${\cal H}_\infty$ Control of Nonlinear Systems: Differential Games and Viscosity Solutions
SIAM Journal on Control and Optimization
Simple necessary and sufficient conditions for the stability of constrained processes
SIAM Journal on Applied Mathematics
L2-Gain and Passivity Techniques in Nonlinear Control
L2-Gain and Passivity Techniques in Nonlinear Control
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We generalize the analysis of J.A. Ball, M.V. Day, and P. Kachroo (Mathematics of Control, Signals, and Systems, vol. 12, pp. 307–345, 1999) to a fluid model of a single server re-entrant queue. The approach is to solve the Hamilton-Jacobi-Isaacs equation associated with optimal robust control of the system. The method of “staged” characteristics is generalized from Ball et al. (1999) to construct the solution explicitly. Formulas are developed allowing explicit calculations for the Skorokhod problem involved in the system equations. Such formulas are particularly important for numerical verification of conditions on the boundary of the nonnegative orthant. The optimal control (server) strategy is shown to be of linear-index type. Dai-type stability properties are discussed. A modification of the model in which new “customers” are allowed only at a specified entry queue is considered in 2 dimensions. The same optimal strategy is found in that case as well.