• Previous Article
    Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property
  • CPAA Home
  • This Issue
  • Next Article
    Advances in the truncated Euler–Maruyama method for stochastic differential delay equations
April  2020, 19(4): 2101-2126. doi: 10.3934/cpaa.2020093

Theoretical and numerical results for some bi-objective optimal control problems

1. 

Departamento EDAN and IMUS, Universidad de Sevilla, Campus Reina Mercedes, Sevilla, SPAIN, 41012

2. 

Departamento EDAN, Universidad de Sevilla, Campus Reina Mercedes Sevilla, SPAIN, 41012

Decicated to Prof. T. Caraballo for his 60th birthday

Received  December 2018 Revised  April 2019 Published  January 2020

Fund Project: The first author is supported by MINECO (Spain), grant MTM2016-76990-P. The second author is supported by CEI (Junta de Andalucía, Spain), Grant FQM-131, FPU 2017 (Ministerio de Educación, Spain).

This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.

Citation: Enrique Fernández-Cara, Irene Marín-Gayte. Theoretical and numerical results for some bi-objective optimal control problems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2101-2126. doi: 10.3934/cpaa.2020093
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dyn., 1 (1990), 303-325.   Google Scholar

[2]

G. Allaire and A. Craig, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, Oxford, London, 2007.  Google Scholar

[3]

L. J. Ávarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Multi-objective Pareto-optimal control: an application to wastewater management, Comput Optim Appl, Berlin, 46 (2010), 135-157.  doi: 10.1007/s10589-008-9190-9.  Google Scholar

[4]

J. L. BoldriniB. M. Calsavara Caretta and E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev Mat Complut, 23 (2010), 49-75.  doi: 10.1007/s13163-009-0012-0.  Google Scholar

[5]

J. L. BoldriniE. Fernández-Cara and M. A. Rojas-Medar, An optimal control problem for a generalized Boussinesq model: The time dependent case, Rev. Mat. Complut., 20 (2007), 339-366.  doi: 10.5209/rev_REMA.2007.v20.n2.16487.  Google Scholar

[6]

H. Brézis, Functional Analysis, Sobolev Sapces and Partial Differential Equations, Springer, London, 2011.  Google Scholar

[7]

P. P. Carvalho and E. Fernández-Cara, On the computation of Nash and Pareto equilibria for some bi-objective control problems, work in progress. Google Scholar

[8]

E. Casas, The Navier-Stokes equations coupled with the heat equation: analysis and control, Control Cybernet., 23 (1994), 605-620.   Google Scholar

[9]

E. CasasJ. P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 39 (2000), 1182-1203.  doi: 10.1137/S0363012998345627.  Google Scholar

[10] Ph. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989.   Google Scholar
[11]

C. Fabre, Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1996), 267-302.  doi: 10.1051/cocv:1996109.  Google Scholar

[12] T. M. Flett, Differential Analysis: Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge, 1980.   Google Scholar
[13]

A. V. Fursikov and O. Pironneau, Finite element methods for Navier-Stokes equations, Annual Review of Fluid Mechanics, 24 (1992), 167-204.   Google Scholar

[14]

A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, American Mathematical Society, Boston, 2000.  Google Scholar

[15]

FI. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 67, 1972.  Google Scholar

[16]

R. Glowinski, Finite Element Methods for Incompressible Viscous Flow, Book of Numerical Analysis, 9, 2003.  Google Scholar

[17]

F. Hecht, http://www.freefem.org. Google Scholar

[18]

J.-L. Lions, Contrȏle de Pareto de systèmes distribués. Le cas d'évolution, C.R. Acad. Sci. Paris, Série I, 302 (1986), 413-417.   Google Scholar

[19]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1, 1971.  Google Scholar

[20]

J. L. Lions, Some remarks on Stackelbergâs optimization, Math. Models Methods Appl. Sci., 4 (1994), 477-487.  doi: 10.1142/S0218202594000273.  Google Scholar

[21]

J. F. Nash, Noncooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[22]

V. Pareto, Cours d'économie politique, Rouge, Laussane, Switzerland, 1896. Google Scholar

[23]

E. Polak, Optimization. Algorithm and Consistent Approximation, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0663-7.  Google Scholar

[24]

H. Von Stalckelberg, Marktform und gleichgewicht, Springer, Berlin, Germany, 1, 1934. Google Scholar

[25]

FR. Témam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and Applications, North-Holland Publishing Co., Amsterdam, 2, 1977.  Google Scholar

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dyn., 1 (1990), 303-325.   Google Scholar

[2]

G. Allaire and A. Craig, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, Oxford, London, 2007.  Google Scholar

[3]

L. J. Ávarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Multi-objective Pareto-optimal control: an application to wastewater management, Comput Optim Appl, Berlin, 46 (2010), 135-157.  doi: 10.1007/s10589-008-9190-9.  Google Scholar

[4]

J. L. BoldriniB. M. Calsavara Caretta and E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev Mat Complut, 23 (2010), 49-75.  doi: 10.1007/s13163-009-0012-0.  Google Scholar

[5]

