Skip to main content

Various Topics in the Theory of Optimal Control of Distributed Systems

  • Conference paper
Optimal Control Theory and its Applications

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 105))

Abstract

These notes correspond to a set of nine lectures given at the 14th Biennial Seminar of the Canadian Math. Congress, London, Ontario, August 1973; in the oral presentation, some well-known facts were more carefully stated than here, where on the contrary we present in fuller details some new results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

Bibliography

  1. M. Amouroux [1] Sur la représentation des systèmes dynamiques. Report LAAS - Toulouse - June 1973.

    Google Scholar 

  2. K. Arrow, T. Harris, and J. Marschak [1] Optimal inventory policy Econometrics, 19 (1951), 250–272.

    MathSciNet  MATH  Google Scholar 

  3. C. Baiocchi [1] C.R. Acad. Sc. Paris, 1971.

    Google Scholar 

  4. C. Baiocchi and E. Magenes [1] Rome, Coll. Accad. Naz. Lincei,1972.

    Google Scholar 

  5. A.V. Balakrishnan [1] On a new computing technique in Optimal Control. SIAM J. on Control, G (1968), 169–173.

    Google Scholar 

  6. A. Bamberger [1] Report Laboria (1974).

    Google Scholar 

  7. A. Bamberger and J.P. Yvon [1] Report Laboria, 1974 idem Banks [1] These proceedings.

    Google Scholar 

  8. J. Baranger [1] Quelques résultats en Optimisation non convexe. Thesis, Grenoble, March 1973.

    Google Scholar 

  9. A. Bensoussan, M. Goursat and J.L. Lions [1] Note C.R. Acad. Sc. Paris, June 1973.

    Google Scholar 

  10. A. Bensoussan, J.L. Lions [1] Notes in the C.R. Acad. Sc., idem 1) May 1973 2) 3 ) June 1973.

    Google Scholar 

  11. A. Bensoussan, J.L. Lions [2] Problèmes de temps d’arrêt Optimal et inéquations variationnelles paraboliques. Applicable Analysis. 1973.

    Google Scholar 

  12. A. Bensoussan, J.L. Lions [3] Paper in preparation.

    Google Scholar 

  13. A. Bensoussan, J.L. Lions [4] Books in preparation.

    Google Scholar 

  14. A. Bensoussan, J.L. Lions [5] On some topics in Optimal Control. (in Russian; in memory of Prof. Petrowski). Ouspechi Mat. Nauk. 1973.

    Google Scholar 

  15. A. Bensoussan, J.L. Lions and R. Temam [1] Report on decomposition Methods. Cahier IRIA, vol. II, (2), 1972; and Chapter 2 of the book Lions-Marchuk, ed. [1].

    Google Scholar 

  16. L.D. Berkowitz [1] to appear.

    Google Scholar 

  17. F. Bidaut [1] Thesis, Paris, June 1973.

    Google Scholar 

  18. A. Bossavit [1] Report, Univ. of Paris, 1969.

    Google Scholar 

  19. C.M. Brauner and P. Penel [1] Un problème de contróle Optimal non linéaire en Biomathématique. Annali de l’Univ. Ferrera, XVIII (1972), 1–44.

    Google Scholar 

  20. C.M. Brauner and P. Penel [2] Thesis 3d Cycle, Univ. Paris. Sud, 1973.

    Google Scholar 

  21. H. Brézis [1] Equations et inéquations dans les espaces vectoriels en 4lualité. Ann. Inst. Fourier, 18 (1968), 115–175.

    Article  MATH  Google Scholar 

  22. H. Brézis [2] Problèmes Unilatéraux. J. Math. Pures et Appl. 51 (1972), 1–168.

    Google Scholar 

  23. H. Brézis and G. Stampacchia [1] Note C.R. Acad. Sci. Paris, 1973.

    Google Scholar 

  24. F. Browder [1] On the unification of the Calculus of variations and the theory of monotone non linear operators in Banach spaces. Proc.Nat.Acad.Sc. USA, 56, (1966), 1080–1086.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Cea, A. Gioan and J. Michel [1] Quelques résultats sur l’identification de domaines. To appear.

    Google Scholar 

  26. G. Chavent [1] Thesis, Paris, 1971.

    Google Scholar 

  27. G.Chavent [2] REPORTS Laboria, 1973, 1974.

    Google Scholar 

  28. M. Delfour and S.K. Mitter [1] Reports Univ. of Montreal, 1972, 1973.

    Google Scholar 

  29. A.S. Demidov [1] On skin effect and asymptotic behaviour of some elliptic pseudo differential operators. (In Russian), Ouspechi Mat. Nank., 97 (1972), 245–246.

    MathSciNet  Google Scholar 

  30. G. Duff [1] Report Laboria, 1973.

    Google Scholar 

  31. G. Duvaut [1] Note on the Stefan’s problem. C.R. Acad. Sc. Paris, June 1973.

    Google Scholar 

  32. G. Duvaut and J.L. Lions [1] Sur les inéquations en Mécanique et en Physique. Dunod, Paris, 1972. English translation, Springer, to appear.

