We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

Classification: 49L25, 49N90, 90C35

Keywords: optimal control, viscosity solutions, differential games, switching, flow control, networks

@article{COCV_2011__17_1_155_0, author = {Bagagiolo, Fabio and Bauso, Dario}, title = {Objective function design for robust optimality of linear control under state-constraints and uncertainty}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {17}, number = {1}, year = {2011}, pages = {155-177}, doi = {10.1051/cocv/2009040}, zbl = {1210.49027}, mrnumber = {2775191}, language = {en}, url = {http://www.numdam.org/item/COCV_2011__17_1_155_0} }

Bagagiolo, Fabio; Bauso, Dario. Objective function design for robust optimality of linear control under state-constraints and uncertainty. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, pp. 155-177. doi : 10.1051/cocv/2009040. http://www.numdam.org/item/COCV_2011__17_1_155_0/

[1] Minimum time for a hybrid system with thermostatic switchings, in Hybrid Systems: Computation and Control, A. Bemporad, A. Bicchi and G. Buttazzo Eds., Lect. Notes Comput. Sci. 4416, Springer-Verlag, Berlin, Germany (2007) 32-45. | MR 2363618 | Zbl 1221.49053

,[2] Singular perturbation of a finite horizon problem with state-space constraints. SIAM J. Contr. Opt. 36 (1998) 2040-2060. | MR 1638940 | Zbl 0953.49031

and ,[3] Robust optimality of linear saturated control in uncertain linear network flows, in Decision and Control, 2008, CDC 2008, 47th IEEE Conference (2008) 3676-3681.

and ,[4] Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. BirkhĂ¤user, Boston, USA (1997). | MR 1484411 | Zbl 0890.49011

and ,[5] Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximation. Discrete Contin. Dyn. Syst. 6 (2000) 361-380. | MR 1739379 | Zbl 1158.91323

, and ,[6] Robust control policies for multi-inventory systems with average flow constraints. Automatica 42 (2006) 1255-1266. | MR 2242916 | Zbl 1097.90002

, and ,[7] The explicit linear quadratic regulator for constrained systems. Automatica 38 (2002) 320. | MR 2110266 | Zbl 0999.93018

, , and ,[8] Robust solutions of uncertain linear programs. Oper. Res. 25 (1998) 1-13. | MR 1702364 | Zbl 0941.90053

and ,[9] Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Automatic Control 16 (1971) 117-128. | MR 297442

and ,[10] A robust optimization approach to inventory theory. Oper. Res. 54 (2006) 150-168. | MR 2201249 | Zbl 1167.90314

and ,[11] Pursuit differential games with state constraints. SIAM J. Contr. Opt. 39 (2001) 1615-1632. | MR 1825595 | Zbl 1140.91320

, and ,[12] On the general inverse problem of optimal control theory. J. Optim. Theory Appl. 32 (1980) 491-497. | MR 610561 | Zbl 0421.49029

,[13] A linear-decision based approximation approach to stochastic programming. Oper. Res. 56 (2008) 344-357. | MR 2410310 | Zbl 1167.90609

, , and ,[14] Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR 732102 | Zbl 0543.35011

, and ,[15] Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl. 128 (2006) 75-102. | MR 2201890 | Zbl 1099.91022

and ,[16] The existence of value in differential games, Mem. Amer. Math. Soc. 126. AMS, Providence, USA (1972). | MR 359845 | Zbl 0262.90076

and ,[17] Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math. 49 (1984) 109-139. | MR 767202 | Zbl 0559.35013

and ,[18] Representation formulas for solutions of HJI equations with discontinuous coefficients and existence of value in differential games. J. Optim. Theory Appl. 130 (2006) 209-229. | MR 2281799 | Zbl 1123.49033

and ,[19] On the state constraint problem for differential games. Indiana Univ. Math. J. 44 (1995) 467-487. | MR 1355408 | Zbl 0840.49016

,[20] Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Math. Program. 107 (2006) 131-153. | MR 2218124 | Zbl 1089.49035

and ,[21] About the inverse problem of optimal control. Appl. Comput. Math 2 (2003) 90-97. | MR 2039151 | Zbl 1209.49045

,[22] The inverse problem of linear optimal control for constant disturbance. Int. J. Control 43 (1986) 233-246. | MR 822399 | Zbl 0584.93027

and ,[23] Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451-477. | MR 1909297 | Zbl 1032.35055

,[24] Optimal control problems with state-space constraints I. SIAM J. Contr. Opt. 31 (1986) 132-146. | Zbl 0597.49023

,[25] Differential Models of Hysteresis. Springer-Verlag, Berlin, Germany (1996). | MR 1329094 | Zbl 0820.35004

,