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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 and 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. 

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

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

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

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. 

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

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. 

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

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

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

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. 

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

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
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