Objective function design for robust optimality of linear control under state-constraints and uncertainty
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 155-177.

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.

DOI : 10.1051/cocv/2009040
Classification : 49L25, 49N90, 90C35
Mots clés : 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},
     pages = {155--177},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2009040},
     mrnumber = {2775191},
     zbl = {1210.49027},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2009040/}
}
TY  - JOUR
AU  - Bagagiolo, Fabio
AU  - Bauso, Dario
TI  - Objective function design for robust optimality of linear control under state-constraints and uncertainty
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2011
SP  - 155
EP  - 177
VL  - 17
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2009040/
DO  - 10.1051/cocv/2009040
LA  - en
ID  - COCV_2011__17_1_155_0
ER  - 
%0 Journal Article
%A Bagagiolo, Fabio
%A Bauso, Dario
%T Objective function design for robust optimality of linear control under state-constraints and uncertainty
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 155-177
%V 17
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2009040/
%R 10.1051/cocv/2009040
%G en
%F 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, Tome 17 (2011) no. 1, pp. 155-177. doi : 10.1051/cocv/2009040. http://www.numdam.org/articles/10.1051/cocv/2009040/

[1] F. Bagagiolo, 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 | Zbl

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

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

[4] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston, USA (1997). | MR | Zbl

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

[6] D. Bauso, F. Blanchini and R. Pesenti, Robust control policies for multi-inventory systems with average flow constraints. Automatica 42 (2006) 1255-1266. | MR | Zbl

[7] A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems. Automatica 38 (2002) 320. | MR | Zbl

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

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

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

[11] P. Cardialaguet, M. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints. SIAM J. Contr. Opt. 39 (2001) 1615-1632. | MR | Zbl

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

[13] X. Chen, M. Sim, P. Sun and J. Zhang, A linear-decision based approximation approach to stochastic programming. Oper. Res. 56 (2008) 344-357. | MR | Zbl

[14] M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR | Zbl

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

[16] R.J. Elliot and N.J. Kalton, The existence of value in differential games, Mem. Amer. Math. Soc. 126. AMS, Providence, USA (1972). | MR | Zbl

[17] L.C. Evans and H. Ishii, Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math. 49 (1984) 109-139. | MR | Zbl

[18] M. Garavello and P. Soravia, 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 | Zbl

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

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

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

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

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

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

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

Cité par Sources :