Abstract
A problem of guaranteed closed-loop control under incomplete information is considered for a linear stochastic differential equation (SDE) from the viewpoint of the method of open-loop control packages worked out earlier for the guidance of a linear control system of ordinary differential equations (ODEs) to a convex target set. The problem consists in designing a deterministic open-loop control providing (irrespective of a realized initial state from a given finite set) prescribed properties of the solution (being a random process) at a terminal point in time. It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problem for the SDE is reduced to an equivalent problem for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. Solvability conditions for the problems in question are written.
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References
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].
N. N. Krasovskii, Control of a Dynamical System: Problem on the Minimum of Guaranteed Result (Nauka, Moscow, 1985) [in Russian].
N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems (Springer, New York, 1988).
A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems (Nauka, Moscow, 1981) [in Russian].
Yu. S. Osipov, “Control packages: An approach to solution of positional control problems with incomplete information,” Russ. Math. Surv. 61 (4), 611–661 (2006).
A. V. Kryazhimskiy and Yu. S. Osipov, “On the solvability of problems of guaranteeing control for partially observable linear dynamical systems,” Proc. Steklov Inst. Math. 277, 144–159 (2012).
A. V. Kryazhimskii and N. V. Strelkovskii, “An open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems,” Tr. Inst. Mat. Mekh. 20 (3), 132–147 (2014).
F. L. Chernous’ko and V. B. Kolmanovskii, Optimal Control under Random Perturbation (Nauka, Moscow, 1978) [in Russian].
V. L. Rozenberg, “Dynamic restoration of the unknown function in the linear stochastic differential equation,” Autom. Remote Control 68 (11), 1959–1969 (2007).
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].
N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1995), Vol. 2.
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Original Russian Text © V.L. Rozenberg, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 3.
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Rozenberg, V.L. A control problem under incomplete information for a linear stochastic differential equation. Proc. Steklov Inst. Math. 295 (Suppl 1), 145–155 (2016). https://doi.org/10.1134/S0081543816090157
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DOI: https://doi.org/10.1134/S0081543816090157