Abstract
We show that the low-regret notion of Lions [C. R. Acad. Sci. Paris Ser. I Math. 315:1253–1257, 1992] is well adapted for the control of the ill-posed heat problem. Passing to the limit, we give a characterization of the no-regret control by a singular optimality system. No Slater hypothesis on the admissible set of controls u ad is necessary, since we use a corrector of order zero argument. The result is a generalization of the low-regret control [Dorville et al., Appl. Math. Lett. 17:549–552, 2004; Dorville et al., C. R. Acad. Sci. Paris Ser. I Math. 338:921–924, 2004] to the no-regret control optimality system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Dorville R., Nakoulima O., Omrane A. (2004) Low-regret control for singular distributed systems: The backwards heat ill-posed problem. Appl. Math. Lett., Vol. 17, No. 5, pp. 549-552.
Dorville R., Nakoulima O., Omrane A. (2004) Contrôle optimal pour les problèmes de contrôlabilité de systèmes distribués à données manquantes. C. R. Acad. Sci. Paris Ser. I Math., Vol. 338, pp. 921-924.
Lions J. L. (1969) Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris.
Lions J. L. (1983) Contrôle optimal pour les systèmes distribués singuliers. Gauthiers-Villard, Paris.
Lions J. L. (1973) Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, Springer, Berlin Heidelberg, New York
Lions J. L. (1992) Contrôle à moindres regrets des systèmes distribués. C. R. Acad. Sci. Paris Ser. I Math., Vol. 315, pp. 1253-1257.
Lions J. L. (1994) No-regret and low-regret control, Environment, Economics and Their Mathematical Models, Masson, Paris.
Lions J. L. (1999) Duality Arguments for Multi Agents Least-Regret Control. Collège de France, Paris.
Nakoulima O., Omrane A., Velin J. (2000) Perturbations à moindres regrets dans les systèmes distribués à données manquantes. C. R. Acad. Sci. Paris Ser. I Math., Vol. 330, pp. 801-806.
Nakoulima O., Omrane A., Velin J. (2003) On the pareto control and no-regret control for distributed systems with incomplete data. SIAM J. Control Optim. Vol. 42, No. 4, pp. 1167-1184.
Savage L. J. (1972) The Foundations of Statistics. (2nd edition). Dover, New York.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Omrane, A. (2008). Optimal Control of Ill-Posed Parabolic Distributed Systems. In: Konaté, D. (eds) Mathematical Modeling, Simulation, Visualization and e-Learning. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74339-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-74339-2_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74338-5
Online ISBN: 978-3-540-74339-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)