OPTIMAL CONTROL FOR A CONTROLLED ILL-POSED WAVE EQUATION WITHOUT REQUIRING THE SLATER HYPOTHESIS

Abdelhak Hafdallah     (Laboratory of Mathematics, Informatics and Systems, University of Larbi Tébessi, Rue de Constantine 12002, Tébessa, Algeria)

Abstract


In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with  ncomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.


Keywords


Ill-posed wave equation, No-regret control, Incomplete data, Carleman estimates, Null- controllability

Full Text:

PDF

References


  1. Baleanu D., Joseph C., Mophou G. Low-regret control for a fractional wave equation with incomplete data. Adv. Difference Equ., 2016. Vol. 2016. Art. no. 240. P. 1–20. DOI: 10.1186/s13662-016-0970-8
  2. Baudouin L., De Buhan M. and Ervedoza S. Global Carleman estimates for waves and applications. Comm. Partial Differential Equations, 2013. Vol. 38, No. 5. P. 823–859. DOI: 10.1080/03605302.2013.771659
  3. Berhail A., Omrane A. Optimal control of the ill-posed Cauchy elliptic problem. Int. J. Differ. Equ., 2015. Vol. 2018. Art. no. 468918. P. 1–9. DOI: 10.1155/2015/468918
  4. Ciprian G., Gal. N.J. A spectral approach to ill-posed problems for wave equations. Ann. Mat. Pura Appl. (4), 2008. Vol. 187, No. 4. P. 705–717. DOI: 10.1007/s10231-007-0063-0
  5. Hafdallah A., Ayadi A., Laouar C. No-regret optimal control characterization for an ill-posed wave equation. Int. J. Math. Trends Tech., 2017. Vol. 45, No. 3. P. 283–287. DOI: 10.14445/22315373/IJMTT-V41P528
  6. Hafdallah A., Ayadi A. Optimal control of electromagnetic wave displacement with an unknown velocity of propagation. Int. J. Control, 2019. Vol. 92, No. 11. P. 2693–2700. DOI: 10.1080/00207179.2018.1458157
  7. Hafdallah A., Ayadi A. Optimal control of a thermoelastic body with missing initial conditions. Int. J. Control, 2020. Vol. 93, No. 7. P. 1570–1576. DOI: 10.1080/00207179.2018.1519258
  8. Hafdallah A. On the optimal control of linear systems depending upon a parameter and with missing data. Nonlinear Stud., 2020. Vol. 27, No. 2. P. 457–469.
  9. Jacob B., Omrane A. Optimal control for age-structured population dynamics of incomplete data. J. Math. Anal. Appl., 2010. Vol. 370, No. 1. P. 42–48. DOI: 10.1016/j.jmaa.2010.04.042
  10. Lions J.L. Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren Math. Wiss., vol. 170. Berlin, Heidelberg: Springer-Verlag, 1971. 400 p.
  11. Lions J.L. Control of Distributed Singular Systems. Paris: Gauthier-Villars, 1985. 552 p.
  12. Lions J.L. Contrôle à moindres regrets des systèmes distribués. C. R. Acad. Sci. Paris. Sér. I, Math., 1992. Vol. 315, No. 12. P. 1253–1257.
  13. Lions J.L. No-Regret and Low-Regret Control, in Environment, Economics and Their Mathematical Models. Paris: Masson, 1994.
  14. Nakoulima O., Omrane A., Velin J. On the Pareto control and no-regret control for distributed systems with incomplete data. SIAM J. Control Optim., 2003. Vol. 42, No. 4. P. 1167–1184. DOI: 10.1137/S0363012900380188
  15. Savage L.J. The Foundations of Statistics. 2nd ed. New York: Dover Publ., 1972. 310 p.



DOI: http://dx.doi.org/10.15826/umj.2020.1.007

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.