Abstract
The problem of reconstructing unknown external actions in a linear stochastic differential equation is investigated on the basis of the approach of the theory of dynamic inversion. We consider the statement when the simultaneous reconstruction of disturbances in the deterministic and stochastic terms of the equation is performed with the use of discrete information on a number of realizations of a part of coordinates of the stochastic process. The problem is reduced to an inverse problem for systems of ordinary differential equations describing the mathematical expectation and covariance matrix of the original process. A finite-step software-oriented solution algorithm based on the method of auxiliary controlled models is proposed. We derive an estimate for its convergence rate with respect to the number of measured realizations.
Similar content being viewed by others
References
Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, London, 1995).
A. V. Kryazhimskii and Yu. S. Osipov, “Modelling of a control in a dynamic system,” Engrg. Cybernetics 21 (2), 38–47 (1984).
V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems (Izd. UrO RAN, Yekaterinburg, 2000; VSP, Utrecht–Boston, 2002).
A. V. Kryazhimskii and Yu. S. Osipov, “On a stable positional recovery of control from measurements of a part of coordinates,” in Some Problems of Control and Stability: Collection of Papers (UNTs AN SSSR, Sverdlovsk, 1989), pp. 33–47 [in Russian].
Yu. S. Osipov, A. V. Kryazhimskii, and V. I. Maksimov, “Some algorithms for the dynamic reconstruction of inputs,” Proc. Steklov Inst. Math. 275 (Suppl. 1), S86–S120 (2011).
M. S. Blizorukova and V. I. Maksimov, “On a reconstruction algorithm for the trajectory and control in a delay system,” Proc. Steklov Inst. Math. 280 (Suppl. 1), S66–S79 (2013).
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Wiley, New York, 1981).
Yu. S. Osipov and A. V. Kryazhimskii, “Positional modeling of a stochastic control in dynamical systems,” in Stochastic Optimization: Proceedings of the International Conference, Kiev, Ukraine, 1984 (Springer, Berlin, 1986), Ser. Lecture Notes in Control and Information Sciences 81, pp. 696–704.
V. L. Rozenberg, “Dynamic restoration of the unknown function in the linear stochastic differential equation,” Autom. Remote Control 68 (11), 1959–1969 (2007).
V. L. Rozenberg, “Reconstructing parameters of a linear stochastic equation under incomplete information,” in Systems Dynamics and Control Processes: Abstracts of International Conference SDCP-2014 Dedicated to the 90th Anniversary of Academician N.N. Krasovskii, Yekaterinburg, Russia, 2014, pp. 159–161.
A. N. Shiryaev, Probability, Statistics, and Random Processes (Izd. Mosk. Gos. Univ., Moscow, 1974) [in Russian].
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications (Springer, Berlin, 1985; Mir, Moscow, 2003).
V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, and A. F. Turbin, Handbook on Probability Theory and Mathematical Statistics (Nauka, Moscow, 1985) [in Russian].
V. S. Pugachev and I. N. Sinitsyn, Stochastic Differential Systems (Nauka, Moscow, 1990) [in Russian].
V. L. Rozenberg, “A control problem under incomplete information for a linear stochastic differential equation,” Ural Math. J. 1 (1), 68–82 (2015).
F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1988) [in Russian].
A. Yu. Vdovin, On the Problem of Perturbation Recovery in a Dynamic System, Candidate’s Dissertation in Physics and Mathematics (Sverdlovsk, 1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.L. Rozenberg, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 2.
Rights and permissions
About this article
Cite this article
Rozenberg, V.L. Reconstruction of external actions under incomplete information in a linear stochastic equation. Proc. Steklov Inst. Math. 296 (Suppl 1), 196–205 (2017). https://doi.org/10.1134/S0081543817020183
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543817020183