State-Constrained Relaxed Problems for Semilinear Elliptic Equations

Journal of Mathematical Analysis and Applications

Volumn 223, Issue 1, 1998, Pages 248-271

  Arada, N ^a   Raymond, J P ^a  

^a CNRS   (France)

Author keywords

Optimal control; relaxed solutions; Young measures; Pontryagin's minimum principle; pointwise state constraints

Indexed keywords

EID: 0001018402     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1998.5976     Document Type: Article

References (25)

1. Alibert J. J., Raymond J. P. Boundary control of semilinear elliptic equations with discontinuous leading coefficient and unbounded controls. Numer. Funct. Anal. Optim. 18:1997;235-250.
2. Balakrishnan A. V. A computational approach to the maximum principle. J. Comput. System Sci. 5:1971;163-191.
3. Ball J. M. A version of the fundamental theorem for Young measures. Rascle M., Serre D., Slemrod M. Partial Differential Equations and Continuum Models of Phase Transitions. Lecture Notes in Physics. 1989;207-215 Springer-Verlag, Berlin.
4. Berkovitz L. D. A penalty function proof of the maximum principle. Appl. Math. Optim. 2:1976;291-303.
5. Berkovitz L. D. A penalty method for deriving necessary conditions in problems of optimal control. Math. Control Theory. Banach Center Publications. 14:1985;Polish Academy of Sciences, Warsaw. p. 59-79.
6. Bonnans J. F., Casas E. A boundary Pontryagin's principle for the optimal control of state constrained elliptic systems. Barbu V. Optimization, Optimal Control and Partial Differential Equations. 1992;241-249 Birkhäuser, Basel.
7. Bonnans J. F., Casas E. An extension of Pontryagin's principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities. SIAM J. Control Optim. 33:1995;274-298.
8. Casas E. Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:1993;993-1006.
9. Casas E. The relaxation theory applied to optimal control problems of semilinear elliptic equations. Dole z J., Fidler J. System Modelling and Optimization. 1996;187-194 Chapman & Hall, London.
10. Fattorini H. O. Existence theory and the maximum principle for relaxed infinite-dimensional optimal control problems. SIAM J. Control Optim. 32:1994;311-331.
11. Li X. J., Yong J. Optimal Control Theory for Infinite Dimensional Systems. 1995;Birkhauser, Basel.
12. McShane E. J. The Lagrange multiplier rule. Amer. Math. Monthly. 80:1973;922-925.
13. Medhin N. G. Necessary conditions for optimal control problems with bounded state by a penalty method. J. Optim. Theory Appl. 52:1987;97-110.
14. Medhin N. Optimal processes governed by integral equations with unilateral constraints. J. Math. Anal. Appl. 129:1988;269-283.
15. Murthy M. K. V., Stampacchia G. A variational inequality with mixed boundary conditions. Israel J. Math. 13:1972;188-224.
16. Olech C. Existence theory in optimal control. Control Theory and Topics in Functional Analysis. 1976;International Atomic Energy Agency, Vienna. p. 291-328.
17. Papageorgiou N. S. On the optimal control of strongly nonlinear equations. J. Math. Anal. Appl. 164:1992;83-103.
18. Raymond J. P. Existence theorems without convexity assumptions for optimal control problems governed by parabolic and elliptic systems. Appl. Math. Optim. 26:1992;29-62.
19. Rockafellar R. T. Integral functionals, normal integrands and measurable selections. Waelbroeck L. Nonlinear Operators and the Calculus of Variations. Lecture Notes in Math. 1976;157-207 Springer-Verlag, New York.
20. Roubi c T. Constrained optimization: A general tolerance approach. Aplikace Matematiky. 35:1990;99-128.
21. Roubi c T. Generalized solutions of constrained optimization problems. SIAM J. Control Optim. 24:1986;951-960.
22. Roubi c T. Relaxation in Optimization Theory and Variational Calculus. 1997;de Gruyter, Berlin.
23. Warga J. Optimal Control of Differential and Functional Equations. 1972;Academic Press, New York.
24. Yezza A. Relaxed optimal control problems governed by integral equations. J. Math. Anal. Appl. 175:1993;126-142.
25. Young L. C. Lectures on the Calculus of Variations and Optimal Control Theory. 1969;Saunders, Philadelphia.