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Differential-geometric and group-theoretical methods in optimal control theory

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Abstract

Problems arising in the geometry of attainable sets of nonlinear control systems are considered. Old and new results are discussed.

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Literature cited

  1. A. A. Agrachev and S. A. Vakhrameev, “Chronological series and the Cauchy-Kovalevskaya theorem,” in: Problems in Geometry [in Russian], Vol. 12 (Itogi Nauki Tekh. VINITI Akad. Nauk SSSR), Moscow (1981), pp. 165–189.

    Google Scholar 

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

    Google Scholar 

  3. A. A. Agrachev and R. V. Gamkrelidze, “Chronological algebras and nonstationary vector fields,” in: Problems in Geometry [in Russian], Vol. 11 (Itogi Nauki Tekh. VINITI Akad. Nauk SSSR), Moscow (1980), pp. 135–176.

    Google Scholar 

  4. V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  5. J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).

    Google Scholar 

  6. R. Gabasov and F. M. Kirillova, “Methods of optimal control,” in: Contemporary Problems in Mathematics [in Russian], Vol. 6 (Itogi Nauki Tekh. VINITI Akad. Nauk SSSR), Moscow (1975), pp. 133–261.

    Google Scholar 

  7. R. Gabasov and F. M. Kirillova, “Mathematical theory of optimal control,” in: Mathematical Analysis [in Russian], Vol. 16 (Itogi Nauki Tekh. VINITI Akad. Nauk SSSR), Moscow (1979), pp. 55–97.

    Google Scholar 

  8. R. V. Gamkrelidze, “On optimal sliding states,” Dokl. Akad. Nauk SSSR,143, No. 6, 1243–1245 (1962).

    Google Scholar 

  9. R. V. Gamkrelidze, “Necessary first-order conditions, and the axiomatics of extremal problems,” Tr. Mat. Inst. Akad. Nauk SSSR,112, Part I, 152–180 (1971).

    Google Scholar 

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

  11. R. V. Gamkrelidze and G. L. Kharatishvili, “Extremal problems in linear topological spaces,” Izv. Akad. Nauk SSSR, Ser. Mat.,33, No. 4, 781–839 (1969).

    Google Scholar 

  12. A. Ya. Dubovitskii and A. A. Milyutin, Necessary Conditions for Weak Extremum in the General Optimal Control Problem [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  13. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  14. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York (1967).

    Google Scholar 

  15. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  16. A. F. Filippov, “On some problems of optimal control theory,” Vestn. Mosk. Gos. Univ., Ser. Mat. Mekh. Astron. Fiz. Khim., No. 2, 25–32 (1959).

    Google Scholar 

  17. D. Aeyel, “Generic observability of differentiable systems,” SIAM J. Control Optimiz.,19, No. 5, 595–603 (1981).

    Google Scholar 

  18. A. Bacciotti, “Structuremetrique des orbites des familles symetriques des champes des vecteurs et theorie du temps minimum,” SIAM J. Control Optimiz.,17, No. 2, 311–319 (1979).

    Google Scholar 

  19. A. Bacciotti and G. Stefani, “The region of attainability of nonlinear systems with unbounded controls,” J. Optimiz. Theory Appl.,35, No. 1, 57–84 (1981).

    Google Scholar 

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

    Google Scholar 

  21. J. Baillieul, “The geometry of homogeneous polynomial dynamical systems,” Nonlinear Anal.: Theory, Meth. Appl.,4, No. 5, 879–900 (1980).

    Google Scholar 

  22. M. P. Bendsoe, “On the existence of observable single-output systems of a simple type,” SIAM J. Control Optimiz., 19, No. 4, 555–559 (1981).

    Google Scholar 

  23. B. Bonnard, “Controlabilite des systemes bilineaires,” Outils et Modeles Math. Autom., Anal. Syst. Trait Signal, Vol. 1, 229–243, Paris (1981).

