DOI: 10.17586/2226-1494-2019-19-1-59-66


A. A. Vedyakov, E. V. Milovanovich, V. Y. Tertychny-Dauri, G. V. Timofeeva

Read the full article  ';
Article in Russian

For citation:
Vedyakov A.A., Milovanovich E.V., Tertychny-Dauri V. Yu., Timofeeva G.V. Optimal control as conditional variational problem with variable right endpoint. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2019, vol. 19, no. 1, pp. 59–66 (in Russian). doi: 10.17586/2226-1494-2019-19-1-59-66


Subject of research. We consider dynamical system optimal control problem relating to a class of conditional variation problems with variable endpoints. Variational method is applied for research of the elastic mechanical system with controllable spring stiffness. Methods. The classical calculus of variational methods is used, which includes a variation of auxiliary functional and corresponding Euler equations. In solving a general conditional variation problem, the obtained differential system of equations in closed form is studied for design of an optimal control system for the initial dynamic object with a given quality functional. Main results. Results of unconstrained optimization are generalized to the case with additional differential (nonholonomic) constraints. Transversality condition in the variational problem is formulated in terms of local programming. An optimal control algorithm is constructed in the elastic oscillator model example, and the value of finite transition period is found. Practical relevance. The proposed approach can be used in optimal control design for dynamical systems. This optimization scheme can be also applied to controlled dynamic systems, when operation time is not fixed in advance.

Keywords: variable endpoint, performance functional, optimal control, functional variation, transversality condition

1. Gel'fand I.M., Fomin S.V. Calculus of Variations. Moscow, Fizmatgiz Publ., 1961, 228 p. (in Russian)
2. El'sgol'ts L.E. Differential Equations and Calculus of Variations. Moscow, Nauka Publ., 1969, 424 p. (in Russian)
3. Yang L. Lectures on Calculus of Variations and Optimal Control Theory. Moscow, Mir Publ., 1974, 488 p. (in Russian)
4. Tertychny-Dauri V.Yu. Galamech. Vol. 4. Optimum Mechanics. Moscow, Fizmatlit Publ., 2008, 608 p. (in Russian)
5. Degtyarev G.L. Synthesis of optimal control in systems with distributed parameters using Lyapunov functions. In Direct Method in Stability Theory and its Applications. Novosibirsk, Nauka Publ., 1981, pp. 75–83. (in Russian)
6. Matveev A.S. Variational analysis in optimization problems of systems with distributed parameters and set vector functions. Siberian Mathematical Journal, 1990, vol. 31, no. 6, pp. 127–141. (in Russian)
7. Panchenkov A.N. Extreme problems of motion control with local functionals. In Problems of Motion Stability, Analytical Mechanics and Motion Control. Novosibirsk, Nauka Publ., 1979, pp. 190–202. (in Russian)
8. Tertychny-Dauri V.Yu. Galamech. Vol. 1. Adaptive Mechanics. Moscow, Fizmatlit Publ., 2008, 544 p. (in Russian)
9. Tertychny-Dauri V.Yu. Solution of variational dynamic problems under parametric uncertainty. Problems of Information Transmission, 2005, vol. 41, no. 1, pp. 45–58. (in Russian)
10. Tertychny-Dauri V.Yu. Variational dynamical problems with parameters and their adaptive interpretation. Automation and Remote Control, 2005, vol. 66, no. 9, pp. 1465–1477. doi: 10.1007/s10513-005-0185-z
11. Tertychnyi-Dauri V.Yu. A conditional optimal control problem and its adaptive solution method. Automation and Remote Control, 2006, vol. 67, no. 3, pp. 393–404.
12. Blatt J.M., Lyness J.N. The practical use of variation principles in non-linear mechanics. Journal of the Australian Mathematical Society, 1962, vol. 2, no. 3, pp. 357. doi: 10.1017/s144678870002694x (in Russian)
13. Mayne D.Q., Polak E. First order strong variation algorithm for optimal control. Journal of Optimization Theory and Applications, 1975, vol. 16, no. 3/4, pp. 277–301. doi: 10.1007/bf01262938
14. Leitmann G. The Calculus of Variations and Optimal Control. NY, Plenum Press, 1986, 312 p. doi: 10.1007/978-1-4899-0333-4
15. Lerner A.Ya., Rozenman E.A. Optimal Control. Moscow, Energiya Publ., 1970, 360 p. (in Russian)

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Copyright 2001-2020 ©
Scientific and Technical Journal
of Information Technologies, Mechanics and Optics.
All rights reserved.