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Introduction

  • Alexander J. Zaslavski
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 82)

Abstract

Let \(-\infty < T_{1} < T_{2} < \infty \), \(A \subset [T_{1},T_{2}] \times {R}^{n}\) be a closed subset of the t x-space R n+1 and let A(t) denote its sections, that is
$$\displaystyle{A(t) =\{ x \in {R}^{n} : (t,x) \in A\},\quad t \in [T_{ 1},T_{2}].}$$
For every (t,x)∈A let U(t,x) be a given subset of the u-space R m , \(x = (x_{1},\ldots x_{n})\), \(u = (u_{1},\ldots u_{m})\).

Keywords

Optimal Control Problem Variational Problem Relative Topology Strong Topology Convexity Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alberti G, Serra Cassano F (1994) Non-occurrence of gap for one-dimensional autonomous functionals, Calculus of variations, homogenization and continuum mechanics. World Scientific Publishing, River Edge, NJ, pp 1–17Google Scholar
  2. 2.
    Angell TS (1976) Existence theorems for hereditary Lagrange and Mayer problems of optimal control. SIAM J Contr Optim 14:1–18MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arkin VI, Evstigneev IV (1987) Stochastic models of control and economic dynamics. Academic Press, LondonGoogle Scholar
  4. 5.
    Aseev SM, Veliov VM (2012) Maximum principle for infinite-horizon optimal control problems with dominating discount. Dynam Contin Discrete Impuls Syst Ser B 19:43–63MathSciNetMATHGoogle Scholar
  5. 8.
    Ball JM, Mizel VJ (1984) Singular minimizers for regular one-dimensional problems in the calculus of variations. Bull Amer Math Soc 11:143–146MathSciNetMATHCrossRefGoogle Scholar
  6. 9.
    Ball JM, Mizel VJ (1985) One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch Ration Mech Anal 90:325–388MathSciNetMATHCrossRefGoogle Scholar
  7. 10.
    Baumeister J, Leitao A, Silva GN (2007) On the value function for nonautonomous optimal control problem with infinite horizon. Syst Contr Lett 56:188–196MathSciNetMATHCrossRefGoogle Scholar
  8. 12.
    Berkovitz LD (1974) Optimal control theory. Springer, New YorkMATHCrossRefGoogle Scholar
  9. 13.
    Berkovitz LD (1974) Lower semicontinuity of integral functionals. Trans Amer Math Soc 192:51–57MathSciNetMATHCrossRefGoogle Scholar
  10. 14.
    Blot J, Cartigny P (2000) Optimality in infinite-horizon variational problems under sign conditions. J Optim Theory Appl 106:411–419MathSciNetMATHCrossRefGoogle Scholar
  11. 15.
    Blot J, Hayek N (2000) Sufficient conditions for infinite-horizon calculus of variations problems. ESAIM Contr Optim Calc Var 5:279–292MathSciNetMATHCrossRefGoogle Scholar
  12. 16.
    Brezis H (1973) Opérateurs maximaux monotones. North Holland, AmsterdamMATHGoogle Scholar
  13. 17.
    Carlson DA, Haurie A, Leizarowitz A (1991) Infinite horizon optimal control. Springer, BerlinMATHCrossRefGoogle Scholar
  14. 18.
    Cartigny P, Michel P (2003) On a sufficient transversality condition for infinite horizon optimal control problems. Automatica J IFAC 39:1007–1010MathSciNetMATHCrossRefGoogle Scholar
  15. 21.
    Cesari L (1983) Optimization - theory and applications. Springer, New YorkMATHCrossRefGoogle Scholar
  16. 22.
    Cinquini S (1936) Sopra l’esistenza della solusione nei problemi di calcolo delle variazioni di ordine n. Ann Scuola Norm Sup Pisa 5:169–190MathSciNetGoogle Scholar
  17. 23.
    Clarke FH (1983) Optimization and nonsmooth analysis. Wiley Interscience, New YorkMATHGoogle Scholar
  18. 24.
    Clarke FH, Loewen PD (1989) An intermediate existence theory in the calculus of variations. Ann Scuola Norm Sup Pisa 16:487–526MathSciNetMATHGoogle Scholar
  19. 25.
    Clarke FH, Vinter RB (1985) Regularity properties of solutions to the basic problem in the calculus of variations. Trans Amer Math Soc 289:73–98MathSciNetMATHCrossRefGoogle Scholar
  20. 26.