    Google Scholar 

  24. R. W. Brockett, “System theory on group manifolds and coset spaces,” SIAM J. Control,10, No. 2, 265–284 (1972).

    Google Scholar 

  25. R. W. Brockett, “Lie algebras and Lie groups in control theory,” in: D. Q. Mayne and R. W. Brockett (eds.), Geometric Methods in System Theory, Reidel, Dordrecht (1973), pp. 43–82.

    Google Scholar 

  26. R. W. Brockett, “On the reachable set for bilinear systems,” in: Lecture Notes in Econ. and Math. Syst.,111 (1975), pp. 54–63.

    Google Scholar 

  27. R. W. Brockett, “Volterra series and geometric control theory,” Automatica,12, No. 2, 167–176 (1976).

    Google Scholar 

  28. R. W. Brockett, “Control theory and singular Riemannian geometry,” in: New Directions and Applications in Mathematics, New York (1982), pp. 11–27.

  29. W. L. Chow, “Über Systeme von liniearen partiellen Differentialgleichungen erster Ordnung,” Math. Ann.,117, 98–105 (1939).

    Google Scholar 

  30. R. Conti, “On relay controllability for bilinear processes,” in: Differential Equations, Stockholm (1977), pp. 32–36.

  31. P. E. Crouch, “Dynamical realizations of finite Volterra series,” SIAM J. Control Optimiz.,18, No. 4, 621–623 (1980).

    Google Scholar 

  32. M. Fliess, “The unobservability ideal for nonlinear systems,” IEEE Trans. Autom. Contr.,AC-26, No. 2, 592 (1981).

    Google Scholar 

  33. M. Fliess, “Realization of nonlinear systems and abstract transitive Lie algebras,” Bull. Am. Math. Soc.,2, 444–446 (1980).

    Google Scholar 

  34. G. W. Haynes and H. Hermes, “Nonlinear controllability via Lie theory,” SIAM J. Control,8, No. 4, 450–460 (1970).

    Google Scholar 

  35. R. Hermann, “On the accessibility problem in control theory,” in: J. P. LaSalle and S. Lefschetz (eds.), Int. Symp. Nonlinear Diff. Equations and Nonlinear Mech., Colorado Springs, 1961, Academic Press (1963), pp. 325–332.

    Google Scholar 

  36. R. Hermann, “Some remarks on geometry of systems,” in: D. Q. Mayne and R. W. Brockett (eds.), Geometric Methods in System Theory, Reidel, Dordrecht (1973), pp. 237–242.

    Google Scholar 

  37. R. Hermann and A. J. Krener, “Nonlinear controllability and observability,” IEEE Trans. Automat. Control.,AC-22, 728–740 (1977).

    Google Scholar 

  38. H. Hermes, “On local and global controllability,” SIAM J. Control.,12, No. 2, 252–261 (1974).

    Google Scholar 

  39. H. Hermes, “Necessary and sufficient conditions for local controllability and time optimality,” in: Proc. Int. Congr. Math., 1974, Vancouver,2, No. 1 (1975), pp. 343–347.

    Google Scholar 

  40. H. Hermes, “High-order algebraic conditions for controllability,” in: Lecture Notes in Econ. and Math. Syst.,131 (1976), pp. 165–171.

    Google Scholar 

  41. H. Hermes, “Local controllability and sufficient conditions in singular problems,” I, J. Diff. Eq.,20, No. 1, 213–232 (1976), and II, SIAM J. Control,14, No. 6, 1049–1069 (1976).

    Google Scholar 

  42. H. Hermes, “On local controllability,” SIAM J. Control Optimiz.,20, No. 2, 211–220 (1982).

    Google Scholar 

  43. H. Hermes and G. Haynes, “On the nonlinear control problem with control appearing linearly,” SIAM J. Control,A1, No. 1, 85–108 (1963).

