Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time
Pages: 3821  3838,
Issue 10,
December
2017
doi:10.3934/dcdsb.2017192 Abstract
References
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Vladimir Gaitsgory  Department of Mathematics, Macquarie University, Macquarie Park, NSW 2113, Australia (email)
Alex Parkinson  Department of Mathematics, Macquarie University, Macquarie Park, NSW 2113, Australia (email)
Ilya Shvartsman  Department of Mathematics and Computer Science, Penn State Harrisburg, Middletown, PA 17057, United States (email)
1 
D. Adelman and D. Klabjan, Duality and existence of optimal policies in generalized joint replenishment, Mathematics of Operations Research, 30 (2005), 2850. 

2 
R. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972. 

3 
J.P. Aubin, Viability Theory, Birkhäuser, 1991. 

4 
M. Bardi and I. CapuzzoDolcetta, Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. 

5 
A. G. Bhatt and V. S. Borkar, Occupation measures for controlled Markov processes: Characterization and optimality, Annals of Probability, 24 (1996), 15311562. 

6 
C. J. Bishop, E. A. Feinberg and J. Zhang, Examples concerning Abel and Cesaro limits, Journal of Mathematical Analysis and Applications, 420 (2014), 16541661. 

7 
J. Blot, A Pontryagin principle for infinitehorizon problems under constraints, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (2012), 267275. 

8 
V. S. Borkar, A convex analytic approach to Markov decision processes, Probability Theory and Related Fields, 78 (1988), 583602. 

9 
R. Buckdahn, D. Goreac and M. Quincampoix, Stochastic optimal control and linear programming approach, Appl. Math. Optim. 63 (2011), 257276. 

10 
D. A. Carlson, A. B. Haurier and A. Leizarowicz, Infinite Horizon Optimal Control. Deterministic and Stochastic Processes, Springer, Berlin, 1991. 

11 
N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley & Sons, Inc., New York, 1988. 

12 
L. Finlay, V. Gaitsgory and I. Lebedev, Duality in linear programming problems related to deterministic long run average problems of optimal control, SIAM J. Control and Optimization, 47 (2008), 16671700. 

13 
W. H. Fleming and D. Vermes, Convex duality approach to the optimal control of diffusions, SIAM J. Control Optimization, 27 (1989), 11361155. 

14 
V. Gaitsgory, On representation of the limit occupational measures set of control systems with applications to singularly perturbed control systems, SIAM J. Control and Optimization, 43 (2004), 325340. 

15 
V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting, SIAM J. Control and Optim., 48 (2009), 24802512. 

16 
V. Gaitsgory and M. Quincampoix, On sets of occupational measures generated by a deterministic control system on an infinite time horizon, Nonlinear Analysis (Theory, Methods & Applications), 88 (2013), 2741. 

17 
V. Gaitsgory and S. Rossomakhine, Linear programming approach to deterministic long run average problems of optimal control, SIAM J. of Control and Optimization, 44 (2006), 20062037. 

18 
D. Goreac and O.S. Serea, Linearization techniques for $L^{\infty} $  control problems and dynamic programming principles in classical and $L^{\infty} $ control problems, ESAIM: Control, Optimization and Calculus of Variations, 18 (2012), 836855. 

19 
L. Grüne, Asymptotic controllability and exponential stabilization of nonlinear control systems at singular points, SIAM J. Control Optim., 36 (1998), 14951503. 

20 
L. Grüne, On the relation between discounted and average optimal value functions, J. Diff. Equations, 148 (1998), 6569. 

21 
D. HernandezHernandez, O. HernandezLerma and M. Taksar, The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 1733. 

22 
O. HernandezLerma and J. B. Lasserre, The linear programmimg approach, in Handbook of Markov Decision Processes: Methods and Applications, (eds. E.A Feinberg and A. Shwartz), Kluwer, 40 (2002), 377407. 

23 
D. Klabjan and D. Adelman, An Infinitedimensional linear programming algorithm for deterministic semiMarkov decision processes on Borel spaces, Mathematics of Operations Research, 32 (2007), 528550. 

24 
T. G. Kurtz and R. H. Stockbridge, Existence of Markov controls and characterization of optimal Markov controls, SIAM J. on Control and Optimization, 36 (1998), 609653. 

25 
J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMIrelaxations, SIAM J. Control Optim., 47 (2008), 16431666. 

26 
E. Lehrer and S. Sorin, A uniform Tauberian theorem in dynamic programming, Mathematics of Operations Research, 17 (1992), 303307. 

27 
M. Quincampoix and O. Serea, The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures, Nonlinear Anal. 72 (2010), 28032815. 

28 
J. Renault, Uniform value in dynamic programming, J. European Mathematical Society, 13 (2011), 309330. 

29 
J. E. Rubio, Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986. 

30 
R. H. Stockbridge, TimeAverage control of a martingale problem. Existence of a stationary solution, Annals of Probability, 18 (1990), 190205. 

31 
R. H. Stockbridge, Timeaverage control of a martingale problem: A linear programming formulation, Annals of Probability, 18 (1990), 206217. 

32 
R. Sznajder and J. A. Filar, Some comments on a theorem of Hardy and Littlewood, J. Optimization Theory and Applications, 75 (1992), 201208. 

33 
R. Vinter, Convex duality and nonlinear optimal control, SIAM J. Control and Optim., 31 (1993), 518538. 

34 
A. Zaslavski, Stability of the Turnpike Phenomenon in DiscreteTime Optimal Control Problems, Springer, 2014. 

35 
A. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. 

36 
A. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, New York, 2014. 

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