Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2019, Article ID 3238462, 16 pages
https://doi.org/10.1155/2019/3238462
Research Article

Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

Department of Mathematics Education, College of Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk, Republic of Korea

Correspondence should be addressed to Jin-soo Hwang; rk.ca.ugead@gnawhsj

Received 22 October 2018; Accepted 29 November 2018; Published 3 February 2019

Academic Editor: Salim Messaoudi

Copyright © 2019 Jin-soo Hwang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, Germany, 1883.
  2. A. Arosio, “Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces,” in Proceedings of the nd Workshop on Functional-Analytic Methods in Complex Analysis, Treste, World Scientific, Singapore, 1993. View at MathSciNet
  3. S. Spagnolo, “The Cauchy problem for Kirchhoff equations,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 62, pp. 17–51, 1992. View at Google Scholar · View at MathSciNet
  4. S. Pohozaev, “On a class of quasilinear hyperbolic equations,” Matematicheskii Sbornik, vol. 96, pp. 152–166, 1975. View at Google Scholar · View at MathSciNet
  5. J. L. Lions, “On some questions in boundary value problem of Mathematical Physics,” in Contemporary developments in Continuum Mechanics and Partial Differential Equations, G. M. de la Penha and L. A. Medeiros, Eds., Math. Studies, North Holland, 1977. View at Google Scholar
  6. K. Nishihara and Y. Yamada, “On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms,” Funkcialaj Ekvacioj, vol. 33, no. 1, pp. 151–159, 1990. View at Google Scholar · View at MathSciNet
  7. M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and exponential decay for a Kirchhoff-Carrier model with viscosity,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 40–60, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. J.-s. Hwang and S.-i. Nakagiri, “Optimal control problems for Kirchhoff type equation with a damping term,” Nonlinear Analysis, Theory, Method and Applications, vol. 72, no. 3-4, pp. 1621–1631, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Germany, 1971. View at MathSciNet
  10. J. Droniou and J.-P. Raymond, “Optimal pointwise control of semilinear parabolic equations,” Nonlinear Analysis, vol. 39, pp. 135–156, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J.-s. Hwang and S.-i. Nakagiri, “Optimal control problems for the equation of motion of membrane with strong viscosity,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 327–342, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. J.-s. Hwang, “Optimal control problems for an extensible beam equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 436–448, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. A. Belmiloudi, “Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 620–642, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. N. Arada and J.-P. Raymond, “Minimax control of parabolic systems with state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 1, pp. 254–271, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories, I, Cambridge University Press, Cambridge, UK, 2000. View at MathSciNet
  16. X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, Mass, USA, 1995. View at Publisher · View at Google Scholar
  17. V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Reidel, Dordrecht, Netherlands, 1986. View at MathSciNet
  18. M. E. Bradley and S. Lenhart, “Bilinear spatial control of the velocity term in a Kirchhoff plate equation,” Electronic Journal of Differential Equations, vol. 27, pp. 1–15, 2001. View at Google Scholar · View at MathSciNet
  19. M. E. Bradley, S. Lenhart, and J. Yong, “Bilinear optimal control of the velocity term in a Kirchhoff plate equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 451–467, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. S. Lenhart and M. Liang, “Bilinear optimal control for a wave equation with viscous damping,” Houston Journal of Mathematics, vol. 3, no. 26, pp. 575–595, 2000. View at Google Scholar · View at MathSciNet
  21. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  22. R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  23. R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5 of Evolution Problems I, Springer-Verlag, 2000. View at Publisher · View at Google Scholar
  24. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, II, Springer-Verlag, Heidelberg, Germany, 1972. View at MathSciNet
  25. L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998. View at MathSciNet
  26. J. Simon, “Compact sets in the space Lp(0,T;B),” Annali di Matematica Pura ed Applicata, vol. 146, no. 4, pp. 65–96, 1987. View at Google Scholar
  27. R. Temam, Navier-Stokes Equations Theory and Numerical Analysis, North-Holland, 1984. View at MathSciNet