Skip to main content
Log in

Higher-order local approximations of smooth control systems and pointwise higher-order optimality conditions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. A. A. Agrachev, “A second-order necessary condition for optimality in the general nonlinear case,”Mat. Sb.,102, No. 4, 551–568 (1977).

    MATH  MathSciNet  Google Scholar 

  2. A. A. Agrachev, “Quadratic mappings in geometrical control theory,” In:Progress in Science and Technology. Series on Problems of Geometry, Vol. 20. All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 111–205.

    Google Scholar 

  3. A. A. Agrachev and S. A. Vakhrameev, “Chronological series and the Cauchy-Kovalevskaya theorem,” In:Progress in Science and Technology. Series on Problems of Geometry, Vol. 12, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1981), pp. 165–189.

    Google Scholar 

  4. A. A. Agrachev, S. A. Vakhrameev, and R. V. Gamkrelidze, “Differential-geometric and group-theoretic methods in optimal control theory,” In:Progress in Science and Technology. Series on Problems of Geometry, Vol. 14, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1983), pp. 3–56.

    Google Scholar 

  5. A. A. Agrachev and R. V. Gamkrelidze, “The principle of second-order optimality for a time-optimal problem,”Mat. Sb.,100, No. 4, 610–643 (1976).

    MATH  MathSciNet  Google Scholar 

  6. A. A. Agrachev and R. V. Gamkrelidze, “Exponential representation of flows and the chronological calculus,”Mat. Sb.,107, No. 4, 467–532 (1978).

    MATH  MathSciNet  Google Scholar 

  7. A. A. Agrachev and R. V. Gamkrelidze, “Chronological algebras and nonstationary vector fields,” In:Progress in Science and Technology. Series on Problems of Geometry, Vol. 11, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1980), pp. 135–176.

    Google Scholar 

  8. A. A. Agrachev and R. V. Gamkrelidze, “Symplectic geometry and necessary optimality conditions,”Mat. Sb.,182, No. 1, 36–54 (1991).

    MATH  MathSciNet  Google Scholar 

  9. A. A. Agrachev and R. V. Gamkrelidze, “Volterra series and groups of permutations,” In:Progress in Science and Technology, Vol. 39, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1991), pp. 3–40.

    Google Scholar 

  10. A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, “Local invariants for smooth control systems,” All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1986). Deposited at VINITI 4.10.86, No. 7020-B (1986).

    Google Scholar 

  11. A. A. Agrachev and A. V. Sarychev, “Filtrations of Lie algebras of vector fields and the nilpotent approximation of control systems,”Dokl. Akad. Nauk SSSR,295, No. 4, 777–781 (1987).

    MATH  Google Scholar 

  12. S. A. Vakhrameev, “Hilbert manifolds with corners of finite codimension and optimal control theory,” In:Progress in Science and Technology. Series on Algebra. Topology. Geometry, Vol. 28, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1990), pp. 96–171.

    Google Scholar 

  13. S. A. Vakhrameev, “Geometrical and topological methods in optimal control theory,” In:Contemporary Mathematics and Its Applications. Thematic Reviews. Analysis-5, Vol. 9, All-Union Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow (1993), pp. 5–240.

    Google Scholar 

  14. S. A. Vakhrameev and A. V. Sarychev, “Geometricla control theory,” In:Progress in Science and Technology. Series on Algebra. Topology. Geometry, Vol. 23, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1985), pp. 197–280.

    Google Scholar 

  15. R. Gabasov and F. M. Kirillova,Singular Optimal Control [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  16. R. V. Gamkrelidze,Foundations of Optimal Control [in Russian], Tbilisi Univ. Press, Tbilisi (1977).

    Google Scholar 

  17. R. V. Gamkrelidze, “First-order necessary conditions and axiomatics of extremal problems,”Tr. Mat. Inst. Akad. Nauk SSSR,112, 152–180 (1971).

