ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2014. Vol. 8

Modern Methods for Solving Nonconvex Optimal Control Problems

A. S. Strekalovsky

The paper presents a few remarks on the evolution of Irkutsk’s school of O. V. Vasiliev on optimal control methods based on Pontryagin principle. Besides, one reviews some features of Pontryagin principle, in particular, its sufficiency and constructive property for linear (on the state) control systems and convex cost functionals. Further, some historical notes on the development of optimal control methods based on Pontryagin principle are considered. In particular, a separated attention has been paid to the impact of Irkutsk school of O. V. Vasiliev in the theory and method of optimal control, and the achievements of the former postgraduate student of O. V. Vasiliev professor V. A. Srochko. The mathematical presentation is concentrated on the story of the invention and investigations of the convergence and substantiation of the consecutive approximate’s method based on Pontryagin principle. In addition, one considers new Global Optimality Conditions in a general nonconvex optimal control problem with Bolza goal functionals. Moreover, together with the necessity proof of global optimality conditions we investigate its relations to Pontryagin principle. Besides, the constructive (algorithmic) property of new optimality conditions is also demonstrated, and an example of nonconvex optimal control problems has been solved by means of global optimality conditions. In this example, we performed an improvement of a feasible control satisfying Pontryagin principle with a corresponding improvement of the cost functional. Finally, employing Pontryagin principle and new Global Optimality Conditions we give a demonstration of construction of a optimal control method and provide for new result on its convergence.

Pontryagin principle, optimal control methods, global optimality conditions

1. Vasiliev O. V. Lections on Optimization Methods (in Russian). Irkutsk, ISU Publ., 1994.

2. Vasiliev O.V., Tyatyushkin A.I. A method for solving optimal control problemsbased on the maximum principle. USSR Computational Mathematics and Mathematical Physics, 1981, vol. 21, no 6, pp. 14-22.

3. Ashchepkov L.T., Belov B.I, Bulatov V.P., Vasiliev O.V., Srochko V.A., Tarasenko N.V. Method for solving problems of mathematical programming and optimal control (in Russian). Novosibirsk, Nauka, 1984.

4. Vasiliev F.P. Optimization methods (in Russian). Мoscow, Factorial Press, 2002.

5. Gabasov R.F., Kirillova F.M. Linear system optimization (in Russian). Minsk, Belorussian University, 1973.

6. Gabasov R., Kirillova F.M. Maximum principle in optimal control theory (inRussian). Minsk, Belorussian University, 1974.

7. Girsanov I.V. Lectures on mathematical theory of extremal problems (in Russian). Moscow, MSU Publ., 1970.

8. Krylov I.A., Chernous’ko F.L. An algorithm for the method of successive approximations in optimal control problems. USSR Computational Mathematicsand Mathematical Physics, 1972, vol. 12, no. 1, pp. 14-34.

9. Lions J.-L. Optimal control of systems described by partial differential equations. Heidelberg, Springer, 1971.

10. Lyubushin A.A. Modifications and convergence of successive approximations foroptimal control problems. USSR Computational Mathematics and Mathematical Physics, 1979, vol. 19, no. 6, pp. 53-61.

11. Lyubushin A.A. Modifications of the method of successive approximations for solving optimal control problems USSR Computational Mathematics and Mathematical Physics, 1982, vol. 22, no. 1, pp. 29-34.

12. Lyubushin A.A. and Chernous’ko F.L. Method of successive approximations forcalculating optimal control (in Russian). Izv. Akad. Nauk SSSR, Tekh. Kibern., 1983, no. 2, 147–159.

13. Pontryagin L.S., Boltyanskij V.G., Gamkrelidze R.V., Mishchenko E.F. Mathematical theory of optimal processes. New York, Interscience Publishers, JohnWiley and Sons, 1962.

14. Srochko V.A. Variational maximum principle and linearization methods foroptimal control problems (in Russian). Irkutsk, ISU Publ., 1989.

15. Srochko V.A. Iterative methods for solving optimal control problems (in Russain). Moscow, Fizmatlit, 2000.

16. Srochko V.A., Aksenyushkina E.V. Linear-quadratic problem of optimal control: justification and convergence of nonlocal methods (in Russian). Izvestia IGU, Ser.Matematika, 2013, vol. 6, no. 1, pp. 89-100.

