ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

The Technique for Accurate and Approximate Synthesis of Optimal Continuous-time Stochastic Control Systems

Author(s):

Konstantin Rybakov

Moscow Aviation Institute (National Research University)

rkoffice@mail.ru

Abstract:

Based on the extension principle the sufficient optimality conditions are formulated to estimate the accuracy of the approximate solution for the optimal control problem with respect to the optimality criterion value for continuous-time stochastic systems. The technique for solving optimal control problems using such conditions is illustrated by the example of finding program control and feedback control for a linear stochastic system.

Keywords

References:

  1. Averbukh V. I., Smolyanov O. G. Theory of differentiation in linear topological spaces. Russian Mathematical Surveys, 1967, vol. 22, no. 6, pp. 201-258
  2. Averbukh V. I., Smolyanov O. G. The various definitions of the derivative in linear topological spaces. Russian Mathematical Surveys, 1968, vol. 24, no. 4, pp. 67-113
  3. Averina T. A. Chislennye metody. Algoritmy modelirovaniya sistem so sluchainoi strukturoi [Numerical Methods. Algorithms for Modeling Systems with Random Structure]. Moscow, Yurait Publ., 2018
  4. Bellman R. Dynamic programming. Princeton University Press, 1957
  5. Bogachev V. I., Krylov N. V., Rockner M., Shaposhnikov S. V. Fokker-Planck-Kolmogorov Equations. AMS, 2016
  6. Gurman V. I. Printsip rasshireniya v zadachakh upravleniya [Principle of Extension in Problems of Control]. Moscow, Nauka Publ., 1997
  7. Krotov V. F. [Methods of solution of variational problems on the basis of sufficient conditions of absolute minimum]. Avtomatika i telemekhanika, 1962, vol. 23, no. 12, pp. 1571-1583; 1963, vol. 24, no. 5, pp. 581-598. (In Russ. )
  8. Krotov V. F., Gurman V. I. Metody i zadachi optimal'nogo upravleniya [Methods and Problems of Optimal Control]. Moscow, Nauka Publ., 1973
  9. Krotov V. F., Lagosha B. A., Lobanov S. M., Danilina N. I., Sergeev S. I. Osnovy teorii optimal'nogo upravleniya [Fundamentals of Optimal Control Theory]. Moscow, Vysshaya Shkola Publ., 1990
  10. Krylov N. V. Controlled Diffusion Processes. Springer, 1980
  11. Krylov N. V., Rozovskii B. L. On conditional distributions of diffusion processes. Mathematics of the USSR - Izvestiya, 1978, vol. 12, no. 2, pp. 336-356
  12. Kuznetsov D. F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1. 5 and 2. 0 orders of strong convergence. Automation and Remote Control, 2018, vol. 79, no. 7, pp. 1240-1254
  13. Kuznetsov D. F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 2. 5 order of strong convergence. Automation and Remote Control, 2019, vol. 80, no. 5, pp. 867-881
  14. Levakov A. A. Metody integrirovaniya stokhasticheskikh differentsial'nykh uravnenii [Methods for Solving Stochastic Differential Equations]. Minsk, BSU Publ., 2010
  15. Ø ksendal B. Stochastic Differential Equations. An Introduction with Applications. Springer-Verlag, 2000
  16. Panteleev A. V. Sufficient conditions for optimality for continuous stochastic control systems on the basis of an incomplete state vector. Soviet Mathematics, 1990, vol. 34, no. 11, pp. 62-75
  17. Panteleev A. V., Letova T. A. Metody optimizatsii v primerakh i zadachakh [Optimization Methods in Examples and Tasks]. Saint Petersburg, Lan Publ., 2008
  18. Panteleev A. V., Rybakov K. A. [Design of optimal nonlinear stochastic systems by spectral method]. Informatika i ee primeneniya, 2011, vol. 5, no. 2, pp. 69-81. (In Russ. )
  19. Panteleev A. V., Rybakov K. A. Metody i algoritmy sinteza optimal'nykh stokhasticheskikh sistem upravleniya pri nepolnoi informatsii [Methods and Algorithms for Synthesis of Optimal Stochastic Control Systems with Incomplete Information]. Moscow, MAI Publ., 2012
