# Boundary-influenced robust controls: two network examples

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

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

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topDay, 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

top- 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.
- 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.
- R. Atar, P. Dupuis and A. Schwartz, Explicit solutions for a network control problem in the large deviation regime, Queueing Systems46 (2004) 159–176.
- 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).
- 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).
- 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.
- 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.
- M. Bardi and I. Cappuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
- T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems – A Dynamic game approach. Birkhäuser, Boston (1991).
- A. Budhiraja and P. Dupuis, Simple necessary and sufficient conditions for the stability of constrained processes. SIAM J. Appl. Math.59 (1999) 1686–1700.
- H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations. Math. Oper. Res.16 (1991) 408–446.
- H. Chen and D.D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization. Springer-Verlag, N.Y. (2001).
- 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.
- M.V. Day, On the velocity projection for polyhedral Skorokhod problems. Appl. Math. E-Notes5 (2005) 52–59.
- 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.
- 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.
- P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities. Annals Op. Res.44 (1993) 9–42.
- P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem, I and II. Prob. Theor. Rel. Fields115 (1999) 153–195, 197–236.
- 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.
- A.F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Kluwer Academic Publishers (1988).
- 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.
- W.H. Fleming and W.M. McEneaney, Risk-sensitive control on an infinite time horizon. SAIM J. Control Opt.33 (1995) 1881–1915.
- J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in Proc. of IMA Workshop on Stochastic Differential Systems. Springer-Verlag (1988).
- P. Hartman, Ordinary Differential Equations (second edition). Birkhauser, Boston (1982).
- R. Isaacs, Differential Games. Wiley, New York (1965).
- P.L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J.52 (1985) 793–820.
- X. Luo and D. Bertsimas, A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Opt.37 (1998) 177–210.
- 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).
- S. Meyn, Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Prob.5 (1995) 946–957.
- S. Meyn, Sequencing and routing in multiclass queueing networks, part 1: feedback regulation. SIAM J. Control Optim.40 (2001) 741–776.
- M.I. Reiman, Open queueing networks in heavy traffic. Math. Oper. Res.9 (1984) 441–458.
- R.T. Rockafellar, Convex Analysis. Princeton Univ. Press, Princeton (1970).
- P. Soravia, H∞ control of nonlinear systems: differential games and viscosity solutions. SIAM J. Control Optim.34 (1996) 071–1097.
- 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).
- G. Weiss, A simplex based algorithm to solve separated continuous linear programs, to appear (preprint available at ). URIhttp://stat.haifa.ac.il/~gweiss/
- P. Whittle, Risk-sensitive Optimal Control. J. Wiley, Chichester (1990).
- R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic Networks, Springer, New York IMA Vol. Math. Appl.71 (1995) 125–137.

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