    Google Scholar 

  44. R. M. Hirschorn, “Invertibility of nonlinear control systems,” SIAM J. Control Optimiz.,17, No. 2, 289–297 (1979).

    Google Scholar 

  45. L. R. Hunt, “Controllability of general nonlinear systems,” Math. Syst. Theory,12, No. 4, 361–370 (1979).

    Google Scholar 

  46. L. R. Hunt, “Controllability of nonlinear systems in two dimensions,” Math. Syst. Theory,13, 361–376 (1980).

    Google Scholar 

  47. B. Jakubczyk, “Existence and uniqueness of realization of nonlinear systems,” SIAM J. Contr. Optimiz.,18, No. 4, 455–471 (1980).

    Google Scholar 

  48. V. Jurdjevic, “Abstract control systems: controllability and observability,” SIAM J. Control,8, No. 3, 424–439 (1970).

    Google Scholar 

  49. V. Jurdjevic, “Certain controllability properties of analytic control systems,” SIAM J. Control,10, No. 2, 354–360 (1972).

    Google Scholar 

  50. V. Jurdjevic, “Causal dynamical systems: irreducible realizations,” in: D. Q. Mayne and R. W. Brockett (eds.), Geometric Methods in System Theory, Reidel, Dordrecht (1973), pp. 253–262.

    Google Scholar 

  51. V. Jurdjevic, “On the structure of irreducible state representations of a causal system,” Math. Syst. Theory,8, No. 1, 77–89 (1974).

    Google Scholar 

  52. V. Jurdjevic, “Attainable sets and controllability: a geometry approach,” in: Lecture Notes in Econ. and Math. Syst.,106, 219–251 (1974).

    Google Scholar 

  53. V. Jurdjevic and J. Kupka, “Control systems on semisimple Lie groups and their homogeneous spaces,” Ann. Inst. Fourier,31, No. 4, 151–179 (1981).

    Google Scholar 

  54. V. Jurdejevic and J. Kupka, “Control systems subordinated to a group action: Accessibility,” J. Diff. Eq.,39, No. 2, 186–211 (1981).

    Google Scholar 

  55. V. Jurdjevic and H. Sussmann, “Controllability of nonlinear systems,” J. Diff. Eq.,12, No. 1, 95–116 (1972).

    Google Scholar 

  56. V. Jurdjevic and H. Sussmann, “Control systems on Lie groups,” J. Diff. Eq.,12, No. 2, 313–329 (1972).

    Google Scholar 

  57. B. Kalitine and C. Lobry, “Complete controlabilite de certains systemes non lineaires,” Rev. Roum. Math. Pures Appl.,24, No. 2, 255–271 (1979).

    Google Scholar 

  58. A. J. Krener, “A generalization of the accessibility problem for control systems,” in: Proc. IEEE Conf. Decis. and Control (Incl. 10th Symp. Adapt. Process), Miami Beach, Florida 1971, New York (1971), pp. 186–187.

  59. A. J. Krener, “The high-order maximal principle,” in: D. Q. Mayne and R. W. Brockett (eds.), Geometric Methods in System Theory, Reidel, Dordrecht (1973), pp. 174–184.

    Google Scholar 

  60. A. J. Krener, “Bilinear and nonlinear realizations of input-output maps,” SIAM J. Control,13, No. 4, 827–834 (1975).

    Google Scholar 

  61. A. J. Krener, “Local approximation of control systems,” J. Diff. Eq.,19, No. 1, 125–133 (1975).

    Google Scholar 

  62. A. J. Krener, “A generalization of Chow's theorem and bang-bang theorem to nonlinear control problems,” SIAM J. Control,12, No. 1, 43–52 (1974).

