Control problems for convection-diffusion equations with control localized on manifolds

Phuong Anh Nguyen; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 467-488
  • ISSN: 1292-8119

Abstract

top
We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of L p -vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.

How to cite

top

Nguyen, Phuong Anh, and Raymond, Jean-Pierre. "Control problems for convection-diffusion equations with control localized on manifolds." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 467-488. <http://eudml.org/doc/90603>.

@article{Nguyen2001,
abstract = {We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of $L^p$-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.},
author = {Nguyen, Phuong Anh, Raymond, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {pointwise control; optimal control; convection-diffusion equation; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions},
language = {eng},
pages = {467-488},
publisher = {EDP-Sciences},
title = {Control problems for convection-diffusion equations with control localized on manifolds},
url = {http://eudml.org/doc/90603},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Nguyen, Phuong Anh
AU - Raymond, Jean-Pierre
TI - Control problems for convection-diffusion equations with control localized on manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 467
EP - 488
AB - We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of $L^p$-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.
LA - eng
KW - pointwise control; optimal control; convection-diffusion equation; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions
UR - http://eudml.org/doc/90603
ER -

References

top
  1. [1] R.A. Adams, Sobolev spaces. Academic Press, New-York (1975). Zbl0314.46030MR450957
  2. [2] H. Amann, Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45 (1983) 225-254. Zbl0535.35017MR719122
  3. [3] S. Anita, Optimal control of parameter distributed systems with impulses. Appl. Math. Optim. 29 (1994) 93-107. Zbl0790.49036MR1239458
  4. [4] M. Berggren, R. Glowinski and J.L. Lions, A Computational Approach to Controllability Issues for Flow-Related Models, Part 1. Int. J. Comput. Fluid Dyn. 7 (1996) 237-252. Zbl0894.76056
  5. [5] M. Berggren, R. Glowinski and J.L. Lions, A Computational Approach to Controllability Issues for Flow-Related Models, Part 2. Int. J. Comput. Fluid Dyn. 6 (1996) 253-247. Zbl0894.76056
  6. [6] E. Casas, J.-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. Zbl0984.49011
  7. [7] E. Casas, M. Mateos and J.-P. Raymond, Pontryagin’s principle for the control of parabolic equations with gradient state constraints. Nonlinear Anal. (to appear). Zbl0998.49014
  8. [8] E.J. Dean and P. Gubernatis, Pointwise Control of Burgers’ Equation – A Numerical Approach. Comput. Math. Appl. 22 (1991) 93-100. Zbl0829.65090
  9. [9] Z. Ding, L. Ji and J. Zhou, Constrained LQR Problems in Elliptic distributed Control systems with Point observations. SIAM 34 (1996) 264-294. Zbl0841.49018MR1372914
  10. [10] J. Droniou and J.-P. Raymond, Optimal pointwise control of semilinear parabolic equations. Nonlinear Anal. 39 (2000) 135-156. Zbl0939.49005MR1722110
  11. [11] J.W. He and R. Glowinski, Neumann control of unstable parabolic systems: Numerical approach. J. Optim. Theory Appl. 96 (1998) 1-55. Zbl0952.49026MR1608018
  12. [12] J.W. He, R. Glowinski, R. Metacalfe and J. Periaux, A numerical approach to the control and stabilization of advection-diffusion systems: Application to viscous drag reduction, Flow control and optimization. Int. J. Comput. Fluid Dyn. 11 (1998) 131-156. Zbl0938.76024MR1682719
  13. [13] H. Henrot and J. Sokolowski, Shape Optimization Problem for Heat Equation. Rapport de recherche INRIA (1997). Zbl0921.35063
  14. [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin/Heidelberg/New-York (1981). Zbl0456.35001MR610244
  15. [15] K.-H. Hoffmann and J. Sokolowski, Interface optimization problems for parabolic equations. Control Cybernet. 23 (1994) 445-451. Zbl0822.49002MR1303363
  16. [16] J.-P. Kernevez, The sentinel method and its application to environmental pollution problems. CRC Press, Boca Raton (1997). Zbl0871.92031MR1428425
  17. [17] J.-L. Lions, Pointwise control for distributed systems, in Control and estimation in distributed parameters sytems, edited by H.T. Banks. SIAM, Philadelphia (1992) 1-39. 
  18. [18] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type. AMS, Providence, RI, Transl. Math. Monographs 23 (1968). Zbl0174.15403
  19. [19] H.-C. Lee and O.Yu. Imanuvilov, Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media. SIAM J. Control Optim. 39 (2000) 457-477. Zbl0968.49007MR1788067
  20. [20] P.A. Nguyen, Optimal Control Localized on Thin Structure for Semilinear Parabolic Equations and the Boussinesq system. Thesis, Toulouse (2000). Zbl1026.49004
  21. [21] P.A. Nguyen and J.-P. Raymond, Control Localized On Thin Structure For Semilinear Parabolic Equations. Sém. Inst. H. Poincaré (to appear). Zbl1026.49004MR1936011
  22. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin/Heidelberg/New-York (1983). Zbl0516.47023MR710486
  23. [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Tome 2, Fourier Analysis, Self-Adjointness. Academic Press, Inc. (1975). Zbl0308.47002
  24. [24] J.-P. Raymond and H. Zidani, Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143-177. Zbl0922.49013
  25. [25] J. Simon, Compact Sets in the Space L p ( 0 , T ; B ) . Ann. Mat. Pura Appl. 196 (1987) 65-96. Zbl0629.46031MR916688
  26. [26] H. Triebel, Interpolation Theory, Functions Spaces, Differential Operators. North Holland Publishing Campany, Amsterdam/New-York/Oxford (1977). Zbl0387.46032MR503903
  27. [27] V. Vespri, Analytic Semigroups Generated in H - m , p by Elliptic Variational Operators and Applications to Linear Cauchy Problems, Semigroup theory and applications, edited by Clemens et al. Marcel Dekker, New-York (1989) 419-431. Zbl0679.47024MR1009411

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.