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Relaxation of optimal control problems in Lp-SPACES

Published online by Cambridge University Press:  15 August 2002

Nadir Arada
Nadir Arada*
Affiliation:
UMR 5640 du CNRS, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex 4, France; [email protected].
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Abstract

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

Keywords

Optimal control problemsrelaxationgeneralized Young measuresstability propertiesPontryagin's principle.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 73 - 95
Copyright
© EDP Sciences, SMAI, 2001

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References

Ahmed, N.U., Properties of relaxed trajectories for a class of nonlinear evolution equations on a Banach space. SIAM J. Control Optim. 21 (1983) 953-967. CrossRef
Arada, N. and Raymond, J.P., State-constrained relaxed control problems for semilinear elliptic equations. J. Math. Anal. Appl. 223 (1998) 248-271. CrossRef
Arada, N. and Raymond, J.P., Stability analysis of relaxed Dirichlet boundary control problems. Control Cybernet. 28 (1999) 35-51.
Ball, J.M., A version of the fundamental theorem for Young's measures, in PDE's and Continuum Models of Phase Transition, edited by M. Rascle, D. Serre and M. Slemrod. Springer, Berlin, Lecture Notes in Phys. 344 (1989) 207-215. CrossRef
E. Casas, The relaxation theory applied to optimal control problems of semilinear elliptic equations, in System Modelling and Optimization, edited by J. Dolezal and J. Fidler. Chapman & Hall, London (1996) 187-194.
E. Di Benedetto, Degenerate Parabolic Equations. Springer-Verlag, New York (1993).
DiPerna, R.J. and Majda, A.J., Oscillation and concentrations in the weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667-689. CrossRef
L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74. American Mathematical Society (1990).
H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications. Cambridge University Press (1998).
R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978).
Ghouila-Houri, A., Sur la généralisation de la notion de commande optimale d'un système guidable. Rev. Franç. Info. Rech. Oper. 1 (1967) 7-32.
Kinderlehrer, D. and Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115 (1991) 329-365. CrossRef
Kinderlehrer, D. and Pedregal, P., Gradients Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. CrossRef
Lions, P.L., The concentration-compactness principle in the calculus of variations: The locally compact case, Parts. 1 and 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109-145, 223-283. CrossRef
McShane, E.J., Necessary conditions in the generalized-curve problems of the calculus of variations. Duke Math. J. 7 (1940) 513-536. CrossRef
Papageorgiou, N.S., Properties of the relaxed trajectories of evolution equations and optimal control. SIAM J. Control Optim. 27 (1989) 267-288. CrossRef
Raymond, J.P., Nonlinear boundary control of semilinear parabolic equations with pointwise state constraints. Discrete Contin. Dynam. Systems 3 (1997) 341-370. CrossRef
Raymond, J.P. and Zidani, H., Hamiltonian Pontryaguin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143-177. CrossRef
T. Roubícek, Relaxation in Optimization Theory and Variational Calculus. De Gruyter Series in Nonlinear Analysis and Applications (1997).
T. Roubícek, Convex locally compact extensions of Lebesgue spaces anf their applications, in Calculus of Variations and Optimal Control, edited by A. Ioffe, S. Reich and I. Shafrir. Chapman & Hall, CRC Res. Notes in Math. 411, CRC Press, Boca Raton, FL (1999) 237-250.
Schonbek, M.E., Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982) 959-1000.
L. Tartar, Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics, Heriott-Watt Symposium IV, Pitmann Res. Notes in Math. 39 (1979).
J. Warga, Optimal control of differential and functional equations. Academic Press, New York (1972).
Xiang, X. and Ahmed, N.U., Properties of relaxed trajectories of evolution equations and optimal control. SIAM J. Control Optim. 31 (1993) 1135-1142. CrossRef
L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. W.B. Saunders, Philadelphia (1969).