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December  2021, 29(6): 4269-4296. doi: 10.3934/era.2021085

Optimal control for the coupled chemotaxis-fluid models in two space dimensions

Department of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Changchun Liu

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program (no. 20210101466JC)

This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.

Citation: Yunfei Yuan, Changchun Liu. Optimal control for the coupled chemotaxis-fluid models in two space dimensions. Electronic Research Archive, 2021, 29 (6) : 4269-4296. doi: 10.3934/era.2021085
References:
[1]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

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B. Chen and C. Liu, Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature, Applied Mathematics and Optimization, 2021. doi: 10.1007/s00245-021-09807-2.  Google Scholar

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B. Chen, H. Li and C. Liu, Optimal distributed control for a coupled phase-field system, Discrete and Continuous Dynamical Systems Series B. doi: 10.3934/dcdsb.2021110.  Google Scholar

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P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.  Google Scholar

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A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar

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S. FrigeriM. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., 81 (2020), 899-931.  doi: 10.1007/s00245-018-9524-7.  Google Scholar

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F. Guillén-González, E. Mallea-Zepeda and M. Rodríguez-Bellido, Optimal bilinear control problem related to a chemo-repulsion system in 2D domains, ESAIM Control Optim. Calc. Var., 26 (2020), 21pp. doi: 10.1051/cocv/2019012.  Google Scholar

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A. Helmut and T. Yutaka, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7.  Google Scholar

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C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24pp. doi: 10.1007/s00033-017-0882-9.  Google Scholar

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C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.  doi: 10.1002/mana.201600180.  Google Scholar

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C. Liu and X. Zhang, Optimal distributed control for a new mechanochemical model in biological patterns, J. Math. Anal. Appl., 478 (2019), 825-863.  doi: 10.1016/j.jmaa.2019.05.057.  Google Scholar

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C. Liu and X. Zhang, Optimal control of a new mechanochemical model with state constraint, Math. Methods Appl. Sci., 44 (2021), 9237-9263.  doi: 10.1002/mma.7350.  Google Scholar

[15]

S. U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66.  doi: 10.1006/jmaa.2000.7254.  Google Scholar

[16]

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

[17]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[18]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), 1-23.  doi: 10.1007/s00033-016-0732-1.  Google Scholar

[19]

X. ZhangH. Li and C. Liu, Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint, Appl. Math. Optim., 82 (2020), 721-754.  doi: 10.1007/s00245-018-9546-1.  Google Scholar

[20]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62.  doi: 10.1007/BF01442543.  Google Scholar

show all references

References:
[1]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[2]

B. Chen and C. Liu, Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature, Applied Mathematics and Optimization, 2021. doi: 10.1007/s00245-021-09807-2.  Google Scholar

[3]

B. Chen, H. Li and C. Liu, Optimal distributed control for a coupled phase-field system, Discrete and Continuous Dynamical Systems Series B. doi: 10.3934/dcdsb.2021110.  Google Scholar

[4]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.  Google Scholar

[5]

P. ColliG. GilardiE. 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

[6]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.  Google Scholar

[7]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar

[8]

S. FrigeriM. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., 81 (2020), 899-931.  doi: 10.1007/s00245-018-9524-7.  Google Scholar

[9]

F. Guillén-González, E. Mallea-Zepeda and M. Rodríguez-Bellido, Optimal bilinear control problem related to a chemo-repulsion system in 2D domains, ESAIM Control Optim. Calc. Var., 26 (2020), 21pp. doi: 10.1051/cocv/2019012.  Google Scholar

[10]

A. Helmut and T. Yutaka, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7.  Google Scholar

[11]

C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24pp. doi: 10.1007/s00033-017-0882-9.  Google Scholar

[12]

C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.  doi: 10.1002/mana.201600180.  Google Scholar

[13]

C. Liu and X. Zhang, Optimal distributed control for a new mechanochemical model in biological patterns, J. Math. Anal. Appl., 478 (2019), 825-863.  doi: 10.1016/j.jmaa.2019.05.057.  Google Scholar

[14]

C. Liu and X. Zhang, Optimal control of a new mechanochemical model with state constraint, Math. Methods Appl. Sci., 44 (2021), 9237-9263.  doi: 10.1002/mma.7350.  Google Scholar

[15]

S. U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66.  doi: 10.1006/jmaa.2000.7254.  Google Scholar

[16]

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

[17]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[18]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), 1-23.  doi: 10.1007/s00033-016-0732-1.  Google Scholar

[19]

X. ZhangH. Li and C. Liu, Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint, Appl. Math. Optim., 82 (2020), 721-754.  doi: 10.1007/s00245-018-9546-1.  Google Scholar

[20]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62.  doi: 10.1007/BF01442543.  Google Scholar

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