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A Survey on Regularity Conditions for State-Constrained Optimal Control Problems and the Non-degenerate Maximum Principle

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Abstract

A survey on the theory of maximum principle for state-constrained optimal control problems is presented. The focus is on such issues as regularity and controllability conditions, non-degeneracy and normality of the maximum principle, and on the continuity of the measure multiplier.

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Notes

  1. The set U can be an arbitrary subset of \(\mathbb {R}^m\). However, with regard to the application aspects of the control theory, in much work on this theory, it is considered closed.

  2. Let us remark that such aspects as reducing the smoothness, or continuity, hypothesis for the data are not the concern of the present survey.

References

  1. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962)

    MATH  Google Scholar 

  2. Gamkrelidze, R.V.: Optimum-rate processes with bounded phase coordinates. Dokl. Akad. Nauk SSSR 125, 475–478 (1959)

    MathSciNet  MATH  Google Scholar 

  3. Gamkrelidze, R.V.: Optimal control processes with restricted phase coordinates. Izv. Akad. Nauk SSSR Scr. Mat. 24, 315–356 (1960)

    Google Scholar 

  4. Berkovitz, L.D.: On control problems with bounded state variables. J. Math. Anal. Appl. 5, 488–498 (1962)

    MathSciNet  MATH  Google Scholar 

  5. Warga, J.: Minimizing variational curves restricted to a preassigned set. Trans. Am. Math. Soc. 112, 432–455 (1964)

    MathSciNet  MATH  Google Scholar 

  6. Gamkrelidze, R.V.: On some extremal problems in the theory of differential equations with applications to the theory of optimal control. SIAM J. Control 3, 106–128 (1965)

    MathSciNet  MATH  Google Scholar 

  7. Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems in the presence of restrictions. Zh. Vychisl. Mat. Mat. Fiz. 5(3), 395–453 (1965). (U.S.S.R. Comput. Math. Math. Phys. 5(3):1–80 (1965))

    MATH  Google Scholar 

  8. Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966)

    MATH  Google Scholar 

  9. Neustadt, L.W.: An abstract variational theory with applications to a broad class of optimization problems. I: General theory. SIAM J. Control 4, 505–527 (1966)

    MathSciNet  MATH  Google Scholar 

  10. Neustadt, L.W.: An abstract variational theory with applications to a broad class of optimization problems. II: Applications. SIAM J. Control 5, 90–137 (1967)

    MathSciNet  MATH  Google Scholar 

  11. Halkin, H.: A satisfactory treatment of equality and operator constraints in the Dubovitskii–Milyutin optimization formalism. J. Optim. Theory Appl. 6(2), 138–149 (1970)

    MathSciNet  MATH  Google Scholar 

  12. Russak, I.B.: On problems with bounded state variables. J. Optim. Theory Appl. 5, 114–157 (1970)

    MathSciNet  MATH  Google Scholar 

  13. Arutyunov, A.V., Tynyanskiy, N.T.: The maximum principle in a problem with phase constraints. Sov. J. Comput. Syst. Sci. 23, 28–35 (1985)

    MathSciNet  Google Scholar 

  14. Arutyunov, A.V.: On necessary optimality conditions in a problem with phase constraints. Sov. Math. Dokl. 31(1), 174–177 (1985)

    MATH  Google Scholar 

  15. Dubovitskii, A.Y., Dubovitskii, V.A.: Necessary conditions for strong minimum in optimal control problems with degeneration of endpoint and phase constraints. Usp. Mat. Nauk 40(2), 175–176 (1985)

    MathSciNet  Google Scholar 

  16. Matveev, A.S.: On the necessary conditions for an extremum in an optimal control problem with phase constraints. Differ. Uravn. 23(4), 629–640 (1987)

    MathSciNet  Google Scholar 

  17. Ferreira, M.M.A., Vinter, R.B.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl. 187, 438–467 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. Elsevier, Amsterdam (1979)

    Google Scholar 

  19. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)

