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.
Mots-clés : robust control, differential game, queueing network
@article{COCV_2006__12_4_662_0,
author = {Day, Martin V.},
title = {Boundary-influenced robust controls : two network examples},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {662--698},
publisher = {EDP-Sciences},
volume = {12},
number = {4},
year = {2006},
doi = {10.1051/cocv:2006016},
mrnumber = {2266813},
zbl = {1114.49029},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2006016/}
}
TY - JOUR AU - Day, Martin V. TI - Boundary-influenced robust controls : two network examples JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 662 EP - 698 VL - 12 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006016/ DO - 10.1051/cocv:2006016 LA - en ID - COCV_2006__12_4_662_0 ER -
%0 Journal Article %A Day, Martin V. %T Boundary-influenced robust controls : two network examples %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 662-698 %V 12 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006016/ %R 10.1051/cocv:2006016 %G en %F COCV_2006__12_4_662_0
Day, Martin V. Boundary-influenced robust controls : two network examples. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 662-698. doi : 10.1051/cocv:2006016. http://www.numdam.org/articles/10.1051/cocv:2006016/
[1] and, A differential game with constrained dynamics and viscosity solutions of a related HJB equation. Nonlinear Anal. 51 (2002) 1105-1130. | Zbl
[2] , and, An escape criterion for queueing networks: Asymptotic risk-sensitive control via differential games. Math. Op. Res. 28 (2003) 801-835. | Zbl
[3] , and, Explicit solutions for a network control problem in the large deviation regime, Queueing Systems 46 (2004) 159-176. | Zbl
[4] , 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). | MR | Zbl
[5] ,,, 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). | MR | Zbl
[6] , and, Robust feedback control of a single server queueing system. Math. Control, Signals, Syst. 12 (1999) 307-345. | Zbl
[7] ,, and, Robust -Gain for nonlinear systems with projection dynamics and input constraints: an example from traffic control. Automatica 35 (1999) 429-444. | Zbl
[8] and, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | MR | Zbl
[9] and, -Optimal Control and Related Minimax Design Problems - A Dynamic game approach. Birkhäuser, Boston (1991). | Zbl
[10] and, Simple necessary and sufficient conditions for the stability of constrained processes. SIAM J. Appl. Math. 59 (1999) 1686-1700. | Zbl
[11] and, Discrete flow networks: bottleneck analysis and fluid approximations. Math. Oper. Res. 16 (1991) 408-446. | Zbl
[12] and, Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization. Springer-Verlag, N.Y. (2001). | MR | Zbl
[13] , On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid models. Ann. Appl. Prob. 5 (1995) 49-77. | Zbl
[14] , On the velocity projection for polyhedral Skorokhod problems. Appl. Math. E-Notes 5 (2005) 52-59. | Zbl
[15] ,,, and, Robust optimal service analysis of single-server re-entrant queues. Comput. Optim. Appl. 22 (2002), 261-302.
[16] and, On Lipschitz continuity of the solution mapping of the Skorokhod problem, with applications. Stochastics and Stochastics Reports 35 (1991) 31-62. | Zbl
[17] and, Dynamical systems and variational inequalities. Annals Op. Res. 44 (1993) 9-42. | Zbl
[18] and, Convex duality and the Skorokhod problem, I and II. Prob. Theor. Rel. Fields 115 (1999) 153-195, 197-236. | Zbl
[19] , and, Fluid network models: linear programs for control and performance bounds in Proc. of the 13th World Congress of International Federation of Automatic Control B (1996) 19-24.
[20] , Differential Equations with Discontinuous Right Hand Sides, Kluwer Academic Publishers (1988). | Zbl
[21] and, The risk-sensitive index and the and morms for nonlinear systems. Math. Control Signals Syst. 8 (1995) 199-221. | Zbl
[22] and, Risk-sensitive control on an infinite time horizon. SAIM J. Control Opt. 33 (1995) 1881-1915. | Zbl
[23] , Brownian models of queueing networks with heterogeneous customer populations, in Proc. of IMA Workshop on Stochastic Differential Systems. Springer-Verlag (1988). | MR | Zbl
[24] , Ordinary Differential Equations (second edition). Birkhauser, Boston (1982). | MR
[25] , Differential Games. Wiley, New York (1965). | MR | Zbl
[26] , Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J. 52 (1985) 793-820. | Zbl
[27] and, A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Opt. 37 (1998) 177-210. | Zbl
[28] , 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). | MR | Zbl
[29] , Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Prob. 5 (1995) 946-957. | Zbl
[30] , Sequencing and routing in multiclass queueing networks, part 1: feedback regulation. SIAM J. Control Optim. 40 (2001) 741-776. | Zbl
[31] , Open queueing networks in heavy traffic. Math. Oper. Res. 9 (1984) 441-458. | Zbl
[32] , Convex Analysis. Princeton Univ. Press, Princeton (1970). | MR | Zbl
[33] , control of nonlinear systems: differential games and viscosity solutions. SIAM J. Control Optim. 34 (1996) 071-1097. | Zbl
[34] , On optimal draining of re-entrant fluid lines, in Stochastic Networks, F.P. Kelly and R.J. Williams, Eds. Springer-Verlag, NY (1995). | MR | Zbl
[35] , A simplex based algorithm to solve separated continuous linear programs, to appear (preprint available at http://stat.haifa.ac.il/~gweiss/). | MR
[36] , Risk-sensitive Optimal Control. J. Wiley, Chichester (1990). | MR | Zbl
[37] , Semimartingale reflecting Brownian motions in the orthant, Stochastic Networks, Springer, New York IMA Vol. Math. Appl. 71 (1995) 125-137. | Zbl
Cité par Sources :
