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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 4, Pages 661–675 (Mi tvp250)

On a stochastic optimality of the feedback control in the LQG-problem

T. A. Belkinaa, Yu. M. Kabanovb, E. L. Presmana

a Central Economics and Mathematics Institute, RAS
b Laboratoire de Mathématiques, Université de Franche-Comté

Abstract: We show that the optimal feedback control $\widehat u$ in the standard nonhomogeneous LQG-problem with infinite horizon has the following property. There is a constant $b_*$ such that, whatever $b> b_*$ is, the deficiency process of optimal control with respect to any possible control $u$, i.e., the difference $J_T(\widehat u\hspace*{0.2pt})- J_T(u)$ between the optimal cost process $J_T(\widehat u\hspace*{0.2pt})$ and the cost process corresponding to control $u$, is majorated at infinity by a deterministic function $b\log T$. In other words, $b\log T$ is an upper function for any deficiency process. This result, combined with an example of an LQG-regulator where, for certain $b>0$, the function $b\log T$ is not an upper function for certain deficiency processes, gives an answer to the long-standing open problem about the best possible rate function for sensitive probabilistic criteria. Our setting covers the optimal tracking problem.

Keywords: linear-quadratic regulator, optimality almost surely, observability, controllability, Riccati equation, martingale law of large numbers, upper functions, Ornstein–Uhlenbeck process.

DOI: https://doi.org/10.4213/tvp250

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English version:
Theory of Probability and its Applications, 2004, 48:4, 592–603

Bibliographic databases:

Citation: T. A. Belkina, Yu. M. Kabanov, E. L. Presman, “On a stochastic optimality of the feedback control in the LQG-problem”, Teor. Veroyatnost. i Primenen., 48:4 (2003), 661–675; Theory Probab. Appl., 48:4 (2004), 592–603

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp250
• https://doi.org/10.4213/tvp250
• http://mi.mathnet.ru/eng/tvp/v48/i4/p661

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. T. A. Belkina, V. I. Rotar', “On optimality in probability and almost surely for processes with a communication property. I. The discrete time case”, Theory Probab. Appl., 50:1 (2006), 16–33
2. T. A. Belkina, M. S. Levochkina, “Stochastic optimality in the problem of a linear controller perturbed by a sequence of dependent random variables”, Discrete Math. Appl., 16:2 (2006), 135–153
3. T. A. Belkina, E. S. Palamarchuk, “On stochastic optimality for a linear controller with attenuating disturbances”, Autom. Remote Control, 74:4 (2013), 628–641
4. E. S. Palamarchuk, “Asymptotic behavior of the solution to a linear stochastic differential equation and almost sure optimality for a controlled stochastic process”, Comput. Math. Math. Phys., 54:1 (2014), 83–96
5. E. S. Palamarchuk, “Stabilization of linear stochastic systems with a discount: modeling and estimation of the long-term effects from the application of optimal control strategies”, Math. Models Comput. Simul., 7:4 (2015), 381–388
6. Palamarchuk E., “On Infinite Time Linear-Quadratic Gaussian Control of Inhomogeneous Systems”, 2016 European Control Conference (Ecc), IEEE, 2016, 2477–2482
7. E. S. Palamarchuk, “Analysis of the asymptotic behavior of the solution to a linear stochastic differential equation with subexponentially stable matrix and its application to a control problem”, Theory Probab. Appl., 62:4 (2018), 522–533
8. Palamarchuk E.S., “On the Generalization of Logarithmic Upper Function For Solution of a Linear Stochastic Differential Equation With a Nonexponentially Stable Matrix”, Differ. Equ., 54:2 (2018), 193–200
9. E. S. Palamarchuk, “O verkhnikh funktsiyakh dlya anomalnykh diffuzii, modeliruemykh protsessom Ornshteina–Ulenbeka s peremennymi koeffitsientami”, Teoriya veroyatn. i ee primen., 64:2 (2019), 258–282
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