Geodesic
Optimal linear-quadratic regulator for a stochastic system under mutually inverse time preferences in the cost
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 1, pp. 38-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate a long-run behavior of a linear stochastic system. It is assumed that the quadratic cost includes a time-varying function and its multiplicative inverse. Such a specification reflects the fact that time preferences used by agents to assess different types of losses evolve in opposite directions. We consider the case when priority is set for the losses associated with state deviations. The optimal control law is derived with respect to extended long-run average cost criteria. We provide conditions for the existence of an alternative control strategy, which is also optimal and is based on a solution of an algebraic Riccati equation.
Keywords: linear regulator of a stochastic system, time preferences, long-run average. 
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E. S. Palamarchuk. Optimal linear-quadratic regulator for a stochastic system under mutually inverse time preferences in the cost. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 1, pp. 38-56. http://geodesic.mathdoc.fr/item/TVP_2023_68_1_a2/

[1] F. Man, H. Smith, “Design of linear regulators optimal for time-multiplied performance indices”, IEEE Trans. Automatic Control, 14:5 (1969), 527–529 | DOI | MR

[2] L. Karp, “Global warming and hyperbolic discounting”, J. Public Econ., 89:2-3 (2005), 261–282 | DOI

[3] E. S. Palamarchuk, “On optimal stochastic linear quadratic control with inversely proportional time-weighting in the cost”, Theory Probab. Appl., 67:1 (2022), 28–43 | DOI | DOI | MR | Zbl

[4] T. A. Belkina, E. S. Palamarchuk, “On stochastic optimality for a linear controller with attenuating disturbances”, Autom. Remote Control, 74:4 (2013), 628–641 | DOI | MR | Zbl

[5] H. Kwakernaak, R. Sivan, Linear optimal control systems, Wiley-Interscience [John Wiley Sons], New York–London–Sydney, xxv+575 pp. | MR | Zbl

[6] A. Ichikawa, H. Katayama, Linear time varying systems and sampled-data systems, Lect. Notes Control Inf. Sci., 265, Springer-Verlag, London, 2001, x+361 pp. | DOI | MR | Zbl

[7] L. Ya. Adrianova, Introduction to linear systems of differential equations, Transl. Math. Monogr., 146, Amer. Math. Soc., Providence, RI, 1995, x+204 pp. | DOI | MR | MR | Zbl

[8] K. W. Gruenberg, A. J. Weir, Linear geometry, Grad. Texts in Math., 49, 2nd ed., Springer, Berlin, 2013, 199 pp. | MR | Zbl

[9] R. W. Brockett, Finite dimensional linear systems, John Wiley and Sons, Inc., New York, 1970, 260 pp. | DOI | MR | Zbl

[10] M. H. A. Davis, Linear estimation and stochastic control, Chapman and Hall Math. Ser., Chapman and Hall, London; Halsted Press [John Wiley Sons], New York, 1977, xii+224 pp. | MR | MR | Zbl | Zbl

[11] E. S. Palamarchuk, “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 | DOI | DOI | MR | Zbl

[12] T. A. Belkina, Yu. M. Kabanov, E. L. Presman, “On a stochastic optimality of the feedback control in the LQG-problem”, Theory Probab. Appl., 48:4 (2004), 592–603 | DOI | DOI | MR | Zbl

[13] 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 | DOI | DOI | MR | Zbl

[14] W. M. Wonham, “On a matrix Riccati equation of stochastic control”, SIAM J. Control, 6:4 (1968), 681–697 | DOI | MR | Zbl

[15] R. Bhatia, Positive definite matrices, Princeton Ser. Appl. Math., Princeton Univ. Press, Princeton, NJ, 2007, x+254 pp. | MR | Zbl

[16] E. S. Palamarchuk, “Time invariance of optimal control in a stochastic linear controller design with dynamic scaling of coefficients”, J. Comput. Syst. Sci. Int., 60:2 (2021), 202–212 | DOI | DOI | MR | Zbl

[17] 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 | DOI | MR | Zbl

[18] C. S. Tapiero, Applied stochastic models and control for finance and insurance, Kluwer Acad. Publ., Boston, MA, 1998, xi+341 pp. | DOI | MR | Zbl

[19] Lixin Cui, Ruiqing Zhao, Wansheng Tang, “Principal-agent problem in a fuzzy environment”, IEEE Trans. Fuzzy Syst., 15:6 (2007), 1230–1237 | DOI