# Objective function design for robust optimality of linear control under state-constraints and uncertainty

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 1, page 155-177
- ISSN: 1292-8119

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topBagagiolo, Fabio, and Bauso, Dario. "Objective function design for robust optimality of linear control under state-constraints and uncertainty." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 155-177. <http://eudml.org/doc/276336>.

@article{Bagagiolo2011,

abstract = {
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.
},

author = {Bagagiolo, Fabio, Bauso, Dario},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control; viscosity solutions; differential games; switching; flow control; networks; optimal control},

language = {eng},

month = {2},

number = {1},

pages = {155-177},

publisher = {EDP Sciences},

title = {Objective function design for robust optimality of linear control under state-constraints and uncertainty},

url = {http://eudml.org/doc/276336},

volume = {17},

year = {2011},

}

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

DA - 2011/2//

PB - EDP Sciences

VL - 17

IS - 1

SP - 155

EP - 177

AB -
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.

LA - eng

KW - Optimal control; viscosity solutions; differential games; switching; flow control; networks; optimal control

UR - http://eudml.org/doc/276336

ER -

## References

top- 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.
- F. Bagagiolo and M. Bardi, Singular perturbation of a finite horizon problem with state-space constraints. SIAM J. Contr. Opt.36 (1998) 2040–2060.
- 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.
- M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston, USA (1997).
- 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.
- D. Bauso, F. Blanchini and R. Pesenti, Robust control policies for multi-inventory systems with average flow constraints. Automatica42 (2006) 1255–1266.
- A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems. Automatica38 (2002) 320.
- A. Ben Tal and A. Nemirovsky, Robust solutions of uncertain linear programs. Oper. Res.25 (1998) 1–13.
- D.P. Bertsekas and I. Rhodes, Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Automatic Control16 (1971) 117–128.
- D. Bertsimas and A. Thiele, A robust optimization approach to inventory theory. Oper. Res.54 (2006) 150–168.
- P. Cardialaguet, M. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints. SIAM J. Contr. Opt.39 (2001) 1615–1632.
- J. Casti, On the general inverse problem of optimal control theory. J. Optim. Theory Appl.32 (1980) 491–497.
- X. Chen, M. Sim, P. Sun and J. Zhang, A linear-decision based approximation approach to stochastic programming. Oper. Res.56 (2008) 344–357.
- 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.
- S. Dharmatti and M. Ramaswamy, Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl.128 (2006) 75–102.
- R.J. Elliot and N.J. Kalton, The existence of value in differential games, Mem. Amer. Math. Soc.126. AMS, Providence, USA (1972).
- L.C. Evans and H. Ishii, Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math.49 (1984) 109–139.
- 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.
- S. Koike, On the state constraint problem for differential games. Indiana Univ. Math. J.44 (1995) 467–487.
- O. Kostyukova and E. Kostina, Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Math. Program.107 (2006) 131–153.
- V.B. Larin, About the inverse problem of optimal control. Appl. Comput. Math2 (2003) 90–97.
- T.T. Lee and G.T. Liaw, The inverse problem of linear optimal control for constant disturbance. Int. J. Control43 (1986) 233–246.
- P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J.51 (2002) 451–477.
- H.M. Soner, Optimal control problems with state-space constraints I. SIAM J. Contr. Opt.31 (1986) 132–146.
- A. Visintin, Differential Models of Hysteresis. Springer-Verlag, Berlin, Germany (1996).