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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
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)
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)
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)
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)
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)
Bryson, A.E., Denham, W.F.: A steepest-ascent method for solving optimum programming problems. J. Appl. Mech. 29(2), 247–257 (1962)
Kelley, H.J.: Method of gradients. Math. Sci. Eng. 5, 205–254 (1962)
Demyanov, V. F., Rubinov, A. M. Foundations of nonsmooth analysis and quasidifferential calculus. Nauka, M. (1990). (in Russian)
Demyanov, V.F., Nikulina, V.N., Shablinskaya, I.R.: Quasidifferentiable functions in optimal control. Math. Program. Study. 29, 160–175 (1986)
Vinter, R.B., Cheng, H.: Necessary conditions for optimal control problems with state constraints. Trans. Am. Math. Soc. 350(3), 1181–1204 (1998)
Vinter, R.B.: Minimax optimal control. SIAM. J. Control Optim. 44(3), 939–968 (2005)
Frankowska, H.: The first order necessary conditions for nonsmooth variational and control problems. SIAM J. Control Optim. 22(1), 1–12 (1984)
Mordukhovich, B.: Necessary conditions for optimality in nonsmooth control problems with nonfixed time. Diff. Equ. 25(1), 290–299 (1989)
Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9(2), 159–189 (1984)
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)
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)
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)
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)
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)
Gorelik, V.A., Tarakanov, A.F.: Penalty method for nonsmooth minimax control problems with interdependent variables. Cybernetics. 25(4), 483–488 (1989)
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.
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
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)
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)
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)
Vasil’ev F. P.: Optimization methods. Moscow, Factorial Press, (2002). 824 p. (in Russian)
Filippov, A.: On certain questions in the theory of optimal control. J. Soc. Ind. Appl. Math. Ser. A Control 1(1), 76–84 (1962)
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)
Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Nauka, Moscow (1979).. ((in Russian))
Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization. 65(6), 1167–1202 (2015)
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)
Vasil’ev L. V., Demyanov V. F.: Nondifferentiable optimization. M.: Nauka, (1981). (in Russian)
Wolfe, P.: The simplex method for quadratic programming. Econom. 27, 382–398 (1959)
Dolgopolik, M.V.: A convergence analysis of the method of codifferential descent. Comput. Optim. Appl. 71(3), 879–913 (2018)
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)
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)
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Fominyh, A.V. The quasidifferential descent method in a control problem with nonsmooth objective functional. Optim Lett 15, 2773–2792 (2021). https://doi.org/10.1007/s11590-021-01710-7
- Nonsmooth optimal control problem
- Parameterization of control
- Method of quasidifferential descent