    Google Scholar 

  20. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II: Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)

    Google Scholar 

  21. Arutyunov, A.V.: On the theory of the maximum principle for optimal control problems with state constraints. Doklady AN SSSR 304(1), 11–14 (1989)

    Google Scholar 

  22. Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Mathematics and Its Application. Kluwer Academic Publisher, Dordrecht (2000)

    MATH  Google Scholar 

  23. Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math. 54(6), 1342–1400 (1991)

    MATH  Google Scholar 

  24. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theory Appl. 149, 474–493 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Dubovitskii, A.Y., Milyutin, A.A.: Necessary conditions for a weak extremum in optimal control problems with mixed constraints of inequality type. Zh. Vychisl. Mat. Mat. Fiz. 8(4), 725–779 (1968)

    MathSciNet  MATH  Google Scholar 

  26. Clarke, F.H., Vinter, R.B.: Optimal multiprocesses. SIAM J. Control Optim. 27(5), 1072–1091 (1989)

    MathSciNet  MATH  Google Scholar 

  27. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Investigation of controllability and regularity conditions for state constrained problems. In: Proceedings of the IFAC Congress in Toulouse, IFAC-PapersOnline, France (2017)

  28. Ferreira, M.M.A., Fontes, F.A.C.C., Vinter, R.B.: Non-degenerate necessary conditions for nonconvex optimal control problems with state constraints. J. Math. Anal. Appl. 233, 116–129 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Arutyunov, A.V., Aseev, S.M., Blagodatskikh, V.I.: First-order necessary conditions in the problem of optimal control of a differential inclusion with phase constraints. Mat. Sb. 184(6), 3–32 (1993)

    MATH  Google Scholar 

  30. Arutyunov, A.V., Aseev, S.M.: State constraints in optimal control: the degeneracy phenomenon. Syst. Control Lett. 26, 267–273 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Arutyunov, A.V., Chernikova, N.Y.: On the controllability of trajectories in optimal control problems with phase constraints, Selected papers presented at the international conference dedicated to the 90th anniversary of the birth of L. S. Pontryagin (Moscow, Russia, August 31–September 9, 1998). Vol. 3. Geometric control theory, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 64, VINITI, Moscow, 1999, 49–56; J. Math. Sci. (New York) 103(6): 664–669 (2001)

  32. Rampazzo, F., Vinter, R.: Degenerate optimal control problems with state constraints. SIAM J. Control Optim. 39(4), 989–1007 (2000)

    MathSciNet  MATH  Google Scholar 

  33. Lopes, S.O., Fontes, F.A.C.C., de Pinho, M.R.: On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discret. Contin. Dyn. Syst. Ser. A 29(2), 559–575 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Lopes, S.O., Fontes, F.A.C.C., Pinho, M.R.: An integral-type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints. Syst. Control Lett. 62(8), 686–692 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Fontes, F.A.C.C., Frankowska, H.: Normality and nondegeneracy for optimal control problems with state constraints. J. Optim. Theory Appl. 166, 115–136 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Alexandrov, V.V., Budninskiy, M.A.: On kinematic control extremals. In: European Control Conference (ECC), Zurich, Switzerland, pp. 210–214 (2013)

  37. Chertovskih, R., Karamzin, D., Khalil, N.T., Pereira, F.L.: An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow. In: Proceedings of IEEE/OES Autonomous Underwater Vehicle Workshop, AUV 2018, Art. No. 8729750 (2018)

  38. Chertovskih, R., Khalil, N.T., Karamzin, D., Pereira, F.L.: Regular path-constrained time-optimal control problems in three-dimensional flow fields. European Journal of Control (2020) (to appear)

