems of Variable Structure. Wiley, New York (1976)

    Google Scholar 

  32. Levaggi, L.: Infinite dimensional systems’ sliding motions. Eur. J. Control 8, 508–516 (2002)

    Article  MATH  Google Scholar 

  33. Levaggi, L.: Existence of sliding motions for nonlinear evolution equations in Banach spaces, Discrete Contin. Dyn. Syst. In: 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications, pp. 477–487 (2013)

  34. Melchionna, S., Rocca, E.: Varifold solutions of a sharp interface limit of a diffuse interface model for tumor growth, Interfaces and Free Bound, pp. 1–29 (2017). to appear (see also preprint arXiv:1610.04478 [math.AP])

  35. Orlov, Y.V.: Application of Lyapunov method in distributed systems. Autom. Remote Control 44, 426–430 (1983)

    MATH  Google Scholar 

  36. Orlov, Y.V.: Discontinuous unit feedback control of uncertain infinite dimensional systems. IEEE Trans. Autom. Control 45, 834–843 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Orlov, Y.V., Utkin, V.I.: Use of sliding modes in distributed system control problems. Autom. Remote Control 43, 1127–1135 (1983)

    MathSciNet  MATH  Google Scholar 

  38. Orlov, Y.V., Utkin, V.I.: Sliding mode control in indefinite-dimensional systems. Autom. J. IFAC 23, 753–757 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  39. Orlov, Y.V., Utkin, V.I.: Unit sliding mode control in infinite-dimensional systems. Adaptive learning and control using sliding modes. Appl. Math. Comput. Sci. 8, 7–20 (1998)

    MathSciNet  MATH  Google Scholar 

  40. Rocca, E., Scala, R.: A rigorous sharp interface limit of a diffuse interface model related to tumor growth. J. Nonlinear Sci. 27, 847–872 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Simon, J.: Compact sets in the space \(L^p(0,T; B)\). Ann. Mater. Pura Appl. (4) 146(7), 65–96 (1987)

    MATH  Google Scholar 

  42. Utkin, V.: Sliding Modes in Control and Optimization, Communications and Control Engineering Series. Springer, Berlin (1992)

    Book  Google Scholar 

  43. Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems. Automation and Control Engineering, 2nd edn. CRC Press, Boca Raton (2009)

    Google Scholar 

  44. Xing, H., Li, D., Gao, C., Kao, Y.: Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay. J. Franklin Inst. 350, 397–418 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method. J. Theor. Biol. 253, 524–543 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Young, K.D., Özgüner, Ü. (eds.): Variable Structure Systems, Sliding Mode and Nonlinear Control. Springer, New York (1999)

    Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia and IMATI-C.N.R., Pavia, via Ferrata 1, 27100, Pavia, Italy

    Pierluigi Colli, Gianni Gilardi & Elisabetta Rocca

  2. “Gheorghe Mihoc-Caius Iacob”, Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711, Bucharest, Romania

    Gabriela Marinoschi

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  1. Pierluigi Colli
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  2. Gianni Gilardi
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  3. Gabriela Marinoschi
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  4. Elisabetta Rocca
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Correspondence to Elisabetta Rocca.

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Colli, P., Gilardi, G., Marinoschi, G. et al. Sliding Mode Control for a Phase Field System Related to Tumor Growth. Appl Math Optim 79, 647–670 (2019). https://doi.org/10.1007/s00245-017-9451-z

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