Boundary-influenced robust controls: two network examples

Martin V. Day

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 662-698
  • ISSN: 1292-8119

Abstract

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We consider the differential game associated with robust control of a system in a compact state domain, using Skorokhod dynamics on the boundary. A specific class of problems motivated by queueing network control is considered. A constructive approach to the Hamilton-Jacobi-Isaacs equation is developed which is based on an appropriate family of extremals, including boundary extremals for which the Skorokhod dynamics are active. A number of technical lemmas and a structured verification theorem are formulated to support the use of this technique in simple examples. Two examples are considered which illustrate the application of the results. This extends previous work by Ball, Day and others on such problems, but with a new emphasis on problems for which the Skorokhod dynamics play a critical role. Connections with the viscosity-sense oblique derivative conditions of Lions and others are noted.

How to cite

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Day, Martin V.. "Boundary-influenced robust controls: two network examples." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 662-698. <http://eudml.org/doc/249675>.

@article{Day2006,
abstract = { We consider the differential game associated with robust control of a system in a compact state domain, using Skorokhod dynamics on the boundary. A specific class of problems motivated by queueing network control is considered. A constructive approach to the Hamilton-Jacobi-Isaacs equation is developed which is based on an appropriate family of extremals, including boundary extremals for which the Skorokhod dynamics are active. A number of technical lemmas and a structured verification theorem are formulated to support the use of this technique in simple examples. Two examples are considered which illustrate the application of the results. This extends previous work by Ball, Day and others on such problems, but with a new emphasis on problems for which the Skorokhod dynamics play a critical role. Connections with the viscosity-sense oblique derivative conditions of Lions and others are noted. },
author = {Day, Martin V.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Robust control; differential game; queueing network.; robust control; queueing network},
language = {eng},
month = {10},
number = {4},
pages = {662-698},
publisher = {EDP Sciences},
title = {Boundary-influenced robust controls: two network examples},
url = {http://eudml.org/doc/249675},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Day, Martin V.
TI - Boundary-influenced robust controls: two network examples
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 662
EP - 698
AB - We consider the differential game associated with robust control of a system in a compact state domain, using Skorokhod dynamics on the boundary. A specific class of problems motivated by queueing network control is considered. A constructive approach to the Hamilton-Jacobi-Isaacs equation is developed which is based on an appropriate family of extremals, including boundary extremals for which the Skorokhod dynamics are active. A number of technical lemmas and a structured verification theorem are formulated to support the use of this technique in simple examples. Two examples are considered which illustrate the application of the results. This extends previous work by Ball, Day and others on such problems, but with a new emphasis on problems for which the Skorokhod dynamics play a critical role. Connections with the viscosity-sense oblique derivative conditions of Lions and others are noted.
LA - eng
KW - Robust control; differential game; queueing network.; robust control; queueing network
UR - http://eudml.org/doc/249675
ER -