  39. Arutyunov, A.V., Karamzin, D.Y.: Properties of extremals in optimal control problems with state constraints. Differ. Equ. 52(11), 1411–1422 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Rampazzo, F., Vinter, R.B.: A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control. IMA J. Math. Control Inf. 16(4), 335–351 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Frankowska, H.: Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern. 38(4), 1327–1340 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Frankowska, H., Tonon, D.: Inward pointing trajectories, normality of the maximum principle and the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints. J. Conv. Anal. 20(4), 1147–1180 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Bettiol, P., Facchi, G.: Linear estimates for trajectories of state-constrained differential inclusions and normality conditions in optimal control. J. Math. Anal. Appl. 414(2), 914–933 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Bettiol, P., Khalil, N., Vinter, R.B.: Normality of generalized euler-lagrange conditions for state constrained optimal control problems. J. Convex Anal. 23, 291–311 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. SIAM Series in Applied Mathematics. Wiley, Hoboken (1967)

    MATH  Google Scholar 

  46. Bryson, E.R., Yu-Chi, H.: Applied Optimal Control. Taylor & Francis, Routledge (1969)

    Google Scholar 

  47. Betts, J.T., Huffman, W.P.: Path-constrained trajectory optimization using sparse sequential quadratic programming. J. Guid. Control Dyn. 16(1), 59–68 (1993)

    MATH  Google Scholar 

  48. Buskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control. J. Comput. Appl. Math. 120, 85–108 (2000)

    MathSciNet  MATH  Google Scholar 

  49. Haberkorn, T., Trelat, E.: Convergence results for smooth regularizations of hybrid nonlinear optimal control problems. SIAM J. Control Optim. 49(4), 1498–1522 (2011)

    MathSciNet  MATH  Google Scholar 

  50. Arutyunov, A., Jacimovic, V., Pereira, F.: Second order necessary conditions for optimal impulsive control problems. J. Dyn. Control Syst. 9, 131–153 (2003)

    MathSciNet  MATH  Google Scholar 

  51. Hager, W.W.: Lipschitz continuity for constrained processes. SIAM J. Control Optim. 17(3), 321–338 (1979)

    MathSciNet  MATH  Google Scholar 

  52. Maurer, H.: Differential stability in optimal control problems. Appl. Math. Optim. 5(1), 283–295 (1979)

    MathSciNet  MATH  Google Scholar 

  53. Afanas’ev, A.P., Dikusar, V.V., Milyutin, A.A., Chukanov, S.A.: Necessary condition in optimal control. Nauka, Moscow (1990). (in Russian)

    MATH  Google Scholar 

  54. Galbraith, G.N., Vinter, R.B.: Lipschitz continuity of optimal controls for state constrained problems. SIAM J. Control Optim. 42(5), 1727–1744 (2003)

    MathSciNet  MATH  Google Scholar 

  55. Bonnans, J.F., Hermant, A.: Revisiting the analysis of optimal control problems with several state constraints. Control Cybern. 38(4), 1021–1052 (2009)

    MathSciNet  MATH  Google Scholar 

  56. Arutyunov, A.V.: Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints. Differ. Equ. 48(12), 1586–1595 (2012)

    MathSciNet  MATH  Google Scholar 

  57. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints. Proc. Steklov Inst. Math. 292, 27–35 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Zakharov, E.V., Karamzin, D.Y.: On the study of conditions for the continuity of the Lagrange multiplier measure in problems with state constraints. Differ. Equ. 51(3), 399–405 (2015)

    MathSciNet  MATH  Google Scholar 

  59. Arutyunov, A.V., Karamzin, D.Y.: On some continuity properties of the measure Lagrange multiplier from the maximum principle for state constrained problems. SIAM J. Control Optim. 53(4), 2514–2540 (2015)

    MathSciNet  MATH  Google Scholar 

  60. Karamzin, D., Pereira, F.L.: On a few questions regarding the study of state-constrained problems in optimal control. J. Optim. Theory Appl. 180(1), 235–255 (2019)

    MathSciNet  MATH  Google Scholar 

  61. Arutyunov, A., Karamzin, D., Pereira, F.L.: A remark on the continuity of the measure Lagrange multiplier in state constrained optimal control problems. In: IEEE 57th Conference on Decision and Control (2019)

  62. Vinter, R.B., Papas, G.: A maximum principle for nonsmooth optimal control problems with state constraints. J. Math. Anal. Appl. 89, 212–232 (1982)

    MathSciNet  MATH  Google Scholar 

  63. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  64. Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9, 159–189 (1984)

    MathSciNet  MATH  Google Scholar 

  65. Vinter, R.B.: Optimal Control. Birkhauser, Boston (2000)

    MATH  Google Scholar 

  66. Arutyunov, A.V., Vinter, R.B.: A simple ’finite approximations’ proof of the Pontryagin maximum principle under reduced differentiability hypotheses. Set-Valued Anal. 12, 5–24 (2004)

    MathSciNet  MATH  Google Scholar 

  67. Russak, I.B.: On general problems with bounded state variables. J. Optim. Theory Appl. 6, 424–452 (1970)

    MathSciNet  MATH  Google Scholar 

  68. Arutyunov, A.V., Karamzin, D.Y.: Maximum principle in an optimal control problem with equality state constraints. Differ. Equ. 51, 33 (2015)

    MathSciNet  MATH  Google Scholar 

  69. Arutyunov, A.V., Karamzin, D.Y.: Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints. J. Glob. Optim. 64, 623–647 (2016)

    MathSciNet  MATH  Google Scholar 

  70. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 19, 117–159 (2012)

    MathSciNet  MATH  Google Scholar 

  71. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-Valued Var. Anal. 23(1), 69–86 (2015)

    MathSciNet  MATH  Google Scholar 

  72. Colombo, G., Henrion, R., Nguyen, D.H., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ. 260(4), 3397–3447 (2016)

    MathSciNet  MATH  Google Scholar 

  73. Cao, T.H., Mordukhovich, B.S.: Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discret. Contin. Dyn. Syst. Ser. B 22(2), 267–306 (2017)

    MathSciNet  MATH  Google Scholar 

  74. Arroud, C.E., Colombo, G.: A maximum principle of the controlled sweeping process. Set-Valued Var. Anal. 26, 607–629 (2018)

    MathSciNet  MATH  Google Scholar 

  75. Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Differ. Equ. 266, 1003–1050 (2019)

    MathSciNet  MATH  Google Scholar 

  76. Colombo, G., Mordukhovich, B.S., Nguyen, D.: Optimal control of sweeping processes in robotics and traffic flow models. J. Optim. Theory Appl. 182, 439–472 (2019)

    MathSciNet  MATH  Google Scholar 

  77. Hoang, N.D., Mordukhovich, B.S.: Extended Euler-Lagrange and Hamiltonian formalisms in optimal control of sweeping processes with controlled sweeping set. J. Optim. Theory Appl. 180, 256–289 (2019)

    MathSciNet  MATH  Google Scholar 

  78. Colombo, G., Mordukhovich, B.S., Nguyen, D.: Optimization of a perturbed sweeping process by discontinuous controls. SIAM J. Control Optim. arXiv:1808.04041(to appear)

  79. Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) New Variational Techniques in Mathematical Physics. Proceedings from CIME, Cremonese, Rome (1974)

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Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable remarks which significantly improved the exposition. The work of the first author (Sects. 2, 3) was supported by the Russian Science Foundation, Project No. 17-11-01168. The work of the second author (Sects. 4, 5) was supported by the Russian Science Foundation, Project No. 19-11-00258 carried out in the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia.

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  1. V.A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russia

    Aram Arutyunov

  2. Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

    Dmitry Karamzin

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  1. Aram Arutyunov
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  2. Dmitry Karamzin
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Correspondence to Aram Arutyunov.

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Communicated by B. Mordukhovich.

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Arutyunov, A., Karamzin, D. A Survey on Regularity Conditions for State-Constrained Optimal Control Problems and the Non-degenerate Maximum Principle. J Optim Theory Appl 184, 697–723 (2020). https://doi.org/10.1007/s10957-019-01623-7

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