No-Regret Optimal Control Characterization for an Ill-Posed Wave Equation

Abdelhak Hafdallah; Abdelhamid Ayadi; Chafia Laouar

doi:https://doi.org/10.14445/22315373/IJMTT-V41P528

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 41 | Number 3 | Year 2017 | Article Id. IJMTT-V41P528 | DOI : https://doi.org/10.14445/22315373/IJMTT-V41P528

No-Regret Optimal Control Characterization for an Ill-Posed Wave Equation

Abdelhak Hafdallah, Abdelhamid Ayadi, Chafia Laouar

Citation :

Abdelhak Hafdallah, Abdelhamid Ayadi, Chafia Laouar, "No-Regret Optimal Control Characterization for an Ill-Posed Wave Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 41, no. 3, pp. 283-288, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V41P528

Abstract

In this paper, we give a characteriz- ation (optimality system) of a quadratic optimal control for an ill-posed wave equation without using the extra hypothesis of Slater (i.e . U_ad set of admissible controls has a non-empty interior). By using a parabolic regularization we get a missing data problem where we associate a no-regret control to obtain a singular optimality system, then we pass to limit and by a corrector of order zero we complete the information.

Keywords

Ill-posed wave equation, no-regret control, optimal control, parabolic regularization.

References

[1] F. Troltzsch, . Optimal Control of Partial Differential Equations: Theory, Methods and Applications, graduate Studies in mathematics, American Mathematical Society, Providence, RI, 112.(2010)
[2] H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, NewYork. (2011)
[3] J. L. Lions, Contrôle à moindres regrets des systèmes distribués. C. R. Acad. Sci. Paris Ser. I Math., Vol. 315, pp 1253-1257(1992).
[4] J. L. Lions, Control of distributed singular systems, Gauthier-villars 1985.
[5] J.L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris, 1968.
[6] J. L. Lions, Perturbations Singulieres dans les Problemes aux Limites et en Contrôle optimal, Lecture Notes in Mathematics, Springer (1973).
[7] L.J. Savage, The foundations of Statistics, 2nd edition, Dover (1972).
[8] R. Dorville, O. Nakoulima and A. Omrane, On the control of ill-posed distributed parameter systems, EDP Sciences , April 2007, Vol.17, 50-66.
[9] M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE constraints. Springer, 2009.