    Google Scholar 

  63. J. Kucera, “Solution in large of control problem: x=(A(1−u)+Bu)x,” Czechosl. Math. J.,16, No. 4, 600–623 (1966).

    Google Scholar 

  64. J. Kucera, “Solution in large of control problem: x=(Au+Bv)x,” Czechosl. Math. J.,17, No. 1, 91–96 (1967).

    Google Scholar 

  65. J. Kucera, “On the accessibility of bilinear systems,” Czechosl. Math. J.,20, No. 1, 160–168 (1970).

    Google Scholar 

  66. G. Langholz and M. Sokolov, “Caratheodory controllability criterion for nonlinear dynamical systems,” Trans. ASME J. Dyn. Syst., Meas., Control,100, No. 3, 209–213 (1978).

    Google Scholar 

  67. N. Levitt and H. J. Sussmann, “On controllability by means of two vector fields,” SIAM J. Control,13, No. 6, 1271–1281 (1975).

    Google Scholar 

  68. C. Lobry, “Application d'un resultat de Chow a la theorie du controle optimal,” C. R. Acad. Sci. Paris,270, No.1, A725-A727 (1970).

    Google Scholar 

  69. C. Lobry, “Controlabilite des systemes nonlineaires,” SIAM J. Control,8, No. 4, 573–605 (1970).

    Google Scholar 

  70. C. Lobry, “Controlabilite des systemes lineaires par des commandes bang-bang,” Rev. Franc. Inf. Rech. Oper. (1970),4, No. R-3, 135–140.

    Google Scholar 

  71. C. Lobry, “Une propriete de l'ensemble des etats accessible d'un systeme quidable,” C. R. Acad. Sci. Paris,272, No. 2, A153-A156 (1971).

    Google Scholar 

  72. C. Lobry, “Une propriete generique des couples des champs de vecteurs,” Czechosl. Math. J.,22, No. 2, 230–237 (1972).

    Google Scholar 

  73. C. Lobry, “Quelques proprietes ‘generiques’ des systemes a commande,” Lect. Notes Math.,280, 120–130 (1972).

    Google Scholar 

  74. C. Lobry, “Dynamical polysystems and control theory,” in: D. Q. Mayne and R. E. Brockett (eds.), Geometric Methods in Control Theory, Reidel, Dordrecht (1973), pp. 1–42.

    Google Scholar 

  75. C. Lobry, “Controllability of nonlinear systems on compact manifolds,” SIAM J. Control,12, No. 1, 1–4 (1974).

    Google Scholar 

  76. C. Lobry, “Controllability of nonlinear control dynamical systems,” in: Control Theory and Topics in Functional Analysis, Vol. 1, International Atomic Energy Agency, Vienna (1976), pp. 361–383.

    Google Scholar 

  77. C. Lobry, “Controlabilite des systemes non lineaires,” in: Outils et Modeles Mathem. Autom. Anal. Syst. et Trait. Signal., Vol. 1, Paris (1981), pp. 187–214.

    Google Scholar 

  78. L. Markus, “Controllability of multitrajectories on Lie groups,” in: Lecture Notes Math.,898, 250–265 (1981).

    Google Scholar 

  79. H. Nijmeijer, “Controlled invariance for affine control systems,” Int. J. Control,34, No. 4, 825–833 (1981).

    Google Scholar 

  80. G. Sallet, “Complete controlabilite sur les groupes lineaires,” in: Outils et Models Math. Autom. Anal. Syst. et Trait. Signal., Vol. 1, Paris (1981), pp. 215–227.

    Google Scholar 

  81. A. J. van der Schaft, “Controllability and observability for affine nonlinear Hamiltonian systems,” IEEE Trans. Autom. Control,27, No. 2, 490–492 (1982).

    Google Scholar 

  82. E. D. Sontag, “On the observability of polynomial systems. I. Finite-time problems,” SIAM J. Control Optimiz.,17, No. 1, 139–151 (1979).

    Google Scholar 

  83. H. J. Sussmann, “Dynamical systems on manifolds: accessibility and controllability,” in: Proc. IEEE Conf. Decis. and Contr. (Incl. 10th Symp. Adapt. Processes), Miami Beach, Florida, 1971, New York (1971), pp. 188–191.

