American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces
  • DCDS Home
  • This Issue
  • Next Article
    A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains
doi: 10.3934/dcds.2020373

Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential

Andrea Signori

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

Received  May 2020 Revised  September 2020 Published  November 2020

A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn–Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimised possesses some objective terms and it also penalises long treatments time, which may affect harm to the patients, and big aggregations of tumourous cells. Hence, the optimisation problem leads to the optimal strategy which reduces the time exposure of the patient to the medication and at the same time allows the doctors to achieve suitable clinical goals.

Keywords: Optimal control, free terminal time, phase field, tumour growth, Cahn–Hilliard equation, adjoint system, necessary optimality conditions.
Mathematics Subject Classification: Primary: 35Q92, 49J20, 49K20; Secondary: 35K86, 92C50.
Citation: Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020373
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.  doi: 10.1002/mma.4548.  Google Scholar

[2]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[3]

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. doi: 10.1007/s00245-019-09562-5.  Google Scholar

[4]

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.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar

[5]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar

[6]

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.  doi: 10.1016/j.nonrwa.2015.05.002.  Google Scholar

[7]

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.  doi: 10.1088/1361-6544/aa6e5f.  Google Scholar

[8]

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.  doi: 10.3934/dcdss.2017002.  Google Scholar

[9]

V. Cristini, X. 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.  doi: 10.1007/s00285-008-0215-x.  Google Scholar

[10] 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
[11]

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.  doi: 10.1088/1361-6544/aa6063.  Google Scholar

[12]

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

[13]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 71, 38 pp. doi: 10.1051/cocv/2019059.  Google Scholar

[14]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar

[15]

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

[16]

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. doi: 10.1007/978-3-319-64489-9_9.  Google Scholar

[17]

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.   Google Scholar

[18]

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.  doi: 10.1017/S0956792516000292.  Google Scholar

[19]

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.  doi: 10.3934/dcds.2017183.  Google Scholar

[20]

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.  doi: 10.3934/Math.2016.3.318.  Google Scholar

[21]

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. doi: 10.1007/978-3-319-75940-1_12.  Google Scholar

[22]

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.  doi: 10.1142/S0218202518500148.  Google Scholar

[23]

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.  doi: 10.1007/s00245-017-9414-4.  Google Scholar

[24]

H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar

[25]

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.  doi: 10.1007/s00285-012-0595-9.  Google Scholar

[26]

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.  doi: 10.1002/cnm.1467.  Google Scholar

[27]

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.  doi: 10.1142/S0218202515500268.  Google Scholar

[28]

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

[29]

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

[30]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.   Google Scholar

[31]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV(eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[32]

J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517.  doi: 10.1142/S0218202510004313.  Google Scholar

[33]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43–66. doi: 10.3233/ASY-191546.  Google Scholar

[34]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305–331. doi: 10.3934/mcrf.2019040.  Google Scholar

[35]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193–217. doi: 10.3934/eect.2020003.  Google Scholar

[36]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[37]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[38]

J. Sprekels and H. Wu, Optimal distributed control of a Cahn–Hilliard–Darcy system with Mass sources, Appl. Math. Optim., (2019), 1–42. Google Scholar

[39]

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

[40]

S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method., J. Theor. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.  Google Scholar

[41]

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.  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.  doi: 10.1002/mma.4548.  Google Scholar

[2]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[3]

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. doi: 10.1007/s00245-019-09562-5.  Google Scholar

[4]

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.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar

[5]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.  Google Scholar

[6]

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.  doi: 10.1016/j.nonrwa.2015.05.002.  Google Scholar

[7]

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.  doi: 10.1088/1361-6544/aa6e5f.  Google Scholar

[8]

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.  doi: 10.3934/dcdss.2017002.  Google Scholar

[9]

V. Cristini, X. 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.  doi: 10.1007/s00285-008-0215-x.  Google Scholar

[10] 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
[11]

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.  doi: 10.1088/1361-6544/aa6063.  Google Scholar

[12]

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

[13]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 71, 38 pp. doi: 10.1051/cocv/2019059.  Google Scholar

[14]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 131, 31 pp. doi: 10.1007/s00526-019-1579-z.  Google Scholar

[15]

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

[16]

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. doi: 10.1007/978-3-319-64489-9_9.  Google Scholar

[17]

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.   Google Scholar

[18]

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.  doi: 10.1017/S0956792516000292.  Google Scholar

[19]

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.  doi: 10.3934/dcds.2017183.  Google Scholar

[20]

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.  doi: 10.3934/Math.2016.3.318.  Google Scholar

[21]

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. doi: 10.1007/978-3-319-75940-1_12.  Google Scholar

[22]

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.  doi: 10.1142/S0218202518500148.  Google Scholar

[23]

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.  doi: 10.1007/s00245-017-9414-4.  Google Scholar

[24]

H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar

[25]

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.  doi: 10.1007/s00285-012-0595-9.  Google Scholar

[26]

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.  doi: 10.1002/cnm.1467.  Google Scholar

[27]

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.  doi: 10.1142/S0218202515500268.  Google Scholar

[28]

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

[29]

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

[30]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.   Google Scholar

[31]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV(eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[32]

J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517.  doi: 10.1142/S0218202510004313.  Google Scholar

[33]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43–66. doi: 10.3233/ASY-191546.  Google Scholar

[34]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305–331. doi: 10.3934/mcrf.2019040.  Google Scholar

[35]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193–217. doi: 10.3934/eect.2020003.  Google Scholar

[36]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[37]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[38]

J. Sprekels and H. Wu, Optimal distributed control of a Cahn–Hilliard–Darcy system with Mass sources, Appl. Math. Optim., (2019), 1–42. Google Scholar

[39]

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

[40]

S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method., J. Theor. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.  Google Scholar

[41]

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.  doi: 10.1002/cnm.2597.  Google Scholar

[1]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[2]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[3]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[4]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[5]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[6]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[7]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[8]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[9]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[10]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067
[11]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[12]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040
[13]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
[14]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[15]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[16]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[17]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[18]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[19]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006
[20]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

2019 Impact Factor: 1.338

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences