# Relaxation of optimal control problems in ${\U0001d5ab}^{\U0001d5c9}$-spaces

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

- Volume: 6, page 73-95
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topArada, Nadir. "Relaxation of optimal control problems in $\sf L^p$-spaces." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 73-95. <http://eudml.org/doc/90614>.

@article{Arada2001,

abstract = {We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p<\infty $). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.},

author = {Arada, Nadir},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {optimal control problems; relaxation; generalized Young measures; stability properties; Pontryagin’s principle; relaxed controls; noncompact control sets; compactification; semilinear control systems; necessary optimality conditions},

language = {eng},

pages = {73-95},

publisher = {EDP-Sciences},

title = {Relaxation of optimal control problems in $\sf L^p$-spaces},

url = {http://eudml.org/doc/90614},

volume = {6},

year = {2001},

}

TY - JOUR

AU - Arada, Nadir

TI - Relaxation of optimal control problems in $\sf L^p$-spaces

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2001

PB - EDP-Sciences

VL - 6

SP - 73

EP - 95

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

LA - eng

KW - optimal control problems; relaxation; generalized Young measures; stability properties; Pontryagin’s principle; relaxed controls; noncompact control sets; compactification; semilinear control systems; necessary optimality conditions

UR - http://eudml.org/doc/90614

ER -

## References

top- [1] N.U. Ahmed, Properties of relaxed trajectories for a class of nonlinear evolution equations on a Banach space. SIAM J. Control Optim. 21 (1983) 953–967. Zbl0524.49008
- [2] N. Arada and J.P. Raymond, State-constrained relaxed control problems for semilinear elliptic equations. J. Math. Anal. Appl. 223 (1998) 248–271. Zbl0911.35018
- [3] N. Arada and J.P. Raymond, Stability analysis of relaxed Dirichlet boundary control problems. Control Cybernet. 28 (1999) 35–51. Zbl0943.49010
- [4] J.M. Ball, 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. Zbl0991.49500
- [5] E. Casas, The relaxation theory applied to optimal control problems of semilinear elliptic equations, in System Modelling and Optimization, edited by J. Doležal and J. Fidler. Chapman & Hall, London (1996) 187-194. Zbl0880.49032MR1471268
- [6] E. Di Benedetto, Degenerate Parabolic Equations. Springer-Verlag, New York (1993). Zbl0794.35090MR1230384
- [7] R.J. DiPerna and A.J. Majda, Oscillation and concentrations in the weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667-689. Zbl0626.35059MR877643
- [8] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74. American Mathematical Society (1990). Zbl0698.35004MR1034481
- [9] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications. Cambridge University Press (1998). Zbl0931.49001MR1669395
- [10] R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978). Zbl0401.49001MR686793
- [11] A. Ghouila-Houri, Sur la généralisation de la notion de commande optimale d’un système guidable. Rev. Franç. Info. Rech. Oper. 1 (1967) 7-32. Zbl0153.20403
- [12] D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115 (1991) 329-365. Zbl0754.49020MR1120852
- [13] D. Kinderlehrer and P. Pedregal, Gradients Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. Zbl0808.46046MR1274138
- [14] P.L. Lions, 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. Zbl0704.49004MR778970
- [15] E.J. McShane, Necessary conditions in the generalized-curve problems of the calculus of variations. Duke Math. J. 7 (1940) 513-536. Zbl0024.32503MR3478
- [16] N.S. Papageorgiou, Properties of the relaxed trajectories of evolution equations and optimal control. SIAM J. Control Optim. 27 (1989) 267-288. Zbl0678.49002MR984828
- [17] J.P. Raymond, Nonlinear boundary control of semilinear parabolic equations with pointwise state constraints. Discrete Contin. Dynam. Systems 3 (1997) 341-370. Zbl0953.49026MR1444199
- [18] J.P. Raymond and H. Zidani, Hamiltonian Pontryaguin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143-177. Zbl0922.49013
- [19] T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. De Gruyter Series in Nonlinear Analysis and Applications (1997). Zbl0880.49002MR1458067
- [20] T. Roubíček, 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. Zbl0964.49012MR1713866
- [21] M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982) 959-1000. Zbl0496.35058MR668586
- [22] 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). Zbl0437.35004MR584398
- [23] J. Warga, Optimal control of differential and functional equations. Academic Press, New York (1972). Zbl0253.49001MR372708
- [24] X. Xiang and N.U. Ahmed, Properties of relaxed trajectories of evolution equations and optimal control. SIAM J. Control Optim. 31 (1993) 1135-1142. Zbl0785.93051MR1233996
- [25] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. W.B. Saunders, Philadelphia (1969). Zbl0177.37801MR259704

## NotesEmbed ?

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