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discounted optimality; non-constant discount factor; time-varying Markov decision processes
In this paper we are concerned with a class of time-varying discounted Markov decision models $\mathcal{M}_n$ with unbounded costs $c_n$ and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation $x_{n+1}=G_n(x_n,a_n,\xi_n), n=0,1,\ldots$, with state-action dependent discount factors of the form $\alpha_n(x_n,a_n)$, where $a_n$ and $\xi_n$ are the control and the random disturbance at time $n$, respectively. Assuming that the sequences of functions $\lbrace\alpha_n\rbrace$,$\lbrace c_n\rbrace$ and $\lbrace G_n\rbrace$ converge, in certain sense, to $\alpha_\infty$, $c_\infty$ and $G_\infty$, our objective is to introduce a suitable control model for this class of systems and then, to show the existence of optimal policies for the limit system $\mathcal{M}_\infty$ corresponding to $\alpha_\infty$, $c_\infty$ and $G_\infty$. Finally, we illustrate our results and their applicability in a class of semi-Markov control models.
[1] Bastin, G., Dochain, D.: On-line Estimation and Adaptive Control of Bioreactors. Elsevier, Amsterdam 2014.
[2] Bertsekas, D. P.: Approximate policy iteration: a survey and some new methods. J. Control Theory Appl. 9 (2011), 310-335. DOI 10.1007/s11768-011-1005-3 | MR 2833999
[3] Dynkin, E. B., Yushkevich, A. A.: Controlled Markov Processes. Springer-Verlag, New York 1979. DOI 10.1007/978-1-4615-6746-2 | MR 0554083
[4] González-Hernández, J., López-Martínez, R. R., Minjárez-Sosa, J. A.: Approximation, estimation and control of stochastic systems under a randomized discounted cost criterion. Kybernetika 45 (2009), 737-754. MR 2599109
[5] Gordienko, E. I., Minjárez-Sosa, J. A.: Adaptive control for discrete-time Markov processes with unbounded costs: discounted criterion. Kybernetika 34 (1998), 217-234. MR 1621512
[6] Hernández-Lerma, O., Lasseerre, J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer, New York 1996. DOI 10.1007/978-1-4612-0729-0 | MR 1363487
[7] Hernández-Lerma, \noindent O., Lasserre, J. B.: Further Topics on Discrete-time Markov Control Processes. Springer-Verlag, New York 1999. DOI 10.1007/978-1-4612-0561-6 | MR 1697198
[8] Hernández-Lerma, O., Hilgert, N.: Limiting optimal discounted-cost control of a class of time-varying stochastic systems. Syst. Control Lett. 40 (2000), 1, 37-42. DOI 10.1016/s0167-6911(99)00121-8 | MR 1829073
[9] Hilgert, N., Minjárez-Sosa, J. A.: Adaptive policies for time-varying stochastic systems under discounted criterion. Math. Meth. Oper. Res. 54 (2001), 3, 491-505. DOI 10.1007/s001860100170 | MR 1890916
[10] Hilgert, N., Minjárez-Sosa, J. A.: Adaptive control of stochastic systems with unknown disturbance distribution: discounted criteria. Math. Meth. Oper. Res. 63 (2006), 443-460. DOI 10.1007/s00186-005-0024-6 | MR 2264761
[11] Hilgert, N., Senoussi, R., Vila, J. P.: Nonparametric estimation of time-varying autoregressive nonlinear processes. C. R. Acad. Sci. Paris Série 1 1996), 232, 1085-1090. DOI 10.1109/.2001.980647 | MR 1423225
[12] Lewis, M. E., Paul, A.: Uniform turnpike theorems for finite Markov decision processes. Math. Oper. Res.
[13] Luque-Vásquez, F., Minjárez-Sosa, J. A.: Semi-Markov control processes with unknown holding times distribution under a discounted criterion. Math. Meth. Oper. Res. 61 (2005), 455-468. DOI 10.1007/s001860400406 | MR 2225824
[14] Luque-Vásquez, F., Minjárez-Sosa, J. A., Rosas-Rosas, L. C.: Semi-Markov control processes with partially known holding times distribution: Discounted and average criteria. Acta Appl. Math. 114 (2011), 3, 135-156. DOI 10.1007/s10440-011-9605-y | MR 2794078
[15] Luque-Vásquez, F., Minjárez-Sosa, J. A., Rosas-Rosas, L. C.: Semi-Markov control processes with unknown holding times distribution under an average criterion cost. Appl. Math. Optim. Theory Appl. 61 (2010), 3, 317-336. DOI 10.1007/s00245-009-9086-9 | MR 2609593
[16] Minjárez-Sosa, J. A.: Markov control models with unknown random state-action-dependent discount factors. TOP 23 (2015), 743-772. DOI 10.1007/s11750-015-0360-5 | MR 3407674
[17] Minjárez-Sosa, J. A.: Approximation and estimation in Markov control processes under discounted criterion. Kybernetika 40 (2004), 6, 681-690. MR 2120390
[18] Powell, W. B.: Approximate Dynamic Programming. Solving the Curse of Dimensionality. John Wiley and Sons Inc, 2007. DOI 10.1002/9780470182963 | MR 2839330
[19] Puterman, M. L.: Markov Decision Processes. Discrete Stochastic Dynamic Programming. John Wiley and Sons 1994. DOI 10.1002/9780470316887 | MR 1270015
[20] Rieder, U.: Measurable selection theorems for optimization problems. Manuscripta Math. 24 (1978), 115-131. DOI 10.1007/bf01168566 | MR 0493590 | Zbl 0385.28005
[21] Robles-Alcaráz, M. T., Vega-Amaya, O., Minjárez-Sosa, J. A.: Estimate and approximate policy iteration algorithm for discounted Markov decision models with bounded costs and Borel spaces. Risk Decision Analysis 6 (2017), 2, 79-95. DOI 10.3233/rda-160116
[22] Royden, H. L.: Real Analysis. Prentice Hall 1968. MR 0928805 | Zbl 1191.26002
[23] Schäl, M.: Conditions for optimality and for the limit on n-stage optimal policies to be optimal. Z. Wahrs. Verw. Gerb. 32 (1975), 179-196. DOI 10.1007/bf00532612 | MR 0378841
[24] Shapiro, J. F.: Turnpike planning horizon for a markovian decision model. Magnament Sci. 14 (1968), 292-300. DOI 10.1287/mnsc.14.5.292
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