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Optimal Distributed Control of a Cahn–Hilliard–Darcy System with Mass Sources

  • Jürgen Sprekels
  • Hao WuEmail author
Article
  • 38 Downloads

Abstract

In this paper, we study an optimal control problem for a two-dimensional Cahn–Hilliard–Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.

Keywords

Cahn–Hilliard–Darcy system Distributed optimal control Necessary optimality condition 

Mathematics Subject Classification

35G25 49J20 49K20 49J50 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading and helpful suggestions. The research of H. Wu is partially supported by NNSFC Grant No. 11631011 and the Shanghai Center for Mathematical Sciences.

References

  1. 1.
    Anderson, D.-M., McFadden, G.-B., Wheeler, A.-A.: Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30(1), 139–165 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellomo, N., Li, N.-K., Maini, P.-K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosia, S., Conti, M., Grasselli, M.: On the Cahn–Hilliard–Brinkman system. Commun. Math. Sci. 13, 1541–1567 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cahn, J.W., Hilliard, J.-E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  5. 5.
    Chen, Y., Wise, S.-M., Shenoy, V.-B., Lowengrub, J.-S.: A stable scheme for a nonlinear multiphase tumor growth model with an elastic membrane. Int. J. Numer. Methods Biomed. Eng. 30, 726–754 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Colli, P., Farshbaf-Shaker, M.-H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53, 2696–2721 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colli, P., Gilardi, G., Hilhorst, D.: On a Cahn–Hilliard type phase field system related to tumor growth. Discret. Contin. Dyn. Syst. 35, 2423–2442 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4, 311–325 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions. Appl. Math. Optim. 72, 195–225 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30, 2518–2546 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions. SIAM J. Control Optim. 56, 1665–1691 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a convective Cahn–Hilliard system with double obstacles and dynamic boundary conditions: a ‘deep quench’ approach. J. Convex Anal. 26 (2019) (see also Preprint arXiv:1709.03892 [math. AP] (2017), 1–30)
  14. 14.
    Conti, M., Giorgini, A.: The three-dimensional Cahn–Hilliard–Brinkman system with unmatched viscosities (2018). https://hal.archives-ouvertes.fr/hal-01559179
  15. 15.
    Cristini, V., Lowengrub, J.-S.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  16. 16.
    Dai, M., Feireisl, E., Rocca, E., Schimperna, G., Schonbek, M.: Analysis of a diffuse interface model for multi-species tumor growth. Nonlinearity 30(4), 1639 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Della Porta, F., Giorgini, A., Grasselli, M.: The nonlocal Cahn–Hilliard–Hele–Shaw system with logarithmic potential. Nonlinearity 31, 4851 (2018)Google Scholar
  18. 18.
    Della Porta, F., Grasselli, M., On the nonlocal Cahn–Hilliard–Brinkman and Cahn–Hilliard–Hele–Shaw systems. Commun. Pure Appl. Anal. 15, 299–317 (2016), Erratum: Commun. Pure Appl. Anal. 16, 369–372 (2017)Google Scholar
  19. 19.
    Fasano, A., Bertuzzi, A., Gandolfi, A.: Mathematical Modeling of Tumour Growth and Treatment, Complex Systems in Biomedicine, pp. 71–108. Springer, Milan (2006)zbMATHGoogle Scholar
  20. 20.
    Feng, X., Wise, S.-M.: Analysis of a Darcy–Cahn–Hilliard diffuse interface model for the Hele–Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50, 1320–1343 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Frieboes, H.-B., Jin, F., Chuang, Y.-L., Wise, S.-M., Lowengrub, J.-S., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis. J. Theor. Biol. 264, 1254–1278 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Friedman, A.: Mathematical analysis and challenges arising from models of tumor growth. Math. Models Methods Appl. Sci. 17, 1751–1772 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Frigeri, S., Grasselli, M., Rocca, E.: On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26, 215–243 (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Frigeri, S., Grasselli, M., Sprekels, J.: Optimal distributed control of two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems with degenerate mobility and singular potential. Appl. Math. Optim. (2018).  https://doi.org/10.1007/s00245-018-9524-7
  25. 25.
    Frigeri, S., Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal Cahn–Hilliard/Navier–Stokes system in two dimensions. SIAM J. Control. Optim. 54, 221–250 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Frigeri, S., Lam, K.-F., Rocca, E., Schimperna, G.: On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials. Commun. Math. Sci. 16, 821–856 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Garcke, H., Lam, K.-F.: Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Math. 1(3), 318–360 (2016)CrossRefGoogle Scholar
  28. 28.
    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(6), 1095–1148 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Garcke, H., Lam, K.-F., Rocca, E.: Optimal control of treatment time in a diffuse interface model for tumour growth. Appl. Math. Optim. (2017).  https://doi.org/10.1007/s00245-017-9414-4
  30. 30.
    Garcke, H., Lam, K.-F.: On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds.) Trends in Applications of Mathematics to Mechanics, Springer INdAM Series, vol. 27. Springer, Berlin (2018)Google Scholar
  31. 31.
    Garcke, H., Lam, K.-F., Nürnberg, R., Sitka, E.: A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Models Methods Appl. Sci. 28(3), 525–577 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Giorgini, A., Grasselli, M., Wu, H.: The Cahn–Hilliard–Hele–Shaw system with singular potential. Ann. Inst. H. Poincare Anal. Non Lineaire 35(4), 1079–1118 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    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. Meth. Biomed. Eng. 28, 3–24 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50, 388–418 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hintermüller, M., Wegner, D.: Optimal control of a semi-discrete Cahn–Hilliard–Navier–Stokes system. SIAM J. Control Optim. 52, 747–772 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hintermüller, M., Wegner, D.: Distributed and boundary control problems for the semi-discrete Cahn–Hilliard/Navier–Stokes system with non-smooth Ginzburg–Landau energies. In: Topological Optimization and Optimal Transport, Radon Series on Computational and Applied Mathematics, vol. 17, pp. 40–63 (2017)Google Scholar
  37. 37.
    Hintermüller, H., Keil, M., Wegner, D.: Optimal control of a semi-discrete Cahn–Hilliard–Navier–Stokes system with non-matched fluid densities. SIAM J. Control Optim. 55, 1954–1989 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259, 3032–3077 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lowengrub, J.-S., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 2617–2654 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lowengrub, J.-S., Titi, E.-S., Zhao, K.: Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24, 691–734 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53, 1654–1680 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. AMS, Providence (2010)zbMATHGoogle Scholar
  44. 44.
    Wang, X.-M., Wu, H.: Long-time behavior for the Hele–Shaw–Cahn–Hilliard system. Asymptot. Anal. 78, 217–245 (2012)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wang, X.-M., Zhang, Z.-F.: Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 367–384 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wise, S.-M.: Unconditionally stable finite difference, nonlinear multigrid simulations of the Cahn–Hilliard–Hele–Shaw system of equations. J. Sci. Comput. 44, 38–68 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Wise, S.-M., Lowengrub, J.-S., Cristini, V.: An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. 53, 1–20 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Zhao, X.-P., Liu, C.-C.: Optimal control of the convective Cahn–Hilliard equation. Appl. Anal. 92, 1028–1045 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Zhao, X.-P., Liu, C.-C.: Optimal control of the convective Cahn–Hilliard equation in 2D case. Appl. Math. Optim. 70, 61–82 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina

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