A formula relating two different variational limits of sequences of optimal control problems governed by ordinary differential equations is found out. It is employed to obtain a new representation theorem for the Γ-limit of a sequence of optimal control problems

Contact details are reproduced from the original publication and may be historical.

Lorenzo Freddi

Dip. di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy

  1. Z. Artstein: Rapid oscillations, chattering systems, and relaxed controls, SIAM J. Control Optim. 27 (1989) 940–948.https://doi.org/10.1137/0327050Google Scholar
  2. Z. Artstein: A variational limit that yields chattering systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 suppl. (1989) 47–71.https://doi.org/10.1016/s0294-1449(17)30035-5Google Scholar
  3. Z. Artstein: Chattering variational limits of control systems, Forum Math. 5(4) (1993) 369–403.https://doi.org/10.1515/form.1993.5.369Google Scholar
  4. H. Attouch: Variational convergence for functions and operators, Appl. Math. Ser., Pitman, Boston, 1984.Google Scholar
  5. J. P. Aubin, A. Cellina: Differential inclusions, Springer-Verlag, Berlin et al., 1984.https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  6. H. Berliocchi, J. M. Lasry: Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973) 129–184.https://doi.org/10.24033/bsmf.1755Google Scholar
  7. D. P. Bertsekas, S. E. Shreve: Stochastic optimal control: the discrete time case, Mathematics in Science and Engineering 139, Academic Press, Berlin, 1978.Google Scholar
  8. P. Billingsley: Convergence of probability measures, John Wiley & Sons, New York et al., 1968.Google Scholar
  9. G. Buttazzo: Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Res. Notes Math. Ser. 207, Longman, Harlow, 1989.Google Scholar
  10. G. Buttazzo, E. Cavazzuti: Limit problems in optimal control theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 151–160.https://doi.org/10.1016/s0294-1449(17)30019-7Google Scholar
  11. G. Buttazzo, G. Dal Maso: Γ-convergence and optimal control problems, J. Optim. Theory Appl. 38 (1982) 385–407.https://doi.org/10.1007/BF00935345Google Scholar
  12. G. Buttazzo, L. Freddi: Sequences of optimal control problems with measures as controls, Adv. Math. Sci. Appl. 2(1) (1993) 215–230.Google Scholar
  13. G. Buttazzo, L. Freddi: Optimal control problems with weakly converging input operators, Discrete Contin. Dynam. Systems 1(3) (1995) 401–420.https://doi.org/10.3934/dcds.1995.1.401Google Scholar
  14. C. Castaing, M. Valadier: Convex analysis and measurable multifunctions, Lecture Notes in Mathematics 8, Springer-Verlag, Berlin et al., 1977.https://doi.org/10.1007/BFb0087685Google Scholar
  15. G. Dal Maso: An introduction to Γ-convergence, Birkhäuser, Boston, 1993.https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  16. Z. Denkowski, S. Migorski: Control problems for parabolic and hyperbolic equations via the theory of G and Γ-convergence, Ann. Mat. Pura Appl. 149 (1987) 23–39.https://doi.org/10.1007/BF01773923Google Scholar
  17. J. Dugundji: An extension of Tietze’s theorem, Pacific J. Math. 1 (1951) 353–367.https://doi.org/10.2140/pjm.1951.1.353Google Scholar
  18. N. Dunford, J. T. Schwartz: Linear operators, vol. 1, general theory, Interscience Publ. Ltd., New York, 1958.Google Scholar
  19. H. Federer: Geometric measure theory, Springer-Verlag, Berlin, 1969.Google Scholar
  20. L. Freddi: Optimal control problems with weakly converging input operators in a nonreflexive framework, Portugal. Math. 57(1) (2000) 97–126.Google Scholar
  21. A. D. Ioffe, V. Tihomirov: Extension of variational problems, Trans. Moscow Math. Soc. 18 (1968) 207–273.Google Scholar
  22. E. Mascolo, L. Migliaccio: Relaxation in optimal control theory, Ann. Univ. Ferrara - sez. VII (N.S.) 34 (1988) 247–263.https://doi.org/10.1007/BF02824985Google Scholar
  23. E. Mascolo, L. Migliaccio: Relaxation methods in control theory, Appl. Math. Optim. 20 (1989) 97–103.https://doi.org/10.1007/BF01447649Google Scholar
  24. S. Migorski: Asymptotic behaviour of optimal solutions in control problems for elliptic equations, Riv. Mat. Pura Appl. 11 (1992) 7–28.Google Scholar
  25. S. Migorski: On asymptotic limits of control problems with parabolic and hyperbolic equations, Riv. Mat. Pura Appl. 12 (1992) 33–50.Google Scholar
  26. J. Neveu: Bases Mathématiques du Calcul des Probabilités, Masson, Paris, 1964.Google Scholar
  27. M. Valadier: Young Measures, ‘Methods of Nonconvex Analysis’, Lecture Notes in Math. 1446 (1990) 152–188.https://doi.org/10.1007/BFb0084935Google Scholar
  28. J. Warga: Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.https://doi.org/10.1016/B978-0-12-735150-6.50011-7Google Scholar

L. Freddi. “Γ-Convergence and Chattering Limits in Optimal Control Theory.” Journal of Convex Analysis 8 (2001), No. 1, 39–70. https://doi.org/10.68381/jca08002