    MATH  MathSciNet  Google Scholar 

  18. R. V. Gamkrelidze, A. A. Agrachev, and S. A. Vakhrameev, “Ordinary differential equations on vector bundles and the chronological calculus,” In:Progress in Science and Technology. Series on Contemporary Problems of Mathematics, Vol. 35, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 3–107.

    Google Scholar 

  19. M. I. Zelikin, “The necessary optimality conditions of trajectories for control-linear problems,” In:Some Problems of Contemporary Analysis [in Russian], Moscow (1984), pp. 35–41.

  20. M. I. Zelikin, “On singular extremals,”Probl. Teor. Upr. Teor. Inf., Hungary,14, No. 2, 75–88 (1985).

    MATH  MathSciNet  Google Scholar 

  21. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,Mathematical Theory of Optimal Processes [in Russian], 3rd edition, Nauka, Moscow (1976).

    Google Scholar 

  22. A. I. Tretiak, “On necessary conditions of arbitrary order for optimality in a time-optimal problem”,Mat. Sb.,132, No. 2, 261–274 (1987).

    Google Scholar 

  23. A. I. Tretiak, “On odd-order necessary conditions for optimality”, In:Abstr. Reports of Resp. Sci. Conf. “Differen. Integral. Equations and Their Applications,” Pt. 2, Odess. Univ. Press, Odessa (1987), pp. 110–111.

    Google Scholar 

  24. A. I. Tretiak,Necessary Conditions of Arbitrary Order for Optimality [in Russian] Odessa Univ. Press, Odessa (1988).

    Google Scholar 

  25. A. I. Tretiak, “On odd-order necessary conditions for optimality,” In:Abstr. Reports of Soviet-Poland International Workshop ‘Mathematical Methods in Optimal Control and Their Applications,’ Minsk, May 16–29, 1989, Minsk (1989), pp. 115–116.

  26. A. I. Tretiak, “On necessary conditions of odd order for optimality in a time-optimal problems”,Kibern. Vychisl. Tekh., Kiev, No. 85, 32–37 (1990).

    Google Scholar 

  27. A. I. Tretiak, “On odd-order necessary conditions for optimality in a time-optimal problem for systems linear in control”,Mat. Sb.,181, No. 5, 625–641 (1990).

    Google Scholar 

  28. A. I. Tretiak, “On even-order necessary conditions for optimality,” In:Abstr. Reports of III All-Union Pontryagin School on Optimal Control, Geometry and Analysis, Kemerov. Univ. Press, Kemerovo (1990), p. 208.

    Google Scholar 

  29. A. I. Tretiak, “On necessary conditions for optimality of even order in the time-optimal problem for a class of control systems”,Dokl. Akad. Nauk SSSR,316, No. 5, 1058–1060 (1991).

    Google Scholar 

  30. A. I. Tretiak, “Pointwise higher-order optimality conditions” In:Contemporary Problems of Mathematics, Vol. 39, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1991), pp. 118–177.

    Google Scholar 

  31. A. I. Tretiak, “The second-order necessary conditions for optimality,” In:Abstr. Reports of Republ. Scientific Methods Conf., Pt. 2, Odessa Univ. Press, Odessa (1992), p. 99.

    Google Scholar 

  32. A. I. Tretiak, “Necessary optimality condition for the case of vector-valued control”, In:Abstr. Reports of Conf. Math. Belarussia, Grodno Univ. Press, Grodno (1992), p. 150.

    Google Scholar 

  33. A. I. Tretiak, “Sufficient conditions for local controllability”,Kibern. Vychisl. Tekh., Kiev No. 95, 3–6 (1992).

    Google Scholar 

  34. A. I. Tretiak, “Two-parameter family of sufficient conditions for local controllability along a trajectory”,Kibern. Vychisl. Tekh., Kiev, No. 99, 49–53 (1993).