17. Srochko V.A., Ushakova S.N. Improvement of extreme controls and the steepestascent method in the norm maximization problem on the reachable set. Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 5, pp.848-859

18. Strekalovsky A.S. Elements of nonconvex optimization (in Russian). Novosibirsk, Nauka, 2003.

19. Strekalovsky A.S. Maximizing a state convex lagrange functional in optimal control. Automation and Remote Control, 2012, vol. 73, no. 6, pp. 949-961.

20. Strekalovsky A.S. Optimal control problems with terminal functionals representedas the difference of two convex functions. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 11, pp. 1788-1801.

21. Strekalovsky A.S. Bimatrix games and bilinear programming (in Russian). Moscow, Fizmatlit, 2007.

22. Strekalovsky A.S., Yanulevich M.V. Global search in a noncovex optimal controlproblem. Journal of Computer and Systems Sciences International, 2013, vol. 52, no. 6, pp. 893-908.

23. Strekalovsky A.S., Yanulevich M.V. On solving nonconvex optimal controlproblems with terminal objective functional (in Russian). Numerical methods and programming, 2010, vol. 11, pp. 269-280.

24. Strekalovsky A.S., Yanulevich M.V. Global search in the optimal control problem with a terminal objective functional represented as the difference of two convex functions. Computational Mathematics and Mathematical Physics, 2008, vol. 48, no. 7, pp. 1119-1132.

25. Tyatyushkin A.I. Multitechnique Technology for optimization of control systems (in Russian). Novosibirsk, Nauka, 2006.

26. Chernous’ko F.L. State estimation for dynamic systems, Florida, Boca Raton, CRCPress, 1994.

27. Chernous’ko F.L., Ananievski I.M., Reshmin S.A. Control of nonlinear dynamical systems: methods and applications. New York, Springer, 2008.

28. Chernous’ko F.L., Banuchuk N.V. Variational problems of mechanics and control (in Russian). Мoscow, Nauka, 1973.

29. Chernous’ko F.L., Melikyan A.A. Game problems of search and control (in Russian). Мoscow, Nauka, 1978.

30. Chernousko F.L., Lyubushin A.A.Method of successive approximations for optimalcontrol problems. Optimal Control Applications and Methods, 1982, vol. 3, no. 2, pp. 101-114.

31. Clarke F. Optimization and nonsmooth analysis. 2nd edn. Philadelphia, SIAM, 1990.

32. Hiriart-Urruty J.-B., Lemar´echal C. Convex analysis and minimization algorithms. Berlin, New York, Springer-Verlag, 1993.

33. Hiriart-Urruty J.-B. Generalized differentiability, duality and optimization for problem dealing with difference of convex functions. In: Ponstein J. (ed.) Convexityand Duality in Optimization, vol. 256. Berlin, Springer-Verlag, 1985, pp. 37-69.

34. Kelley H.J., Kopp R.E., Moyer H.G. Successive approximation techniques for trajectory optimization. Proc. of Symp. on Vehicle System Optimization, New York, 1961.

35. Mordukhovich B.S. Variational analysis and generalized differentiation. I: Basic Theory. II : Applications. Berlin, Springer, 2006.

36. Nocedal J., Wright St. Numerical optimization. 2nd edn. New York, Springer, 2006.

37. Strekalovsky A.S. Global optimality conditions for optimal control problems with functions of A.D. Alexandrov. Journal of Optimization Theory and Applications, 2013, vol. 159, no. 6, pp. 297-321.

38. Strekalovsky A.S. On global maximum of a convex terminal functional in optimal control problems. J. Global Optimization, 1995, vol. 7, no. 1, pp. 75-91.

39. Strekalovsky A.S., Orlov A.V., Malyshev A.V. On Computational Search for optimistic solution in bilevel problems. J. Global Optimization, 2010, vol. 48, no. 1, pp. 159-172.

40. Vasiliev O.V. Optimization methods. Atlanta, World Federation Publishers Company Inc., 1996.

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