    Clarke FH, Vinter RB (1986) Regularity of solutions to variational problems with polynomial Lagrangians. Bull Polish Acad Sci 34:73–81MathSciNetMATHGoogle Scholar
  21. 34.
    Evstigneev IV, Flam SD (1998) Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators. Set-Valued Anal 6:61–81MathSciNetMATHCrossRefGoogle Scholar
  22. 35.
    Ferriero A (2005) The approximation of higher-order integrals of the calculus of variations and the Lavrentiev phenomenon. SIAM J Contr Optim 44:99–110MathSciNetMATHCrossRefGoogle Scholar
  23. 36.
    Gaitsgory V, Rossomakhine S, Thatcher N (2012) Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting. Dynam Contin Discrete Impuls Syst Ser B 19:65–92MathSciNetMATHGoogle Scholar
  24. 38.
    Guo X, Hernandez-Lerma O (2005) Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates. Bernoulli 11:1009–1029MathSciNetMATHCrossRefGoogle Scholar
  25. 39.
    Ioffe AD (1972) An existence theorem for problems of the calculus of variations. Sov Math Dokl 13:919–923MATHGoogle Scholar
  26. 40.
    Ioffe AD (1976) An existence theorem for a general Bolza problem. SIAM J Contr Optim 14:458–466MathSciNetMATHCrossRefGoogle Scholar
  27. 43.
    Jasso-Fuentes H, Hernandez-Lerma O (2008) Characterizations of overtaking optimality for controlled diffusion processes. Appl Math Optim 57:349–369MathSciNetMATHCrossRefGoogle Scholar
  28. 44.
    Kelley JL (1955) General topology. Van Nostrand, Princeton, NJMATHGoogle Scholar
  29. 45.
    Lavrentiev M (1926) Sur quelques problemes du calcul des variations. Ann Math Pura Appl 4:107–124Google Scholar
  30. 46.
    Leizarowitz A (1985) Infinite horizon autonomous systems with unbounded cost. Appl Math Opt 13:19–43MathSciNetMATHCrossRefGoogle Scholar
  31. 49.
    Loewen PD (1987) On the Lavrentiev phenomenon. Canad Math Bull 30:102–108MathSciNetMATHCrossRefGoogle Scholar
  32. 50.
    Lykina V, Pickenhain S, Wagner M (2008) Different interpretations of the improper integral objective in an infinite horizon control problem. J Math Anal Appl 340:498–510MathSciNetMATHCrossRefGoogle Scholar
  33. 52.
    Malinowska AB, Martins N, Torres DFM (2011) Transversality conditions for infinite horizon variational problems on time scales. Optim Lett 5:41–53MathSciNetMATHCrossRefGoogle Scholar
  34. 53.
    Mania B (1934) Sopra un esempio di Lavrentieff. Boll Un Mat Ital 13:146–153MathSciNetGoogle Scholar
  35. 59.
    McShane EJ (1934) Existence theorem for the ordinary problem of the calculus of variations. Ann Scoula Norm Pisa 3:181–211Google Scholar
  36. 60.
    Mizel VJ (2000) New developments concerning the Lavrentiev phenomenon, Calculus of variations and differential equations. CRC Press, Boca Raton, FL, pp 185–191Google Scholar
  37. 62.
    Mordukhovich BS (1990) Minimax design for a class of distributed parameter systems. Automat Remote Contr 50:1333–1340MathSciNetGoogle Scholar
  38. 64.
    Mordukhovich BS, Shvartsman I (2004). Optimization and feedback control of constrained parabolic systems under uncertain perturbations, Optimal control, stabilization and nonsmooth analysis. Lecture Notes Control Information Sciences. Springer, New York, pp 121–132CrossRefGoogle Scholar
  39. 65.
    Morrey CH (1967) Multiple integrals in the calculus of variations. Springer, BerlinGoogle Scholar
  40. 68.
    Pickenhain S, Lykina V, Wagner M (2008) On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems. Contr Cybern 37:451–468MathSciNetMATHGoogle Scholar
  41. 74.
    Rockafellar RT (1971) Existence and duality theorems for convex problems of Bolza. Trans Amer Math Soc 159:1–40MathSciNetMATHCrossRefGoogle Scholar
  42. 75.
    Rockafellar RT (1975) Existence theorems for general control problems of Bolza and Lagrange. Adv Math 15:312–333MathSciNetMATHCrossRefGoogle Scholar
  43. 77.
