Geodesic
On optimal stochastic linear quadratic control with inversely proportional time-weighting in the cost
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 1, pp. 37-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider an optimal linear-quadratic control problem for a control system where the matrices corresponding to the state in the controlled process equation and the cost functional are absolutely integrable over an infinite time interval. The integral quadratic performance index includes two mutually inversely proportional time-weighting functions. It is shown that a well-known linear stable feedback law turns out to be optimal with respect to criteria from the class of the extended long-run averages. The results are applied to studying a control system under time-varying dynamic scaling of its parameters.
Keywords: stochastic linear-quadratic regulator, pathwise optimality, inversely proportional time-weighting of costs, absolutely integrable state matrix. 
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E. S. Palamarchuk. On optimal stochastic linear quadratic control with inversely proportional time-weighting in the cost. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 1, pp. 37-56. http://geodesic.mathdoc.fr/item/TVP_2022_67_1_a2/

[1] D. Chakraborty, “Persistence of a Brownian particle in a time-dependent potential”, Phys. Rev. E, 85:5 (2012), 051101 | DOI

[2] F. Lillo, R. N. Mantegna, “Drift-controlled anomalous diffusion: a solvable Gaussian model”, Phys. Rev. E, 61:5 (2000), R4675 | DOI

[3] Jae-Hyung Jeon, A. V. Chechkin, R. Metzler, “Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion”, Phys. Chem. Chem. Phys., 16:30 (2014), 15811–15817 | DOI

[4] S. C. Lim, S. V. Muniandy, “Self-similar Gaussian processes for modeling anomalous diffusion”, Phys. Rev. E, 66:2 (2002), 021114 | DOI

[5] Ruzong Fan, Bin Zhu, Yuedong Wang, “Stochastic dynamic models and Chebyshev splines”, Canad. J. Statist., 42:4 (2014), 610–634 | DOI | MR | Zbl

[6] M. Lefebvre, P. Whittle, “Survival optimization for a dynamic system”, Ann. Sci. Math. Québec, 12:1 (1988), 101–119 | MR | Zbl

[7] Jiatu Cai, M. Rosenbaum, P. Tankov, “Asymptotic lower bounds for optimal tracking: a linear programming approach”, Ann. Appl. Probab., 27:4 (2017), 2455–2514 | DOI | MR | Zbl

[8] 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

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

[10] 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 | DOI | DOI | MR | Zbl

[11] A. Locatelli, Optimal control. An introduction, Birkhäuser Verlag, Basel, 2001, x+294 pp. | MR | Zbl

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

[13] 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

[14] 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

[15] H. Cramér, M. R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, John Wiley Sons, Inc., New York–London–Sydney, 1967, xii+348 pp. | MR | Zbl | Zbl

[16] B. Øksendal, Stochastic differential equations. An introduction with applications, Universitext, 6th ed., Springer-Verlag, Berlin, 2003, xxiv+360 pp. | DOI | MR | Zbl

[17] E. S. Palamarchuk, “Optimal controller for a nonautonomous linear stochastic system with a two-sided cost functional”, Autom. Remote Control, 81:1 (2020), 53–63 | DOI | DOI | MR | Zbl

[18] E. S. Palamarchuk, “Optimization of the superstable linear stochastic system applied to the model with extremely impatient agents”, Autom. Remote Control, 79:3 (2018), 439–450 | DOI | MR | Zbl

[19] P. Dai Pra, G. B. Di Masi, B. Trivellato, “Pathwise optimality in stochastic control”, SIAM J. Control Optim., 39:5 (2000), 1540–1557 | DOI | MR | Zbl

[20] 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

[21] 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

[22] E. S. Palamarchuk, “On upper functions for integral quadratic functionals based on time-varying Ornstein–Uhlenbeck process”, Theory Probab. Appl., 65:1 (2020), 17–31 | DOI | DOI | MR | Zbl

[23] 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 | Zbl