March  2020, 9(1): 193-217. doi: 10.3934/eect.2020003

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125 Milano, Italy

Received  November 2018 Revised  April 2019 Published  August 2019

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn–Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a different strategy which in the literature is known as the "deep quench" approach in order to obtain some optimality conditions that have to be interpreted in a more general framework. We establish the existence of optimal controls and some first-order optimality conditions for the system are derived by employing suitable approximation schemes.

Citation: Andrea Signori. Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach. Evolution Equations & Control Theory, 2020, 9 (1) : 193-217. doi: 10.3934/eect.2020003
References:
[1]

A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani, A Cahn–Hilliard–type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598–7626. Available from: https://doi.org/10.1002/mma.4548. doi: 10.1002/mma.4548. Google Scholar

[2]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566–597. Available from: https://doi.org/10.1016/0022-247X(81)90125-6. doi: 10.1016/0022-247X(81)90125-6. Google Scholar

[3]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud., North-Holland, Amsterdam, 1973. Google Scholar

[4]

C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., (2019), 1–49. Available from: https://doi.org/10.1007/s00245-019-09562-5. doi: 10.1007/s00245-019-09562-5. Google Scholar

[5]

P. Colli, M. H. Farshbaf-Shaker, G. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696–2721. Available from: https://doi.org/10.1137/140984749. doi: 10.1137/140984749. Google Scholar

[6]

P. Colli, M. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1–24. Available from: https://doi.org/10.1007/s00245-014-9250-8. doi: 10.1007/s00245-014-9250-8. Google Scholar

[7]

P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423–2442. Available from: https://doi.org/10.3934/dcds.2015.35.2423. doi: 10.3934/dcds.2015.35.2423. Google Scholar

[8]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93–108. Available from: https://doi.org/10.1016/j.nonrwa.2015.05.002. doi: 10.1016/j.nonrwa.2015.05.002. Google Scholar

[9]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518–2546. Available from: https://doi.org/10.1088/1361-6544/aa6e5f. doi: 10.1088/1361-6544/aa6e5f. Google Scholar

[10]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modeling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37–54. Available from: https://doi.org/10.3934/dcdss.2017002. doi: 10.3934/dcdss.2017002. Google Scholar

[11]

P. Colli, G. Gilardi and J. Sprekels, 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), 485–514. Available from: http://www.heldermann.de/JCA/JCA26/JCA262/jca26024.htm. Google Scholar

[12]

P. Colli, G. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35–58. Available from: https://doi.org/10.3934/eect.2017003. doi: 10.3934/eect.2017003. Google Scholar

[13]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn-Hilliard system with dynamic boundary condition and double obstacle inclusions, in Solvability, Regularity, Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Milan, 22, 2017,151–182. Available from: https://doi.org/10.1007/978-3-319-64489-9_7. Google Scholar

[14]

V. Cristini, X. R. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723–763. Available from: https://doi.org/10.1007/s00285-008-0215-x. doi: 10.1007/s00285-008-0215-x. Google Scholar

[15] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Leiden, 2010. doi: 10.1017/CBO9780511781452. Google Scholar
[16]

M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639–1658. Available from: https://doi.org/10.1088/1361-6544/aa6063. doi: 10.1088/1361-6544/aa6063. Google Scholar

[17]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, preprint, arXiv: 1903.00333.Google Scholar

[18]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019). Available from: https://doi.org/10.1007/s00526-019-1579-z. doi: 10.1007/s00526-019-1579-z. Google Scholar

[19]

M. Ebenbeck and H. Garcke, Analysis of a Cahn Hilliard Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998–6036, Available from: https://doi.org/10.1016/j.jde.2018.10.045. doi: 10.1016/j.jde.2018.10.045. Google Scholar

[20]

S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215–243. Available from: https://doi.org/10.1017/S0956792514000436.Google Scholar

[21]

S. Frigeri, K. F. Lam, E. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821–856. Available from: http://dx.doi.org/10.4310/CMS.2018.v16.n3.a11. doi: 10.4310/CMS.2018.v16.n3.a11. Google Scholar

[22]

S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, In Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Cham, 22 (2017), 217–254. Available from: https://doi.org/10.1007/978-3-319-64489-9_9. Google Scholar

[23]

H. Garcke and K. F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European. J. Appl. Math., 28 (2017), 284–316. Available from: https://doi.org/10.1017/S0956792516000292. doi: 10.1017/S0956792516000292. Google Scholar

[24]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277–4308. Available from: https://doi.org/10.3934/dcds.2017183. doi: 10.3934/dcds.2017183. Google Scholar

[25]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318–360. Available from: https://doi.org/10.3934/Math.2016.3.318. doi: 10.3934/Math.2016.3.318. Google Scholar

