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

Yunfei Yuan; Changchun Liu; Yunfei Yuan; Changchun Liu

doi:10.3934/era.2021085

Electronic Research Archive

2021, Volume 29, Issue 6: 4269-4296. doi: 10.3934/era.2021085

Previous Article Next Article

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

Yunfei Yuan ,
Changchun Liu ^,

Department of Mathematics, Jilin University, Changchun 130012, China

Received: 01 August 2021 Revised: 01 September 2021 Published: 26 October 2021
Primary: 92C17, 49J20; Secondary: 49K20, 35K51

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

Related Papers:

Abstract

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.

References

[1]	The Debye system: Existence and large time behavior of solutions. Nonlinear Anal. (1994) 23: 1189-1209.
[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
[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
[4]	Optimal control for a conserved phase field system with a possibly singular potential. Evol. Equ. Control Theory (2018) 7: 95-116.
[5]	Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity (2017) 30: 2518-2546.
[6]	Reaction terms avoiding aggregation in slow fluids. Nonlinear Anal. Real World Appl. (2015) 21: 110-126.
[7]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[8]	Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential. Appl. Math. Optim. (2020) 81: 899-931.
[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
[10]	On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. (2009) 344: 381-429.
[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
[12]	Large time periodic solution to the coupled chemotaxis-Stokes model. Math. Nachr. (2017) 290: 1701-1715.
[13]	Optimal distributed control for a new mechanochemical model in biological patterns. J. Math. Anal. Appl. (2019) 478: 825-863.
[14]	Optimal control of a new mechanochemical model with state constraint. Math. Methods Appl. Sci. (2021) 44: 9237-9263.
[15]	Optimal control of Keller-Segel equations. J. Math. Anal. Appl. (2001) 256: 45-66.
[16]	Compact sets in the space $L^p(0, T;B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96.
[17]	Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. (2015) 66: 2555-2573.
[18]	Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system. Z. Angew. Math. Phys. (2016) 67: 1-23.
[19]	Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint. Appl. Math. Optim. (2020) 82: 721-754.
[20]	Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. (1979) 5: 49-62.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)