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Objective function design for robust optimalityof linear controlunder state-constraints and uncertainty

Published online by Cambridge University Press:  30 October 2009

Fabio Bagagiolo  and
Dario Bauso
Fabio Bagagiolo
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy. bagagiol@science.unitn.it
Dario Bauso
Affiliation:
DINFO, Università di Palermo, 90128 Palermo, Italy. dario.bauso@unipa.it
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Abstract

We consider a model for the control of a linear network flow system with unknown but bounded demandand polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective functionthat 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.

Keywords

Optimal controlviscosity solutionsdifferential gamesswitchingflow controlnetworks
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 155 - 177
Copyright
© EDP Sciences, SMAI, 2009

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References

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.
Bagagiolo, F. and Bardi, M., Singular perturbation of a finite horizon problem with state-space constraints. SIAM J. Contr. Opt. 36 (1998) 2040–2060. CrossRef
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).
Bardi, M., Koike, S. and Soravia, P., Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximation. Discrete Contin. Dyn. Syst. 6 (2000) 361–380.
Bauso, D., Blanchini, F. and Pesenti, R., Robust control policies for multi-inventory systems with average flow constraints. Automatica 42 (2006) 1255–1266. CrossRef
Bemporad, A., Morari, M., Dua, V. and Pistikopoulos, E.N., The explicit linear quadratic regulator for constrained systems. Automatica 38 (2002) 320. CrossRef
Ben Tal, A. and Nemirovsky, A., Robust solutions of uncertain linear programs. Oper. Res. 25 (1998) 1–13.
Bertsekas, D.P. and Rhodes, I., Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Automatic Control 16 (1971) 117–128. CrossRef
Bertsimas, D. and Thiele, A., A robust optimization approach to inventory theory. Oper. Res. 54 (2006) 150–168. CrossRef
Cardialaguet, P., Quincampoix, M. and Saint-Pierre, P., Pursuit differential games with state constraints. SIAM J. Contr. Opt. 39 (2001) 1615–1632. CrossRef
Casti, J., On the general inverse problem of optimal control theory. J. Optim. Theory Appl. 32 (1980) 491–497. CrossRef
Chen, X., Sim, M., Sun, P. and Zhang, J., A linear-decision based approximation approach to stochastic programming. Oper. Res. 56 (2008) 344–357. CrossRef
Crandall, M.G., Evans, L.C. and Lions, P.L., Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487–502. CrossRef
Dharmatti, S. and Ramaswamy, M., Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl. 128 (2006) 75–102. CrossRef
R.J. Elliot and N.J. Kalton, The existence of value in differential games, Mem. Amer. Math. Soc. 126. AMS, Providence, USA (1972).
Evans, L.C. and Ishii, H., Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math. 49 (1984) 109–139. CrossRef
Garavello, M. and Soravia, P., 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. CrossRef
Koike, S., On the state constraint problem for differential games. Indiana Univ. Math. J. 44 (1995) 467–487. CrossRef
Kostyukova, O. and Kostina, E., Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Math. Program. 107 (2006) 131–153. CrossRef
Larin, V.B., About the inverse problem of optimal control. Appl. Comput. Math 2 (2003) 90–97.
Lee, T.T. and Liaw, G.T., The inverse problem of linear optimal control for constant disturbance. Int. J. Control 43 (1986) 233–246. CrossRef
Soravia, P., Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451–477. CrossRef
Soner, H.M., Optimal control problems with state-space constraints I. SIAM J. Contr. Opt. 31 (1986) 132–146. CrossRef
A. Visintin, Differential Models of Hysteresis. Springer-Verlag, Berlin, Germany (1996).