    Google Scholar 

  35. A. I. Tretiak, “The second-order necessary codition, for optimality,” In:Abstr. Reports of All-Union Sci. Conf. on Dynamic Systems: Stability, Control, Optimization Minsk (1993), p. 75.

  36. A. I. Tretiak, “On a certain even-order necessary condition for optimality,” In:Abstr. Reports of I Ukr. Conf. on Automatic Control, Avtomatika-94, Pt. I, Inst. Kibernetiki Akad. Nauk Ukrainy, Kiev (1994), p. 34.

    Google Scholar 

  37. A. I. Tretiak, “Sufficient conditions for local controllability along a trajectory”, In:Abstr. Reports of I Ukr. Conf. on Automatic Control, Avtomatika-94, Pt. I, Inst. Kibernetiki Akad. Nauk Ukrainy, Kiev (1994), p. 35.

    Google Scholar 

  38. A. I. Tretiak, “Local approximations of high order to smooth control systems and some of their applications”, In:Progress in Science and Technology. Modern Mathematics and Its Applications. Thematic Reviews, Vol. 7, Analysis-4, All-Union institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow (1994).

    Google Scholar 

  39. A. I. Tretiak, “Local approximations of high order to smooth control systems” In:Progress in Science and Technology. Modern Mathematics and Its Applications. Thematic Reviews, Vol. 17, Analysis-7, All-Russian Institute for Scientific and Technical Informaiton (VINITI), Ross. Akad. Nauk, Moscow (1994).

    Google Scholar 

  40. A. I. Tretiak, “Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach”, In:Progress in Science and Technology. Modern Mathematics and Its Applications. Thematic Reviews, Vol. 24, Dynamical Systems-4, All-Russian Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow (1995).

    Google Scholar 

  41. A. A. Agrachev, “Newton diagrams and tangent cones to attainable sets,” In:Anal. Contr. Dyn. Syst.: Proc. Conf. Lyon, July, 1990, Boston (1991), pp. 11–20.

  42. A. A. Agrachev and R. V. Gamkrelidze, “Sympletic geometry for optimall control,” In:Nonlinear Controllab. and Optim. Contr.: Lect. Workshop Finite Dimens. Controllab. and Optim. Contr., New Brunswick, N. J., May 18–22, 1987, New York-Basel (1990), pp. 263–277.

  43. A. A. Agrachev and R. V. Gamkrelidze, “Local controllability for families of diffeomorphisms”,Syst. Contr. Lett.,20, 67–76 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  44. A. A. Agrachev and R. V. Gamkrelidze, “Local controllability and semigroups of diffeomorphisms”,Acta Appl. Math.,32, 1–57 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  45. A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, “Local invariants of smooth control systems”,Acta Appl. Math.,14, No. 3, 191–237 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  46. J. Baillieul, “Geometric methods for nonlinear optimal control problems”,J. Optimiz. Theory Appl.,25, No. 4, 519–548 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  47. D. J. Bell and D. H. Jacobson,Singular Optimal Control Problems, Academic Press, London (1975).

    MATH  Google Scholar 

  48. R. M. Bianchini and G. Stefani, “Sufficient conditions for local controllability,” In:Proc. 25th IEEE Conf. Decis. and Contr., Athens, Dec. 10–12, 1986, Vol. 2, New York, N.Y. (1986), pp. 967–970.

  49. R. M. Bianchini and G. Stefani, “A high order maximum principle,” In:Anal. and Contr. Nonlinear Systems: Selec. Pap. 8th Int. Symp. Math. Networks and Syst. Phoenix, June 15–19, 1987, Amsterdam (1988), pp. 131–136.

  50. R. M. Bianchini and G. Stefani, “Graded approximations and controllability along a trajectory,”SIAM J. Contr. Optimiz.,28, No. 4, 903–924 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  51. R. M. Bianchini and G. Stefani, “Controllability along a trajectory: a variational approach,”SIAM J. Contr. Optimiz.,31, No. 3 (1993).