    Google Scholar 

  33. A. Dvoretzky, J. Kiefer and J. Wolfowitz [1] The inventory problem. Econometrica, 20 (1952), 187–222.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Ekeland and R. Temam [1] Analyse Convexe et problèmes variationnels. Paris, Dunod-Gauthier Villars. 1973.

    Google Scholar 

  35. W. Fleming [1] Optimal continuous parameter stochastic control. SIAM Rev. 11 (4), 1969.

    Google Scholar 

  36. A. Friedman [1] Stochastic games and variational inequalities. Archive Rat. Mech. Archive Rat. Mech.’Anal. 1973.

    Google Scholar 

  37. A. Friedman [2] Regularity theorems for variational inequalities. Archive Rat. Mech. Anal. 1973. idem.

    Google Scholar 

  38. E. de Giorgi and S. Spagnolo [1] Sulla convergenza degli integrali dell/energia. To appear.

    Google Scholar 

  39. R. Glowinski, J.L. Lions and R. Trémolieres [1] Méthodes de resolution numérique des inéquations variationnelles. Paris, Dunod, 1974.

    Google Scholar 

  40. M. Goursat [1] Reports Laboria, 1974.

    Google Scholar 

  41. J. Hadamard [1] Reports Laboria, 1974.

    Google Scholar 

  42. G. Henry [1] Report Laboria. 1974.

    Google Scholar 

  43. D. Huet [1] Perturbations singulières d’inégalités variationnelles. C.R. Acad. Sc. Paris, 267 (1968), 932–934.

    MathSciNet  MATH  Google Scholar 

  44. Hullet [1] Ph.D. Thesis, UCLA, 1973.

    Google Scholar 

  45. D. Iglehart and S. Karlin [1] Optimality of (s,S) policies in the infinite horizon dynamic inventory problem. Management Sc. 9 (1963), 259–267.

    Google Scholar 

  46. Jaffré [1] Report Laboria, 1974.

    Google Scholar 

  47. T. Johnson [1] Thesis, MIT, 1973.

    Google Scholar 

  48. J.P. Kernevez [1] Thesis, Paris, 1972.

    Google Scholar 

  49. J.P. Kernevez [2] Control of the flux of substrate entering an enzymatic membrane by an inhibitor at the boundary. J. Opt. theory and Appl. 1973.

    Google Scholar 

  50. J.P. Kernevez [3] IFIP. Optimization Conference. Rome. May 1973.

    Google Scholar 

  51. J.P. Kernevez and D. Thomas [1] Book to appear, Dunod, 1975.

    Google Scholar 

  52. D. Leroy [1] Thesis 3d cycle, Paris, 1973 and Reports Laboria. D. Leroy and J.P. Yvon [1] Report Laboria, to appear.

    Google Scholar 

  53. J.L. Lions [1] Contrôle optimal des systemes gouvernés pour des équations aux dérivehs partielles. Paris, Dunod, 1968. English translation by S.K. Mitter, Springer, 1971.

    Google Scholar 

  54. J.L. Lions [2] Some aspects of the optimal control of distributed parameter systems. Reg. Conf. Series in Appl. Math., August 1971. SIAM, vol. 6, 1972.

    Google Scholar 

  55. J.L. Lions [3] Sur le contrôle optimal des systemes distribués. Conferences UMI. L’Enseignement Math. 1973.

    Google Scholar 

  56. J.L. Lions [4] Perturbations singulières dans les problémes aux limites et en contrôle optimal. Lecture Notes in Math., 323, 1973. Springer.

    Google Scholar 

  57. J.L. Lions [5] Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier Villars, 1969.

    Google Scholar 

  58. J.L. Lions [6] Contrôle optimal des systèmes distribués. Propriétés de comparaison et perturbations singulières. Rome, Acad. dei lincei, 1972.

    Google Scholar 

  59. J.L. Lions [7] Equations différentielles et problèmes aux limites. Springer, III (1961).

    Google Scholar 

  60. J.L. Lions [8] Problèmes aux limites en théorie des distributions. Acta. Math. 94 (1955), 13–153.

    Article  MathSciNet  MATH  Google Scholar 

  61. J.L. Lions and E. Magenes [1] Problèmes aux limites non homogènes et applications. Vol. 1,2,3. Dunod, 1968, 1970. English translation by P. Kenneth, Springer, 1, 2, 1972; 3, 1973.

    Google Scholar 

  62. J.L. Lions and G. Marchuk, ed. [1] Book, Dunod, 1974.

    Google Scholar 

  63. J.L. Lions and G. Stampacchia [1] Variational inequalities. Comm. Pure Appl. Math., XX (1967), 493–519.

    Google Scholar 

  64. K.A. Luré [1] Optimum control of Conductivity of a fluid moving in a Channel in a magnetic field. P.M.M. 28 (1966), 258–267.

    Google Scholar 

  65. A. Marino and S. Spagnolo [1] Un tipo di approxximazione dell/operatore. Annali Scuola N. Pisa, XXIII (1969), 657–673.

