The quasidifferential descent method in a control problem with nonsmooth objective functional


The paper is devoted to the problem of optimal control of an object described by a system with a continuously differentiable right-hand side and a nondifferentiable (but only quasidifferentiable) quality functional. We consider a problem in the form of Mayer with both a free and a fixed right end. Admissible controls are piecewise continuous (with a finite number of discontinuity points) and bounded vector-functions, which belong to certain polyhedron at each moment of time. Standard discretization of the initial system and control parameterization are performed, and theorems on the convergence of the discrete system solution obtained to the desired solution of the continuous problem are presented. Further, for the discrete system obtained, the necessary and, in some cases, sufficient minimum conditions are written in terms of quasidifferential. The quasidifferential descent method is applied to this problem. The algorithm developed is demonstrated by examples.

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  1. 1.

    Loxton, R..C., Teo, K..L., Rehbocka, V., Yiu, K..F..C.: Optimal control problems with a continuous inequality constraint on the state and the control. Automatica 45, 2250–2257 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Yuan, J., Xie, J., Xu, H., Feng, E., Xiu, Z.: Optimization for nonlinear uncertain switched stochastic systems with initial state difference in batch culture process. Complexity 2019, 1–15 (2019)

    MATH  Google Scholar 

  3. 3.

    Li, B., Xu, C., Teo, K.L., Chu, J.: Time optimal Zermelo’s navigation problem with moving and fixed obstacles. Appl. Math. Comput. 224, 866–875 (2013)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Li, B., Yu, C.J., Teo, K.L., Duan, G.R.: An exact penalty function method for continuous inequality constrained optimal control problem. J. Optim. Theory Appl. 151(2), 260–291 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Yang, F., Teo, K.L., Loxton, R., Rehbock, V., Li, B., Yu, C., Jennings, L.: VISUAL MISER: an efficient user-friendly visual program for solving optimal control problems. J. Ind. Manag. Optim. 12(2), 781–810 (2016)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bryson, A.E., Denham, W.F.: A steepest-ascent method for solving optimum programming problems. J. Appl. Mech. 29(2), 247–257 (1962)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kelley, H.J.: Method of gradients. Math. Sci. Eng. 5, 205–254 (1962)

    Article  Google Scholar 

  8. 8.

    Demyanov, V. F., Rubinov, A. M. Foundations of nonsmooth analysis and quasidifferential calculus. Nauka, M. (1990). (in Russian)

  9. 9.

    Demyanov, V.F., Nikulina, V.N., Shablinskaya, I.R.: Quasidifferentiable functions in optimal control. Math. Program. Study. 29, 160–175 (1986)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Vinter, R.B., Cheng, H.: Necessary conditions for optimal control problems with state constraints. Trans. Am. Math. Soc. 350(3), 1181–1204 (1998)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Vinter, R.B.: Minimax optimal control. SIAM. J. Control Optim. 44(3), 939–968 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Frankowska, H.: The first order necessary conditions for nonsmooth variational and control problems. SIAM J. Control Optim. 22(1), 1–12 (1984)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mordukhovich, B.: Necessary conditions for optimality in nonsmooth control problems with nonfixed time. Diff. Equ. 25(1), 290–299 (1989)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9(2), 159–189 (1984)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Shvartsman, I.A.: New approximation method in the proof of the Maximum Principle for nonsmooth optimal control problems with state constraints. J. Math. Anal. Appl. 326(2), 974–1000 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fominyh, A.V.: Open-loop control of a plant described by a system with nonsmooth right-hand side. Comput. Math. Math. Phys. 59(10), 1639–1648 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Fominyh, A.V., Karelin, V.V., Polyakova, L.N.: Application of the hypodifferential descent method to the problem of constructing an optimal control. Opt. Lett. 12(8), 1825–1839 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Fominykh, A.V.: Methods of subdifferential and hypodifferential descent in the problem of constructing an integrally constrained program control. Autom. Remote Control 78(4), 608–617 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Gorelik, V.A., Tarakanov, A.F.: Penalty method and maximum principle for nonsmooth variable-structure control problems. Cybern. Syst. Anal. 28(3), 432–437 (1992)

    Article  Google Scholar 

  20. 20.

    Gorelik, V.A., Tarakanov, A.F.: Penalty method for nonsmooth minimax control problems with interdependent variables. Cybernetics. 25(4), 483–488 (1989)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Morzhin O. V.: On Approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control. Avtomatika i Telemekhanika. (2009). (5) 24–34.

  22. 22.

    Mayne D. Q., Polak E.: An exact penalty function algorithm for control problems with state and control constraints // 24th IEEE Conference on Decision and Control. (1985). P. 1447–1452

  23. 23.

    Mayne, D.Q., Smith, S.: Exact penalty algorithm for optimal control problems with control and terminal constraints. Int. J. Control. 48(1), 257–271 (1988)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Noori, Skandari M., H., Kamyad, A.. V., Effati, S.: Smoothing approach for a class of nonsmooth optimal control problems. Appl. Math. Modell. 40(2), 886–903 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Noori, Skandari M., H., Kamyad, A.. V., Erfanian, H.. R.: Control of a class of nonsmooth dynamical systems. J. Vib. Control 21(11), 2212–2222 (2013)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Vasil’ev F. P.: Optimization methods. Moscow, Factorial Press, (2002). 824 p. (in Russian)

  27. 27.

    Filippov, A.: On certain questions in the theory of optimal control. J. Soc. Ind. Appl. Math. Ser. A Control 1(1), 76–84 (1962)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems (Pitman Monographs and Surveys in Pure and Applied Mathematics). Longman Scientific and Technical, New York (1991)

    Google Scholar 

  29. 29.

    Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Nauka, Moscow (1979).. ((in Russian))

    Google Scholar 

  30. 30.

    Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization. 65(6), 1167–1202 (2015)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Dolgopolik, M.V.: Exact penalty functions for optimal control problems II: exact penalization of terminal and pointwise state constraints. Opt. Control Appl. Methods 41(3), 898–947 (2020)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Vasil’ev L. V., Demyanov V. F.: Nondifferentiable optimization. M.: Nauka, (1981). (in Russian)

  33. 33.

    Wolfe, P.: The simplex method for quadratic programming. Econom. 27, 382–398 (1959)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Dolgopolik, M.V.: A convergence analysis of the method of codifferential descent. Comput. Optim. Appl. 71(3), 879–913 (2018)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Flugge-Lotz, I., Hales, K.A., Minimum-fuel attitude control of a rigid body in orbit by an extended method of steepest descent , , Dept. of Aeron. and Astron. : Stanford University, Stanford, p. 257. Calif, Rept (1966)

  36. 36.

    Dyer, P.E.T.E.R., McReynolds, S.R.: Optimization of control systems with discontinuities and terminal constraints. IEEE Trans. Autom. Control 14(3), 223–229 (1969)

    MathSciNet  Article  Google Scholar 

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The author is sincerely grateful to his colleague M. Dolgopolik for numerous fruitful discussions on the problem considered and to the anonymous referees, whose comments helped to improve the paper.

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Correspondence to A. V. Fominyh.

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Fominyh, A.V. The quasidifferential descent method in a control problem with nonsmooth objective functional. Optim Lett (2021).

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  • Nonsmooth optimal control problem
  • Quasidifferential
  • Parameterization of control
  • Method of quasidifferential descent