[26]

H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, in Trends on Applications of Mathematics to Mechanics, E. Rocca, U. Stefanelli, L. Truskinovski, A. Visintin (ed.), Springer INdAM Series, Springer, Cham, 27 (2018), 243–264. Available from: https://doi.org/10.1007/978-3-319-75940-1_12. Google Scholar

[27]

H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495–544. Available from: https://doi.org/10.1007/s00245-017-9414-4. doi: 10.1007/s00245-017-9414-4. Google Scholar

[28]

H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525–577. Available from: https://doi.org/10.1142/S0218202518500148. doi: 10.1142/S0218202518500148. Google Scholar

[29]

A. Hawkins-Daarud, S. Prudhomme, K. G. van der Zee and J. T. Oden, Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, J. Math. Biol., 67 (2013), 1457–1485. Available from: https://doi.org/10.1007/s00285-012-0595-9. doi: 10.1007/s00285-012-0595-9. Google Scholar

[30]

A. Hawkins-Daruud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng., 28 (2012), 3–24. Available from: https://doi.org/10.1002/cnm.1467. doi: 10.1002/cnm.1467. Google Scholar

[31]

D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011–1043. Available from: https://doi.org/10.1142/S0218202515500268. doi: 10.1142/S0218202515500268. Google Scholar

[32]

J.-L. Lions, Contrôle Optimal de Systèmes Gouverneś par des Equations aux Dérivées Partielles, Dunod, Paris, 1968. Google Scholar

[33]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479–544. Available from: https://doi.org/10.3934/Math.2017.2.479 doi: 10.3934/Math.2017.2.479. Google Scholar

[34]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., (2018), 1–33. Available from: https://doi.org/10.1007/s00245-018-9538-1. doi: 10.1007/s00245-018-9538-1. Google Scholar

[35]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1986) 65–96. Available from: https://doi.org/10.1007/BF01762360.Google Scholar

[36]

J. Sprekels and H. Wu, Optimal Distributed Control of a Cahn-Hilliard Darcy System with Mass Sources, Appl. Math. Optim., (2019), 1–42. Available from: https://doi.org/10.1007/s00245-019-09555-4.Google Scholar

[37]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Grad. Stud. in Math., 112, AMS, Providence, RI, 2010. doi: 10.1090/gsm/112. Google Scholar

[38]

X. Wu, G. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahn-Hilliard models with applications to diffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180–203. Available from: https://doi.org/10.1002/cnm.2597. doi: 10.1002/cnm.2597. Google Scholar

show all references

References:
[1]

A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani, A Cahn–Hilliard–type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598–7626. Available from: https://doi.org/10.1002/mma.4548. doi: 10.1002/mma.4548. Google Scholar

[2]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566–597. Available from: https://doi.org/10.1016/0022-247X(81)90125-6. doi: 10.1016/0022-247X(81)90125-6. Google Scholar

[3]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud., North-Holland, Amsterdam, 1973. Google Scholar

[4]

C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., (2019), 1–49. Available from: https://doi.org/10.1007/s00245-019-09562-5. doi: 10.1007/s00245-019-09562-5. Google Scholar

[5]

P. Colli, M. H. Farshbaf-Shaker, G. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696–2721. Available from: https://doi.org/10.1137/140984749. doi: 10.1137/140984749. Google Scholar

[6]

P. Colli, M. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1–24. Available from: https://doi.org/10.1007/s00245-014-9250-8. doi: 10.1007/s00245-014-9250-8. Google Scholar

[7]

P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423–2442. Available from: https://doi.org/10.3934/dcds.2015.35.2423. doi: 10.3934/dcds.2015.35.2423. Google Scholar

[8]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93–108. Available from: https://doi.org/10.1016/j.nonrwa.2015.05.002. doi: 10.1016/j.nonrwa.2015.05.002. Google Scholar

[9]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518–2546. Available from: https://doi.org/10.1088/1361-6544/aa6e5f. doi: 10.1088/1361-6544/aa6e5f. Google Scholar

[10]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modeling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37–54. Available from: https://doi.org/10.3934/dcdss.2017002. doi: 10.3934/dcdss.2017002. Google Scholar

[11]

P. Colli, G. Gilardi and J. Sprekels, 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), 485–514. Available from: http://www.heldermann.de/JCA/JCA26/JCA262/jca26024.htm. Google Scholar

[12]

P. Colli, G. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35–58. Available from: https://doi.org/10.3934/eect.2017003. doi: 10.3934/eect.2017003. Google Scholar

[13]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn-Hilliard system with dynamic boundary condition and double obstacle inclusions, in Solvability, Regularity, Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Milan, 22, 2017,151–182. Available from: https://doi.org/10.1007/978-3-319-64489-9_7. Google Scholar