    Google Scholar 

  52. B. Bonnard, “Remarques sur les extrémals singulières en contrôle en temps minimal,” In:Outils et Models Math. Autom. Anal. Syst. et Trait. Signal, Vol. 3, Paris (1983), pp. 519–531.

  53. B. Bonnard, “On singular extremals in the time-minimal control problem in ℝ3”,SIAM J. Contr. Optimiz.,23, No. 5, 794–802 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  54. B. Bonnard, “On singular extremals in the time-minirnal control problem” In:Theory and Appl. Nonlinear Contr. Syst.: Selec. Pap. 7th Int. Symp. Math. Theory Networks and Syst., Stockholm, June 10–14, 1985, Amsterdam (1986), pp. 439–452.

  55. B. Bonnard, “Generic properties of singular extremals”, In:Proc. 27th IEEE Conf. Decis. and Contr., Austin, Tex., Dec. 7–9, 1988, Vol. 3, New York, N.Y. (1988), pp. 2369–2374.

  56. A. Bressan, “A high order test for optimality of bang-bang controls,”SIAM J. Contr. Optimiz.,23, No. 1, 38–48 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  57. A. Bressan, “Local asymptotic approximation of non-linear control systems,”Int. J. Contr 41, No. 5, 1331–1336 (1985).

    MATH  MathSciNet  Google Scholar 

  58. R. W. Brockett, “Functional expansions and higher order necessary conditions in optimal control,”Lect. Notes Econ. Math. Syst.,131, 111–121 (1976).

    MATH  MathSciNet  Google Scholar 

  59. D. J. Clements and B. D. O. Anderson, “Singular optimal control — the linear-quadratic problem,”Lect. Notes Contr. Inf. Sci.,5 (1978).

  60. P. E. Crouch, “Solvable approximations to control systems,”SIAM J. Contr. Optimiz.,22, No. 1, 40–54 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  61. P. E. Crouch, “Graded vector spaces and applications to the approximation of nonlinear systems,”Rend. Semin. Mat. Univ. e Politecn. Torino, num. spec., 1–54 (1987).

  62. M. Fliess, “An algebraic approach to functional expansions, application to a singular control problem,” In:Contr. Sci. and Technol. Progr., Proc 8th. Trienni World Congr. Int. Fed. Autom. Contr., Kyoto, Aug. 24–28, 1981, Vol. 1, Oxford (1982), pp. 331–336.

  63. R. V. Gamkrelidze, “Exponential representation of solutions of ordinary differential equations,” In:Proc. Equadiff. V Conf. Prague, 1977: Lect. Notes Math.,703, 117–129 (1979).

  64. H. Hermes, “Controllability and the singular problem,”J. SIAM Contr. Ser. A,2, No. 2, 241–260 (1965).

    MathSciNet  Google Scholar 

  65. H. Hermes, “Local controllability and sufficient conditions in singular problems,”J. Differ. Equat.,20, No. 1, 213–232 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  66. H. Hermes, “Local controllability and sufficient conditions in singular problems, II,”SIAM J. Contr. Optimiz.,14, No. 6, 1049–1062 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  67. H. Hermes, “Control systems which generate decomposable Lie algebras,”J. Differ. Equat.,44, No. 2, 166–187 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  68. A. Isidori,Nonlinear Control Systems: An Introduction, 2nd edition, Springer-Verlag, Berlin (1989).

    Google Scholar 

  69. M. Kawski, “Control variations with an increasing number of switchings,”Bull. Am. Math. Soc.,18, No. 2, 149–152 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  70. M. Kawski, “High-order small-time local controllability,” In:Nonlinear Controllab. and Optim. Contr.: Lect. Workshop Finite Dimens. Controllab. and Optim. Contr., New Brunswick, New Jersey, May 18–22, 1987, New York-Basel (1990), pp. 431–467.