  84. H. J. Sussmann, “The control problem x=(A(1−u)+Bu)x: A comment on an article by J. Kucera,” Czechosl. Math. J.,22, 423–426 (1972).

    Google Scholar 

  85. H. J. Sussmann, “The control problem x=A(u)x,” Czechosl. Math. J.,22, No. 3, 490–494 (1972).

    Google Scholar 

  86. H. J. Sussmann, “The bang-bang problem for certain control systems in GL(n, R),” SIAM J. Control,10, No. 3, 470–476 (1972).

    Google Scholar 

  87. H. J. Sussmann, “An extension of a theorem of Nagano on transitive Lie algebras,” Proc. Am. Math. Soc.,45, No. 3, 349–356 (1974).

    Google Scholar 

  88. H. J. Sussmann, “Orbits of families of vector fields and integrability of systems with singularities,” Bull. Am. Math. Soc.,79, No. 1, 197–199 (1973).

    Google Scholar 

  89. H. J. Sussmann, ”Orbits of families of vector fields and integrability of distributions,” Trans. Am. Math. Soc.,180, June, 171–188 (1973).

    Google Scholar 

  90. H. J. Sussmann, ”Minimal realizations of nonlinear systems,” in: D. Q. Mayne and R. W. Brockett (eds.), Geometric Methods in System Theory, Reidel, Dordrecht (1973), pp. 243–252.

    Google Scholar 

  91. H. J. Sussmann, “On quotients of manifolds: a generalization of the closed subgroup theorem,” Bull. Am. Math. Soc.,80, No. 3, 573–575 (1974).

    Google Scholar 

  92. H. J. Sussmann, “A generalization of the closed subgroup theorem to quotients of arbitrary manifolds,” J. Diff. Geom.,10, No. 1, 151–166 (1975).

    Google Scholar 

  93. H. J. Susmann, “On the number of directions needed to achieve controllability,” SIAM J. Control,13, No. 2, 414–419 (1975).

    Google Scholar 

  94. H. J. Sussmann, “Some properties of vector field systems that are not altered by small perturbations,” J. Diff. Eq.,20, No. 2, 292–315 (1976).

    Google Scholar 

  95. H. J. Sussmann, “Minimal realizations and canonical forms for bilinear systems,” J. Franklin Inst.,301, No. 6, 593–604 (1976).

    Google Scholar 

  96. H. J. Sussmann, “Semigroup representation, bilinear approximation of input-output maps, and generalized inputs,” in: Lect. Notes Econ. Math. Syst.,131, 172–191 1976).

    Google Scholar 

  97. H. J. Sussmann, “Existence and uniqueness of minimal realization of nonlinear systems,” Math. Syst. Theory,10, No. 3, 263–284 (1976–1977).

    Google Scholar 

  98. H. J. Sussmann, “Generic single-input observability of continuous time polynomial systems,” in: Proc. IEEE Conf. Decis. and Contr. and 17th Symp. Adapt. Processes, San Diego, California, 1979, New York (1979), pp. 566–571.

  99. H. J. Sussmann, “Single-input observability of continuous time systems,” Math. Syst. Theory,12, No. 4, 371–393 (1979).

    Google Scholar 

  100. H. J. Sussmann, “A bang-bang theorem with bounds on the number of switchings,” SIAM J. Control and Optimiz.,17, No. 5, 629–651 (1979).

    Google Scholar 

  101. H. J. Sussmann, “Bounds on the number of switchings for trajectories of piecewise-analytic vector fields,” J. Diff. Eq.,43, No. 2, 399–418 (1982).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 3–56, 1983.

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Agrachev, A.A., Vakhrameev, S.A. & Gamkrelidze, R.V. Differential-geometric and group-theoretical methods in optimal control theory. J Math Sci 28, 145–182 (1985). https://doi.org/10.1007/BF02105209

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