    Samuelson PA (1965) A catenary turnpike theorem involving consumption and the golden rule. Amer Econ Rev 55:486–496Google Scholar
  44. 78.
    Sarychev AV (1997) First-and second order integral functionals of the calculus of variations which exhibit the Lavrentiev phenomenon. J Dynam Contr Syst 3:565–588MathSciNetMATHGoogle Scholar
  45. 79.
    Sarychev AV, Torres DFM (2000) Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. Appl Math Optim 41:237–254MathSciNetMATHCrossRefGoogle Scholar
  46. 80.
    Sychev MA, Mizel VJ (1998) A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems. Trans Amer Math Soc 350:119–133MathSciNetMATHCrossRefGoogle Scholar
  47. 81.
    Tonelli L (1921) Fondamenti di calcolo delle variazioni. Zanicelli, BoloniaGoogle Scholar
  48. 86.
    Zaslavski AJ (1996) Generic existence of solutions of optimal control problems without convexity assumptions, preprintGoogle Scholar
  49. 87.
    Zaslavski AJ (2000) Generic well-posedness of optimal control problems without convexity assumptions. SIAM J Contr Optim 39:250–280MathSciNetMATHCrossRefGoogle Scholar
  50. 88.
    Zaslavski AJ (2001) Existence of solutions of optimal control problems without convexity assumptions. Nonlinear Anal 43:339–361MathSciNetMATHCrossRefGoogle Scholar
  51. 89.
    Zaslavski AJ (2001) Well-posedness and porosity in optimal control without convexity assumptions. Calc Var 13:265–293MathSciNetMATHCrossRefGoogle Scholar
  52. 90.
    Zaslavski AJ (2001) Generic well-posedness for a class of optimal control problems. J Nonlinear Convex Anal 2:249–263MathSciNetMATHGoogle Scholar
  53. 92.
    Zaslavski AJ (2003) Generic well-posedness of variational problems without convexity assumptions. J Math Anal Appl 279:22–42MathSciNetMATHCrossRefGoogle Scholar
  54. 93.
    Zaslavski AJ (2003) Well-posedness and porosity in the calculus of variations without convexity assumptions. Nonlinear Anal 53:1–22MathSciNetMATHCrossRefGoogle Scholar
  55. 94.
    Zaslavski AJ (2004) Generic well-posedness of nonconvex optimal control problems. Nonlinear Anal 59:1091–1124MathSciNetMATHGoogle Scholar
  56. 95.
    Zaslavski AJ (2004) Existence and uniform boundedness of approximate solutions of variational problems without convexity assumptions. Dynam Syst Appl 13:161–178MathSciNetMATHGoogle Scholar
  57. 96.
    Zaslavski AJ (2004) The turnpike property for approximate solutions of variational problems without convexity. Nonlinear Anal 58:547–569MathSciNetMATHCrossRefGoogle Scholar
  58. 97.
    Zaslavski AJ (2005) Nonoccurrence of the Lavrentiev phenomenon for nonconvex variational problems. Ann Inst H Poincaré Anal Non linéaire 22:579–596MathSciNetMATHCrossRefGoogle Scholar
  59. 99.
    Zaslavski AJ (2006) Turnpike properties in the calculus of variations and optimal control. Springer, New YorkMATHGoogle Scholar
  60. 100.
    Zaslavski AJ (2006) Nonoccurrence of the Lavrentiev phenomenon for many optimal control problems. SIAM J Contr Optim 45:1116–1146MathSciNetMATHCrossRefGoogle Scholar
  61. 101.
    Zaslavski AJ (2007) Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems. Calc Var 28:351–381MathSciNetMATHCrossRefGoogle Scholar
  62. 103.
    Zaslavski AJ (2008) Generic nonoccurrence of the Lavrentiev phenomenon for a class of optimal control problems. J Dynam Contr Syst 14:95–119MathSciNetMATHCrossRefGoogle Scholar
  63. 104.
    Zaslavski AJ (2008) Nonoccurrence of the Lavrentiev phenomenon for many infinite dimensional linear control problems with nonconvex integrands. Dynam Syst Appl 17:407–434MathSciNetMATHGoogle Scholar
  64. 107.
    Zaslavski AJ (2012) A generic turnpike result for a class of discrete-time optimal control systems. Dynam Contin Discrete Impuls Syst Ser B 19:225–265MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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