References

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  1. R. Atar and P. Dupuis, A differential game with constrained dynamics and viscosity solutions of a related HJB equation. Nonlinear Anal.51 (2002) 1105–1130.  Zbl1025.49020
  2. R. Atar, P. Dupuis and A. Shwartz, An escape criterion for queueing networks: Asymptotic risk-sensitive control via differential games. Math. Op. Res.28 (2003) 801–835.  Zbl1082.60520
  3. R. Atar, P. Dupuis and A. Schwartz, Explicit solutions for a network control problem in the large deviation regime, Queueing Systems46 (2004) 159–176.  Zbl1044.60018
  4. F. Avram, Optimal control of fluid limits of queueing networks and stochasticity corrections, in Mathematics of Stochastic Manufacturing Systems, G. Yin and Q. Zhang Eds., AMS, Lect. Appl. Math.33 (1996).  
  5. F. Avram, D. Bertsimas, M. Ricard, Fluid models of sequencing problems in open queueing networks; and optimal control approach, in Stochastic Networks, F.P. Kelly and R.J. Williams Eds., Springer-Verlag, NY (1995).  Zbl0837.60083
  6. J.A. Ball, M.V. Day and P. Kachroo, Robust feedback control of a single server queueing system. Math. Control, Signals, Syst.12 (1999) 307–345.  Zbl0940.93028
  7. J.A. Ball, M.V. Day, P. Kachroo and T. Yu, RobustL2-Gain for nonlinear systems with projection dynamics and input constraints: an example from traffic control. Automatica35 (1999) 429–444.  Zbl0946.93015
  8. M. Bardi and I. Cappuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).  Zbl0890.49011
  9. T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems – A Dynamic game approach. Birkhäuser, Boston (1991).  Zbl0751.93020
  10. A. Budhiraja and P. Dupuis, Simple necessary and sufficient conditions for the stability of constrained processes. SIAM J. Appl. Math.59 (1999) 1686–1700.  Zbl0934.93068
  11. H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations. Math. Oper. Res.16 (1991) 408–446.  Zbl0735.60095
  12. H. Chen and D.D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization. Springer-Verlag, N.Y. (2001).  Zbl0992.60003
  13. J.G. Dai, On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid models. Ann. Appl. Prob. 5 (1995) 49–77.  Zbl0822.60083
  14. M.V. Day, On the velocity projection for polyhedral Skorokhod problems. Appl. Math. E-Notes5 (2005) 52–59.  Zbl1073.90050
  15. M.V. Day, J. Hall, J. Menendez, D. Potter and I. Rothstein, Robust optimal service analysis of single-server re-entrant queues. Comput. Optim. Appl.22 (2002), 261–302.  Zbl1161.90364
  16. P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping of the Skorokhod problem, with applications. Stochastics and Stochastics Reports35 (1991) 31–62.  Zbl0721.60062
  17. P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities. Annals Op. Res.44 (1993) 9–42.  Zbl0785.93044
  18. P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem, I and II. Prob. Theor. Rel. Fields115 (1999) 153–195, 197–236.  Zbl0944.60061
  19. D. Eng, J. Humphrey and S. Meyn, Fluid network models: linear programs for control and performance bounds in Proc. of the 13th World Congress of International Federation of Automatic ControlB (1996) 19–24.  
  20. A.F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Kluwer Academic Publishers (1988).  Zbl0664.34001
  21. W.H. Fleming and M.R. James, The risk-sensitive index and the H2 and H∞ morms for nonlinear systems. Math. Control Signals Syst.8 (1995) 199–221.  Zbl0854.93045
  22. W.H. Fleming and W.M. McEneaney, Risk-sensitive control on an infinite time horizon. SAIM J. Control Opt.33 (1995) 1881–1915.  Zbl0949.93079
  23. J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in Proc. of IMA Workshop on Stochastic Differential Systems. Springer-Verlag (1988).  Zbl0658.60123
  24. P. Hartman, Ordinary Differential Equations (second edition). Birkhauser, Boston (1982).  Zbl0476.34002
  25. R. Isaacs, Differential Games. Wiley, New York (1965).  
  26. P.L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J.52 (1985) 793–820.  
  27. X. Luo and D. Bertsimas, A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Opt.37 (1998) 177–210.  Zbl0921.49023
  28. S. Meyn, Stability and optimizations of queueing networks and their fluid models, in Mathematics of Stochastic Manufacturing Systems, G. Yin and Q. Zhang Eds., Lect. Appl. Math.33, AMS (1996).  
  29. S. Meyn, Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Prob.5 (1995) 946–957.  Zbl0865.60079
  30. S. Meyn, Sequencing and routing in multiclass queueing networks, part 1: feedback regulation. SIAM J. Control Optim.40 (2001) 741–776.  Zbl1060.90043
  31. M.I. Reiman, Open queueing networks in heavy traffic. Math. Oper. Res.9 (1984) 441–458.  Zbl0549.90043
  32. R.T. Rockafellar, Convex Analysis. Princeton Univ. Press, Princeton (1970).  Zbl0193.18401
  33. P. Soravia, H∞ control of nonlinear systems: differential games and viscosity solutions. SIAM J. Control Optim.34 (1996) 071–1097.  Zbl0926.93019
  34. G. Weiss, On optimal draining of re-entrant fluid lines, in Stochastic Networks, F.P. Kelly and R.J. Williams, Eds. Springer-Verlag, NY (1995).  Zbl0823.60084
  35. G. Weiss, A simplex based algorithm to solve separated continuous linear programs, to appear (preprint available at ).  Zbl1165.90011URIhttp://stat.haifa.ac.il/~gweiss/
  36. P. Whittle, Risk-sensitive Optimal Control. J. Wiley, Chichester (1990).  Zbl0718.93068
  37. R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic Networks, Springer, New York IMA Vol. Math. Appl.71 (1995) 125–137.  Zbl0827.60031

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