Informatics and Applications
2019, Volume 13, Issue 3, pp 41-49
STOCHASTIC DIFFERENTIAL SYSTEM OUTPUT CONTROL BY THE QUADRATIC CRITERION. III. OPTIMAL CONTROL PROPERTIES ANALYSIS
- A. V. Bosov
- A. I. Stefanovich
Abstract
The investigation of the optimal control problem for the Ito diffusion process and linear controlled output with a quadratic quality criterion is continued. The properties of the optimal solution defined by the Bellman function of the form Vt(y, z) = atz2 + @t(y)z + Yt(y), whose coefficients @t(y) and Yt(y) are described by linear parabolic equations, are studied. For these coefficients, alternative equivalent descriptions are defined in the form of stochastic differential equations and a theoretical-to-probabilistic representation of their solutions, known as the Kolmogorov equation. It is shown that the obtained differential representation is equivalent to the Feynman-Kac integral formula. In the future, the obtained description of the coefficients and, as a result, the solutions of the original control problem can be used to implement an alternative numerical method for calculating them as a result of computer simulation of the solution of a stochastic differential equation.
[+] References (11)
- Bosov, A.V., and A.I. Stefanovich. 2018. Upravlenie vykhodom stokhasticheskoy differentsial'noy siste my po kvadratichnomu kriteriyu [Stochastic differential system output control by the quadratic criterion.
I. Dynamic programming optimal solution]. Informati- ka i ee Primeneniya - Inform. Appl. 12(3):99-106. doi: 10.14357/19922264180314.
- Bosov, A. V., and A. I. Stefanovich. 2019. Upravlenie vykhodom stokhasticheskoy differentsial'noy sistemy po kvadratichnomu kriteriyu [Stochastic differential system output control by the quadratic criterion. II. Dynamic programming equations numerical solution]. Informatika i ee Primeneniya - Inform. Appl. 13(1):9-15. doi: 10.14357/19922264190102.
- Gihman, 1.1., and A.V. Skorohod. 2012. The theory of stochastic processes III. New York, NY: Springer-Verlag. 388 p.
- Bismut, J.-M. 1973. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44(2):384- 404. doi: 10.1016/0022-247X(73)90066-8.
- Pardoux, E., and S. G. Peng. 1990. Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1):55-61. doi: 10.1016/0167-6911(90)90082-6.
- Delong, L. 2013. Backward stochastic differential equations with jumps and their actuarial and financial applications. New York, NY: Springer-Verlag. 288 p. doi: 10.1007/9781-4471-5331-3.
- Touzi, N. 2013. Optimal stochastic control, stochastic target problems, and backward SDEs. New York, NY: Springer- Verlag. 214 p. doi: 10.1007/978-1-4614-4286-8.
- Fahim, A., N. Touzi, andX. Warin. 2011. A probabilistic numerical method for fully nonlinear parabolic pdes. Ann. Appl. Probab. 21(4):1322-1364. doi: 10.1214/10-AAP723.
- 0ksendal, B. 2003. Stochastic differential equations. An introduction with applications. New York, NY: Springer- Verlag. 379 p. doi: 10.1007/978-3-642-14394-6.
- Fleming, W. H., and R. W. Rishel. 1975. Deterministic and stochastic optimal control. New York, NY: Springer-Verlag. 222 p.
- Bosov, A. V. 2016. Discrete stochastic system linear output control with respect to a quadratic criterion. J. Comput. Syst. Sc. Int. 55(3):349-364.
[+] About this article
Title
STOCHASTIC DIFFERENTIAL SYSTEM OUTPUT CONTROL BY THE QUADRATIC CRITERION. III. OPTIMAL CONTROL PROPERTIES ANALYSIS
Journal
Informatics and Applications
2019, Volume 13, Issue 3, pp 41-49
Cover Date
2019-09-30
DOI
10.14357/19922264190307
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
stochastic differential equation; optimal control; Bellman function; linear differential equations of parabolic type; Kolmogorov equation; Feynman-Kac formula
Authors
A. V. Bosov  and A. I. Stefanovich
Author Affiliations
Institute of Informatics Problems, Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
|