  71. H. W. Knobloch, “Higher order necessary conditions in optimal control theory,”Lect. Notes Contr. Inf. Sci.,34 (1981).

  72. A. J. Krener, “The high order maximum principle.” In:Geom. Meth. Syst. Theory, Dordrecht-Boston (1973), pp 174–184.

  73. A. J. Krener, “The high order maximum principle and its application to singular extremals,”SIAM J. Contr. Optimiz.,15, No. 2, 256–293 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  74. F. Lamnabhi-Lagarrigue, “Sur les conditions nécessaires d’optimalité du deuxième et troisième ordre dans les problèmes de commande optimale singulière,”Lect. Notes Contr. Inf. Sci.,63, 525–541 (1983).

    MathSciNet  Google Scholar 

  75. F. Lamnabhi-Lagarrigue “A Volterra series interpretation of some higher order conditions in optimal control,”Lect. Notes Contr. Inf. Sci.,58, 615–627 (1984).

    MATH  MathSciNet  Google Scholar 

  76. F. Lamnabhi-Lagarrigue and G. Stefani, “Singular optimal control problems: on the necessary conditions of optimality,”SIAM J. Contr. Optimiz.,28, No. 4, 823–840 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  77. Y. Peraire, “Une démonstration brève d’un principe du maximum d’ordre supérieur,”C.R. Acad. Sci., Ser. 1,299, No. 7, 273–276 (1984).

    MATH  MathSciNet  Google Scholar 

  78. C. Rockland, “Intrinsic nilpotent approximation,”Acta Appl. Math.,8, 213–270 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  79. G. Stefani, “Local properties of nonlinear systems,”Sci. Pap. Inst. Techn. Cybern. Univ. Tech. Wrocl., No. 29, 219–226 (1985).

    MATH  MathSciNet  Google Scholar 

  80. G. Stefani, “On the local controllability of a scalar-input control system,” In:Theory and Appl. Nonlinear Contr. Syst: Selec. Pap. 7th Int. Symp. Math. Theory Networks and Syst., Stockholm June 10–14, 1985, Amsterdam (1986), pp. 167–179.

  81. G. Stefani, “Polynomial approximations to control systems and local controllability,” In:Proc. 24th IEEE Conf. Decis. and Contr., Fort Lauderdale, Fla., Dec. 11–13, 1985, Vol. 1, New York, N.Y. (1985), pp. 33–38.

  82. G. Stefani, “A sufficient condition for extremality,”Lect. Notes Contr. Inf. Sci.,111, 1–12 (1988).

    Article  Google Scholar 

  83. G. Stefani, “On the minimum time-problem,” In:Anal. and Contr. Nonlinear Systems, Amsterdam (1988), pp. 213–220.

  84. H. J. Sussmann, “Lie brackets, real analyticity and geometric control,” In:Differen. Geom. Contr. Theory: Proc. Mich. Technol. Univ. June 28–July 2, 1982, Boston (1983), pp. 1–116.

  85. H. J. Sussmann, “A general theorem on local controllability,”SIAM J. Contr. Optimiz.,25, No. 1, 158–194 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  86. A. I. Tretiak, “On necessary conditions of odd order for optimality and sufficient conditions for local controllability,” In:Proc. XV Int. Conf. Math. Optimiz., Theory and Appl., Eisenach, Dec. 11–15, 1989, Berlin (1989), pp. 242–245.

  87. A. I. Tretiak, “A second order necessary condition for optimality,” In:Fourth Int. Colloquium on Differential Equations, Plovdiv, Bulgaria, Aug. 18–23, 1993, Sofia (1993), p. 253.

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 32, Dynamical Systems-5, 1996.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tretiak, A.I. Higher-order local approximations of smooth control systems and pointwise higher-order optimality conditions. J Math Sci 90, 2150–2191 (1998). https://doi.org/10.1007/BF02433489

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02433489

Keywords

Navigation