    Google Scholar 

  66. M. Mercier and R. Temam [1] To appear.

    Google Scholar 

  67. J.C. Miellou [1] Thesis, Grenoble, 1970.

    Google Scholar 

  68. F. Mignot [1] To appear.

    Google Scholar 

  69. G. Minty [1] Monotone (non linear) operators in Hilbert space. Duke Math. J. 29 (1962), 341–346.

    MathSciNet  MATH  Google Scholar 

  70. S.K. Nit-ter [1] Lectures IRIA. 1973.

    Google Scholar 

  71. J. Mossino [1] Une application de la dualité au contróle optimal5 contrainte portant sur le contrôle et sur l’état. To appear.

    Google Scholar 

  72. F. Murat [1] C.R. Acad. Sc. Paris, 273 (1971), 708–711; 274 (1972), 395–398.

    MathSciNet  Google Scholar 

  73. F. Murat and J. Simon [1] to appear. idem.

    Google Scholar 

  74. J.C. Nbdelec [1] Thesis, Paris, 1970.

    Google Scholar 

  75. M.N. Ogurtrelli [1] A class of non linear integro-differential equations. Bul. Inst. Pol. Din. Iasi, 14 (1968), 43–49.

    Google Scholar 

  76. M.N. Ogurtrelli [2] Un problema misto. Rend. di Mat. 2 (1969), 245–294.

    Google Scholar 

  77. R.E. O’Malley [1] The singularly perturbed linear state regulator problem. SIAM J., Control, 10 (1972).

    Google Scholar 

  78. R.E. O’Malley [2] Singular perturbation of the time-invariant linear state regulator problem. J. Diff. Eq. 12 (1972), 117–128.

    Article  MathSciNet  MATH  Google Scholar 

  79. R.E. O’Malley [3] These proceedings.

    Google Scholar 

  80. Pironneau [1] On optimum profiles in Stokes flow.

    Google Scholar 

  81. J. Fluid Mech. (1973), 59, 117–128.

    Google Scholar 

  82. O. Pironneau [2] On optimum drag in fluid mechanics. To appear.

    Google Scholar 

  83. L.L. Pokrovski [1] Doklady Akad. Nauk., 188 (1969), 528–531.

    Google Scholar 

  84. Pritchard [1] To appear.

    Google Scholar 

  85. D.L. Russell [1] Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory. J. Math. Anal. Appl. 40 (1972), 336–368.

    Article  MathSciNet  MATH  Google Scholar 

  86. Saguez [1] Report Laboria, 1974.

    Google Scholar 

  87. Saint-Jean-Paulin [1] To appear.

    Google Scholar 

  88. H. Scarf [1] The optimality of (S,$) policies in the Dynamic Inventory oroblem. Chap. 13 in Arrow, Karlin, Suppes (ed.), Math. Methods in the Social Sciences, Stanford U. Press, 1960.

    Google Scholar 

  89. L. Schwartz [1] La transformée de Laplace des distributions, 1952.

    Google Scholar 

  90. S. Spagnolo [1] Sulla convergenza di soluzioni di equazioni paraboliche. ed ellitiche. Annal. Sc. Norm. Sup. Pisa, XXII (1968), 574–597.

    Google Scholar 

  91. Spagnolo [2] Sul limite delle soluzioni. Id. XXI (1967), 657–699.

    Google Scholar 

  92. G. Stampacchia [1] Formes bilinéaires coercitives sur les ensembles convexes. C.R. Acad. Sc. Paris, 258 (1964), 4613–4416.

    MathSciNet  Google Scholar 

  93. L. Tartar [1] To appear in the Archive Rat. Mech. Anal.

    Google Scholar 

  94. Van de Vielle [1] To appear.

    Google Scholar 

  95. A. Veinott Jr. [1] The status of Mathematical inventory theory. Management Science 12 (1966), 745–777.

    Article  MathSciNet  MATH  Google Scholar 

  96. A. Veinott Jr. and H. Wagner [1] Computing Optimal (s,S) inventory policies. Management Science. 11 (1965), 525–552.

    Article  MathSciNet  MATH  Google Scholar 

  97. I.M. Visik and L.A. Liousternik. [1] Uspechi Mat. Nauk., 12 (1957), 1–121 (Amer.Math.Soc.Translations, (2), 20, 1962, 239–364 ).

    Google Scholar 

  98. P.K.C. Wang [1] Paper on optimal fusion.

    Google Scholar 

  99. J.P. Yvon [1] Contróle optimal d’un probléme de fusion. Calcolo. 1974.

    Google Scholar 

  100. J.P. Yvon [2] Report Laboria, 1973.

    Google Scholar 

  101. J.P. Yvon [3] Thesis 3 Cycle, Paris, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1974 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lions, J.L. (1974). Various Topics in the Theory of Optimal Control of Distributed Systems. In: Kirby, B.J. (eds) Optimal Control Theory and its Applications. Lecture Notes in Economics and Mathematical Systems, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01569-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-01569-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07018-4

  • Online ISBN: 978-3-662-01569-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics