UDC 519.21
AVERAGING IN THE CONTROL PROBLEM FOR THE DIFFUISON
TRANSFER PROCESS WITH SEMI-MARKOV SWITCHES
Abstract. The limit generators were constructed for the system of stochastic differential equations with semi-Markov switches and diffusion perturbance under the conditions of the existence of a single equilibrium point of the performance criterion. Assuming the existence of a single control on each interval, we solve a two-level problem. The article examines how the behavior of the limit process depends on the pre-limit normalization of the stochastic system in the ergodic semi-Markov environment.
Keywords: random evolution, stochastic optimization, semi-Markov switches.
full text
REFERENCES
- Anisimov V.V. Switching Processes in Gueuing Models. London: Wiley-ISTE, 2008. 352 p.
- Korolyuk V.S., Korolyuk V.V. Stochastic Models of Systems. Kluwer, Dordrecht, 1999.
- Korolyuk V.S., Limnios N. Stochastic Systems in Merging Phase Space. World Scientific, 2005.
- Chabanyuk Y.M., Nikitin A.V., Khimka U.T. Asymptotic properties of the impulse perturbation process with control function under Levy approximation conditions. Mathematychni Studii. 2019. Vol. 52, N 1. P. 96–104. doi.org/10.30970/ms.52.1.96-104.
- Nikitin A.V., Khimka U.T. Asymptotics of normalized control with Markov switchings. Ukrainian Mathematical Journal. 2017. Vol. 68, N 8. P. 1252–1262. doi.org/10.1007/ s11253-017-1291-0 .
- Nikitin A.V. Asymptotic properties of a stochastic diffusion transfer process with an equilibrium point of a quality criterion. Cybernetics and Systems Analysis. 2015. Vol. 51, N 4. P. 650–656. doi.org/10.1007/s10559-015-9756-3 .
- Nikitin A.V. Asymptotic dissipativity of stochastic processes with impulsive perturbation in the Levy approximation scheme. Journal of Automation and Information Sciences. 2018. Vol. 50, Iss. 4. P. 54–63. doi.org/10.1615/jautomatinfscien.v50.i4.50.
- Samoilenko I.V., Chabanyuk Y.M., Nikitin A.V. Asymptotic dissipativity of random processes with impulse perturbation in the Poisson approximation scheme. Cybernetics and Systems Analysis. 2018. Vol. 54, N 2. P. 205–211. doi.org/10.1007/s10559-018-0021-4.
- Khimka U.T., Chabanyuk Ya.M. A difference stochastic optimization procedure with impulse perturbation. Cybernetics and Systems Analysis. 2013. Vol. 49, N 5. P. 768–773. doi.org/10.1007/s10559-013-9564-6.
- Chabanyuk Y., Nikitin A., Khimka U. Asymptotic Analyses for Complex Evolutionary Systems with Markov and Semi-Markov Switching Using Approximation Schemes. Wiley Online Library, London, 2020. 240 р. doi.org/10.1002/9781119779759.
- Nevelson M.B., Khasminsky R.Z. Stochastic approximation and recurrent estimation [in Russian]. Moscow: Nauka, 1972. 304 c.
- Jacod J., Shiryaev A.N. Limit Theorems for Stochastic Processes. Berlin; Heidelberg: Springer Link, 2003. P. 664. doi.org/10.1007/978-3-662-05265-5.
- Papanicolaou G., Stroock D., Varadhan S.R.S. Martingale approach to some limit theorems. Duke turbulence conference (Durham, NC, April 23–25, 1976). Duke University Mathematics Series III, New York: Duke University, 1977. 120 p.
