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The maximum principle for the nonlinear stochastic optimal control problem of switching systems

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Abstract

The aim of this paper is to present a stochastic maximum principle for an optimal control problem of switching systems. It presents necessary conditions of optimality in the form of a maximum principle for stochastic switching systems, in which the dynamic of the constituent processes takes the form of stochastic differential equations. The restrictions on transitions for the system are described through equality constraints.

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References

  1. Afanasyev V.N., Kolmanovsky V.B., Nosov V.P.: Mathematical Theory for Synthesizing Control Systems. Higher school, Moscow (2003)

    Google Scholar 

  2. Agayeva, C., Abushov, G.: Necessary condition of optimality for stochastic control systems with variable structure. In: Proceedings of the 20th International Conference on Continuous, Optimization and Knowledge-Based Technologies, pp. 77–81 (2008)

  3. Agayeva, C., Abushov, Q.: The maximum principle for some stochastic control problem of switching systems. In: Kasmbeyli, R., Diner, C., zpeynirci, S., Sakalauskas, L. (eds.) Selected Papers 24th Mini EURO Conference on Continuous Optimization and Information-Based Technologies in the Financial Sector, pp. 100–105. Izmir University of Economics, Izmir, Turkey, Vilnius Technika (2010)

  4. Aghayeva C.A., Abushov Q.U.: The maximum principle for some nonlinear stochastic control system with variable structure. Theory Stoch. Process. 36, 1–11 (2010)

    Google Scholar 

  5. Antsaklis P.J., Nerode A.: Hybrid control systems: an introductory discussion to the special issue. IEEE Trans. Autom. Control 43(4), 457–460 (1998)

    Article  Google Scholar 

  6. Arkin V.I., Saksonov M.T.: The necessary conditions of optimality in control problems of stochastic differential equations. DAN SSSR 244(4), 11–16 (1979)

    Google Scholar 

  7. Arnold L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)

    Google Scholar 

  8. Bengea S.C., Raymond A.C.: Optimal control of switching systems. Automatica 41, 11–27 (2005)

    Google Scholar 

  9. Bensoussan, A.: Lectures on stochastic control in nonlinear filtering and stochastic control. In: Springer Lectures Notes in Mathematics. Springer, Berlin (1982)

  10. Bensoussan A.: Stochastic maximum principle for distributed parameter systems. J. Frankl. Inst. 315(5/6), 387–406 (1983)

    Article  Google Scholar 

  11. Bismut J.M.: Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control 14, 419–444 (1976)

    Article  Google Scholar 

  12. Bismut J.M.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20(1), 62–78 (1978)

    Article  Google Scholar 

  13. Borkar V.: Controlled diffusion processes. Probab. Surv. 2(4), 213–244 (2005)

    Google Scholar 

  14. Boukas E.: Stochastic Switching Systems: Analysis and Design. Birkhauser, Canada (2006)

    Google Scholar 

  15. Branicky, M., Mitter, S.: Algorithms for optimal hybrid control. In: Proceedings of the 34th IEEE Conference on Decision and Control, vol. 3, pp. 2661–2666 (1995)

  16. Burachik R.S., Gasimov R.N., Ismayilova N.A., Kaya C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Glob. Optim. 34, 55–78 (2006)

    Article  Google Scholar 

  17. Capuzzo D.I., Evans L.C.: Optimal Switching for ordinary differential equations. SIAM J. Control Optim. 22, 143–161 (1984)

    Article  Google Scholar 

  18. Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  Google Scholar 

  19. Fleming W.H., Rishel R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)

    Book  Google Scholar 

  20. Gao R., Liu X., Yang J.: On optimal control problems of a class of impulsive switching systems with terminal states constraints. Nonlinear Anal. Theory Methods Appl. 73, 1940–1951 (2010)

    Article  Google Scholar 

  21. Gikhman I.I., Skorokhod A.V.: Controlled Stochastic Processes. Springer, New York (1979)

    Book  Google Scholar 

  22. Hante F., Leugering G., Seidman T.: Modeling and analysis of modal switching in networked transport systems. Appl. Math. Optim. 59, 275–292 (2009)

    Article  Google Scholar 

  23. Haussman U.G.: General necessary conditions for optimal control of stochastic systems, stochastic systems: modeling, identification and optimization, II. Math. Program. Stud. 6, 30–48 (1976)

    Article  Google Scholar 

  24. Kushner H.J.: Necessary conditions for continuous parameter stochastic optimization problems. SIAM 10, 550–565 (1976)

    Google Scholar 

  25. Mahmudov N.I., Bashirov A.E.: First Order and Second Order Necessary Conditions of Optimality for Stochastic Systems. Statistics and Control of Random Processes, pp. 283–295. Nauka, Moscow (1995)

    Google Scholar 

  26. Makhmudov N.I.: General Necessary Optimality Conditions for Stochastic Systems with Controllable Diffusion. Statistics and Control of Random Processes, pp. 135–138. Nauka, Moscow (1989)

    Google Scholar 

  27. Shige P.: A general stochastic maximum principle for optimal control problem. SIAM J. Control. Optim. 28, 966–979 (1990)

    Article  Google Scholar 

  28. Picolli, B.: Hybrid systems and optimal control. In: Proceedings of the 37st IEEE Conference on Decision and Control, pp. 13–18 (1998)

  29. Seidmann, T.I.: Optimal control for switching systems. In: Proceedings of the 21st Annual Conference on Informations Science and Systems, pp. 485–489 (1987)

  30. Witsenhausen H.S.: A class of hybrid state continuous-time dynamic systems. IEEE Trans. Autom. Control 11(2), 161–167 (1966)

    Article  Google Scholar 

  31. Xu, X., Antsaklis, P.J.: Results and perspectives on computational methods for optimal control of switched systems. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2623, pp. 540–556 (2003)

  32. Yong J., Zhou X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    Google Scholar 

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Correspondence to Charkaz Aghayeva.

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Aghayeva, C., Abushov, Q. The maximum principle for the nonlinear stochastic optimal control problem of switching systems. J Glob Optim 56, 341–352 (2013). https://doi.org/10.1007/s10898-011-9825-8

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