  20. Panteleev A. V., Rybakov K. A. Continuous optimal stochastic control systems with incomplete feedback: approximate synthesis. Automation and Remote Control, 2018, vol. 79, no. 1, pp. 103-116
  21. Panteleev A. V., Semenov V. V. Sintez optimal'nykh sistem upravleniya pri nepolnoi informatsii [Design of Optimal Control Systems with Incomplete Information]. Moscow, MAI Publ., 1992
  22. Panteleev A. V., Yakimova A. S., Rybakov K. A. Obyknovennye differentsial'nye uravneniya. Praktikum [Ordinary Differential Equations. Practical Work]. Moscow, Infra-M Publ., 2016
  23. Plotnikov M. Y., Khrustalev M. M. Dynamic programming method in control problem for diffusion processes with possible trajectory cut-off with incomplete state information. Journal of Computer and Systems Sciences International, 2005, vol. 44, no. 5, pp. 701-705
  24. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. The Mathematical Theory of Optimal Processes. John Wiley & Sons, 1962
  25. Rumyantsev D. S., Khrustalev M. M. Optimal control of quasi-linear systems of the diffusion type under incomplete information on the state. Journal of Computer and Systems Sciences International, 2006, vol. 45, no. 5, pp. 718-726
  26. Rumyantsev D. S., Khrustalev M. M., Tsarkov K. A. An algorithm for synthesis of the suboptimal control law for quasi-linear stochastic dynamical systems. Journal of Computer and Systems Sciences International, 2014, vol. 53, no. 1, pp. 71-83
  27. Rybakov K. A. [Sufficient optimality conditions in the problem of centralized control for switching diffusions]. Vestnik MAI, 2008, vol. 15, no 2, pp. 123-131. (In Russ. )
  28. Rybakov K. A. [Sufficient optimality conditions in the control problem for jump-diffusion systems]. XII Vserossiiskoe soveshchanie po problemam upravleniya (VSPU-2014) [Proc. 12th All-Russian Conference on Control Problems], Moscow, 2014, pp. 734-744. (In Russ. )
  29. Rybakov K. A. [Optimal control of stochastic systems with random sampling period]. Trudy MFTI, 2015, vol. 7, no. 1 (25), pp. 145-165. (In Russ. )
  30. Rybakov K. A., Sotskova I. L. An optimal control for random-structure nonlinear systems under incomplete state vector information. Automation and Remote Control, 2006, vol. 67, no. 7, pp. 1070-1081
  31. Savastyuk S. V., Khrustalev M. M. Optimization of stochastic diffusion systems with constraints on the control-observation process. Automation and Remote Control, 1991, vol. 52, no. 7, pp. 958-963; no. 8, pp. 1109-1114
  32. Fleming W. H., Rishel R. W. Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975
  33. Khrustalev M. M. [Nash equilibrium conditions in stochastic differential games under incomplete information about the state]. Izv. RAN. Teoriya i sistemy upravleniya. 1995, no. 6, pp. 194-208; 1996, no. 1, pp. 72-79. (In Russ. )
  34. Khrustalev M. M., Khalina A. S. Optimal controller synthesis for linear stochastic systems with incomplete information regarding the state. Necessary conditions and numerical methods. Automation and Remote Control, 2014, vol. 75, no. 11, pp. 1948-1963
  35. Elliott R. J. Stochastic Calculus and Applications. Springer-Verlag, 1982
  36. Burrage K., Burrage P. M., Tian T. Numerical methods for strong solutions of stochastic differential equations: an overview. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004, vol. 460, no. 2041, pp. 373-402
  37. Fleming W. H. Optimal control of partially observable diffusions. SIAM Journal on Control, 1968, vol. 6, no. 2, pp. 194-214
  38. Fleming W. H., Soner H. M. Controlled Markov Processes and Viscosity Solutions. Springer, 2006
  39. Kloeden P. E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer, 1995
  40. Stroock D. W., Varadhan S. R. S. Multidimensional Diffusion Processes. Springer, 2006

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