[14]

V. Cristini, X. R. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723–763. Available from: https://doi.org/10.1007/s00285-008-0215-x. doi: 10.1007/s00285-008-0215-x. Google Scholar

[15] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Leiden, 2010. doi: 10.1017/CBO9780511781452. Google Scholar
[16]

M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639–1658. Available from: https://doi.org/10.1088/1361-6544/aa6063. doi: 10.1088/1361-6544/aa6063. Google Scholar

[17]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, preprint, arXiv: 1903.00333.Google Scholar

[18]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019). Available from: https://doi.org/10.1007/s00526-019-1579-z. doi: 10.1007/s00526-019-1579-z. Google Scholar

[19]

M. Ebenbeck and H. Garcke, Analysis of a Cahn Hilliard Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998–6036, Available from: https://doi.org/10.1016/j.jde.2018.10.045. doi: 10.1016/j.jde.2018.10.045. Google Scholar

[20]

S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215–243. Available from: https://doi.org/10.1017/S0956792514000436.Google Scholar

[21]

S. Frigeri, K. F. Lam, E. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821–856. Available from: http://dx.doi.org/10.4310/CMS.2018.v16.n3.a11. doi: 10.4310/CMS.2018.v16.n3.a11. Google Scholar

[22]

S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, In Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels (ed.), Springer INdAM Series, Springer, Cham, 22 (2017), 217–254. Available from: https://doi.org/10.1007/978-3-319-64489-9_9. Google Scholar

[23]

H. Garcke and K. F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European. J. Appl. Math., 28 (2017), 284–316. Available from: https://doi.org/10.1017/S0956792516000292. doi: 10.1017/S0956792516000292. Google Scholar

[24]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277–4308. Available from: https://doi.org/10.3934/dcds.2017183. doi: 10.3934/dcds.2017183. Google Scholar

[25]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318–360. Available from: https://doi.org/10.3934/Math.2016.3.318. doi: 10.3934/Math.2016.3.318. Google Scholar

[26]

H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, in Trends on Applications of Mathematics to Mechanics, E. Rocca, U. Stefanelli, L. Truskinovski, A. Visintin (ed.), Springer INdAM Series, Springer, Cham, 27 (2018), 243–264. Available from: https://doi.org/10.1007/978-3-319-75940-1_12. Google Scholar

[27]

H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495–544. Available from: https://doi.org/10.1007/s00245-017-9414-4. doi: 10.1007/s00245-017-9414-4. Google Scholar

[28]

H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525–577. Available from: https://doi.org/10.1142/S0218202518500148. doi: 10.1142/S0218202518500148. Google Scholar

[29]

A. Hawkins-Daarud, S. Prudhomme, K. G. van der Zee and J. T. Oden, Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, J. Math. Biol., 67 (2013), 1457–1485. Available from: https://doi.org/10.1007/s00285-012-0595-9. doi: 10.1007/s00285-012-0595-9. Google Scholar

[30]

A. Hawkins-Daruud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng., 28 (2012), 3–24. Available from: https://doi.org/10.1002/cnm.1467. doi: 10.1002/cnm.1467. Google Scholar

[31]

D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011–1043. Available from: https://doi.org/10.1142/S0218202515500268. doi: 10.1142/S0218202515500268. Google Scholar

[32]

J.-L. Lions, Contrôle Optimal de Systèmes Gouverneś par des Equations aux Dérivées Partielles, Dunod, Paris, 1968. Google Scholar

[33]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479–544. Available from: https://doi.org/10.3934/Math.2017.2.479 doi: 10.3934/Math.2017.2.479. Google Scholar

[34]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., (2018), 1–33. Available from: https://doi.org/10.1007/s00245-018-9538-1. doi: 10.1007/s00245-018-9538-1. Google Scholar

[35]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1986) 65–96. Available from: https://doi.org/10.1007/BF01762360.Google Scholar

[36]

J. Sprekels and H. Wu, Optimal Distributed Control of a Cahn-Hilliard Darcy System with Mass Sources, Appl. Math. Optim., (2019), 1–42. Available from: https://doi.org/10.1007/s00245-019-09555-4.Google Scholar

[37]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Grad. Stud. in Math., 112, AMS, Providence, RI, 2010. doi: 10.1090/gsm/112. Google Scholar

[38]

X. Wu, G. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahn-Hilliard models with applications to diffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180–203. Available from: https://doi.org/10.1002/cnm.2597. doi: 10.1002/cnm.2597. Google Scholar

[1]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002

[2]

Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019040

[3]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423

[4]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

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Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101

[6]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential. Evolution Equations & Control Theory, 2017, 6 (1) : 35-58. doi: 10.3934/eect.2017003

[7]

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