J. L. BoldriniE. Fernández-Cara and M. A. Rojas-Medar, An optimal control problem for a generalized Boussinesq model: The time dependent case, Rev. Mat. Complut., 20 (2007), 339-366.  doi: 10.5209/rev_REMA.2007.v20.n2.16487.  Google Scholar

[6]

H. Brézis, Functional Analysis, Sobolev Sapces and Partial Differential Equations, Springer, London, 2011.  Google Scholar

[7]

P. P. Carvalho and E. Fernández-Cara, On the computation of Nash and Pareto equilibria for some bi-objective control problems, work in progress. Google Scholar

[8]

E. Casas, The Navier-Stokes equations coupled with the heat equation: analysis and control, Control Cybernet., 23 (1994), 605-620.   Google Scholar

[9]

E. CasasJ. P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 39 (2000), 1182-1203.  doi: 10.1137/S0363012998345627.  Google Scholar

[10] Ph. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989.   Google Scholar
[11]

C. Fabre, Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1996), 267-302.  doi: 10.1051/cocv:1996109.  Google Scholar

[12] T. M. Flett, Differential Analysis: Differentiation, Differential Equations and Differential Inequalities, Cambridge University Press, Cambridge, 1980.   Google Scholar
[13]

A. V. Fursikov and O. Pironneau, Finite element methods for Navier-Stokes equations, Annual Review of Fluid Mechanics, 24 (1992), 167-204.   Google Scholar

[14]

A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, American Mathematical Society, Boston, 2000.  Google Scholar

[15]

FI. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 67, 1972.  Google Scholar

[16]

R. Glowinski, Finite Element Methods for Incompressible Viscous Flow, Book of Numerical Analysis, 9, 2003.  Google Scholar

[17]

F. Hecht, http://www.freefem.org. Google Scholar

[18]

J.-L. Lions, Contrȏle de Pareto de systèmes distribués. Le cas d'évolution, C.R. Acad. Sci. Paris, Série I, 302 (1986), 413-417.   Google Scholar

[19]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1, 1971.  Google Scholar

[20]

J. L. Lions, Some remarks on Stackelbergâs optimization, Math. Models Methods Appl. Sci., 4 (1994), 477-487.  doi: 10.1142/S0218202594000273.  Google Scholar

[21]

J. F. Nash, Noncooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[22]

V. Pareto, Cours d'économie politique, Rouge, Laussane, Switzerland, 1896. Google Scholar

[23]

E. Polak, Optimization. Algorithm and Consistent Approximation, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0663-7.  Google Scholar

[24]

H. Von Stalckelberg, Marktform und gleichgewicht, Springer, Berlin, Germany, 1, 1934. Google Scholar

[25]

FR. Témam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and Applications, North-Holland Publishing Co., Amsterdam, 2, 1977.  Google Scholar

Figure 1.  The domain and the "rough" mesh; $ \Omega $ is composed of the band $ \omega $, the large rectangle $ \mathcal{O}_1 $ and the small rectangle $ \mathcal{O}_2 $. Number of nodes: 1519. Number of triangles: 2876
Figure 2.  The function $ u_{1d} $
Figure 3.  The final velocity fields computed with ALG 8 for various $ \alpha $ in the case $ a = 1.5 $ and $ \nu = 0.06 $. Number of nodes: 3449. Number of triangles: 6658
Figure 4.  The final velocity field, the adjoint and the control computed with ALG 9 for $ \alpha = 0.5 $ in the case $ a = 0.8 $ and $ \nu = 0.1 $. Number of nodes: 6003. Number of triangles: 11684
Figure 5.  The final velocity field, the adjoint and the control computed with ALG 10 for $ \alpha = 0.5 $ in the case $ a = 1.8 $ and $ \nu = 0.00204 $. Number of nodes: 1519. Number of triangles: 2876
Figure 6.  The logarithms of the functionals $ J_1 $, $ J_2 $ for $ \alpha = 0.5 $ and $ J_{(\alpha)} $ for $ \alpha = 0.1, $ $ 0.5 $ and $ 0.9 $, with $ a = 1.5 $ and $ \nu = 0.06 $. Number of nodes: 1519. Number of triangles: 2876
Figure 7.  Number of iterates with $ \alpha = 0.5, 1519 $ nodes and $ \varepsilon = 10^{-6} $ for various values of $ a $ and $ \nu $
Figure 8.  Precision in iteration 50 for $ \alpha = 0.5 $ and $ a = 1.5 $ (NP is the number of nodes)
Figure 9.  Precision in iteration 50 for $ \alpha = 0.5 $ and $ \nu = 0.06 $
Figure 10.  Computation times (in seconds) and numbers of iterates to reach an error less than $ \varepsilon = 10^{-6}, $ for $ \alpha = 0.5 $, $ a = 1.5 $ and $ \nu = 0.06 $
Figure 11.  The cost $ J_{(\alpha)} $ for various methods and parameters
[1]

Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095

[2]

Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200

[3]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[4]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[5]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021

[6]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[7]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052

[8]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[9]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[10]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[11]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39

[12]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[13]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

[14]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[15]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[16]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[17]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

[18]

Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

[19]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[20]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (113)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]