Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints

SIAM Journal on Control and Optimization

Volumn 40, Issue 5, 2002, Pages 1431-1454

  Casas, Eduardo ^a   Mateos, Mariano ^a  

^a UNIVERSITY OF CANTABRIA   (Spain)

Author keywords

Control of elliptic equations; Necessary and sufficient optimality conditions; State constraints

Indexed keywords

BOUNDARY CONDITIONS; CONSTRAINT THEORY; HAMILTONIANS; INTEGRAL EQUATIONS; LAGRANGE MULTIPLIERS; LINEAR CONTROL SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; THEOREM PROVING;

ABSTRACT OPTIMIZATION; SECOND ORDER OPTIMALITY CONDITIONS; SEMILINEAR ELLIPTIC CONTROL; STATE CONSTRAINTS;

OPTIMAL CONTROL SYSTEMS;

EID: 0036403306     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012900382011     Document Type: Article

References (21)

1. J. Bonnans and E. Casas, Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l'état, in Nonlinear Partial Differential Equations and Their Applications, vol. 8, Collège de France Seminar, H. Brezis and J. Lions, eds., Longman Scientific and Technical, New York, 1988, pp. 69-86.
2. J. F. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim., 37 (1999), pp. 1726-1741.
3. H. Cartan, Calcul Différentiel, Hermann, Paris, 1967.
4. E. Casas, Pontryagin's principle for optimal control problems governed by semilinear elliptic equations, in International Conference on Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, vol. 118, Internat. Ser. Numer. Math. 118, Birkhäuser, Basel, 1994, pp. 97-114.
5. E. Casas, M. Mateos, and L. Fernández, Second-order optimality conditions for semilinear elliptic control problems with constraints on the gradient of the state, Control Cybernet., 28 (1999), pp. 463-479.
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), pp. 1182-1203.
7. E. Casas and F. Tröltzsch, Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., to appear.
8. E. Casas and F. Tröltzsch, Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations, Appl. Math. Optim., 39 (1999), pp. 211-227.
9. E. Casas, F. Tröltsch, and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic control problem, Z. Anal. Anwendungen, 15 (1996), pp. 687-707.
10. E. Casas, F. Tröltzsch, and A. Unger, Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, SIAM J. Control Optim., 38 (2000), pp. 1369-1391.
11. F. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res., 1 (1976), pp. 165-174.
12. J. C. Dunn, On second order sufficient optimality conditions for structured nonlinear programs in infinite-dimensional function spaces, in Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Appl. Math. 195, Marcel Dekker, New York, 1998, pp. 83-107.
13. H. Goldberg and F. Tröltzsch, Second-order sufficient optimality conditions, for a class of nonlinear parabolic boundary control problems, SIAM J. Control Optim., 31 (1993), pp. 1007-1025.
14. E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1965.
15. A. D. Ioffe, Necessary and sufficient conditions for a local minimum 3: Second order conditions and augmented duality, SIAM J. Control Optim., 17 (1979), pp. 266-288.
16. X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser Boston, Boston, 1995.
17. M. Mateos, Problemas de control óptimo gobernados por ecuaciones semilineales con restriciones de tipo integral sobre el gradiente del estado, Ph.D. thesis, University of Cantabria, Santander, Spain, 2000.
18. H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control, Math. Programming Study, 14 (1981), pp. 163-177.
19. C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966.
20. J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints, Discrete Contin. Dynam. Systems, 6 (2000), pp. 431-450.
21. G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.