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

Published online by Cambridge University Press:  15 August 2002

Phuong Anh Nguyen  and
Jean-Pierre Raymond
Phuong Anh Nguyen
Affiliation:
Université Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; [email protected]. and
Jean-Pierre Raymond
Affiliation:
Université Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; [email protected]. and
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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 Lp-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.

Keywords

Pointwise controloptimal controlconvection-diffusion equationcontrol localized on manifolds.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 467 - 488
Copyright
© EDP Sciences, SMAI, 2001

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References

R.A. Adams, Sobolev spaces. Academic Press, New-York (1975).
Amann, H., Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45 (1983) 225-254. CrossRef
Anita, S., Optimal control of parameter distributed systems with impulses. Appl. Math. Optim. 29 (1994) 93-107.
Berggren, M., Glowinski, R. and Lions, J.L., Computational Approach, A to Controllability Issues for Flow-Related Models, Part 1. Int. J. Comput. Fluid Dyn. 7 (1996) 237-252. CrossRef
Berggren, M., Glowinski, R. and Lions, J.L., Computational Approach, A to Controllability Issues for Flow-Related Models, Part 2. Int. J. Comput. Fluid Dyn. 6 (1996) 253-247.
Casas, E., Raymond, J.-P. and Zidani, H., Pontryagin's principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim. 39 (2000) 1182-1203. CrossRef
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).
Dean, E.J. and Gubernatis, P., Pointwise Control of Burgers' Equation - A Numerical Approach. Comput. Math. Appl. 22 (1991) 93-100. CrossRef
Ding, Z., Ji, L. and Zhou, J., Constrained LQR Problems in Elliptic distributed Control systems with Point observations. SIAM 34 (1996) 264-294. CrossRef
Droniou, J. and Raymond, J.-P., Optimal pointwise control of semilinear parabolic equations. Nonlinear Anal. 39 (2000) 135-156. CrossRef
He, J.W. and Glowinski, R., Neumann control of unstable parabolic systems: Numerical approach. J. Optim. Theory Appl. 96 (1998) 1-55. CrossRef
He, J.W., Glowinski, R., Metacalfe, R. and Periaux, J., 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. CrossRef
H. Henrot and J. Sokolowski, Shape Optimization Problem for Heat Equation. Rapport de recherche INRIA (1997).
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin/Heidelberg/New-York (1981).
Hoffmann, K.-H. and Sokolowski, J., Interface optimization problems for parabolic equations. Control Cybernet. 23 (1994) 445-451.
J.-P. Kernevez, The sentinel method and its application to environmental pollution problems. CRC Press, Boca Raton (1997).
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.
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).
Lee, H.-C. and Imanuvilov, O.Yu., Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media. SIAM J. Control Optim. 39 (2000) 457-477. CrossRef
P.A. Nguyen, Optimal Control Localized on Thin Structure for Semilinear Parabolic Equations and the Boussinesq system. Thesis, Toulouse (2000).
P.A. Nguyen and J.-P. Raymond, Control Localized On Thin Structure For Semilinear Parabolic Equations. Sém. Inst. H. Poincaré (to appear).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin/Heidelberg/New-York (1983).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Tome 2, Fourier Analysis, Self-Adjointness. Academic Press, Inc. (1975).
Raymond, J.-P. and Zidani, H., Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143-177. CrossRef
J. Simon, Compact Sets in the Space L p (0,T;B). Ann. Mat. Pura Appl. 196 (1987) 65-96.
H. Triebel, Interpolation Theory, Functions Spaces, Differential Operators. North Holland Publishing Campany, Amsterdam/New-York/Oxford (1977).
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.