Abstract and Applied Analysis

State Trajectories Analysis for a Class of Tubular Reactor Nonlinear Nonautonomous Models

B. Aylaj, M. E. Achhab, and M. Laabissi

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Abstract

The existence and uniqueness of global mild solutions are proven for a class of semilinear nonautonomous evolution equations. Moreover, it is shown that the system, under considerations, has a unique steady state. This analysis uses, essentially, the dissipativity, a subtangential condition, and the positivity of the related C 0 -semigroup.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 127394, 13 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969146

Digital Object Identifier
doi:10.1155/2008/127394

Mathematical Reviews number (MathSciNet)
MR2393111

Zentralblatt MATH identifier
1160.47330

Citation

Aylaj, B.; Achhab, M. E.; Laabissi, M. State Trajectories Analysis for a Class of Tubular Reactor Nonlinear Nonautonomous Models. Abstr. Appl. Anal. 2008 (2008), Article ID 127394, 13 pages. doi:10.1155/2008/127394. https://projecteuclid.org/euclid.aaa/1220969146


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References

  • D. Dochain, Contribution to the analysis and control of distributed parameter systems with application to (bio)chemical processes and robotics, Ph.D. thesis, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium, 1994.
  • W. H. Ray, Advanced Process Control, Butterworths Series in Chemical Engineering, McGraw-Hill, Boston, Mass, USA, 1981.
  • S. Renou, Commande Non-Linéaire d'un Systeme Décrit par des Equations Paraboliques: Application au Procédé de Blanchiment, Ph.D. thesis, Génie Chimique, Ecole Polytechnique de Montreal, Montreal, QC, Canada, 2000.
  • S. Renou, M. Perrier, D. Dochain, and S. Gendron, “Solution of the convection-dispersion-reaction equation by a sequencing method,” Computers & Chemical Engineering, vol. 27, no. 5, pp. 615–629, 2003.
  • J. J. Winkin, D. Dochain, and P. Ligarius, “Dynamical analysis of distributed parameter tubular reactors,” Automatica, vol. 36, no. 3, pp. 349–361, 2000.
  • M. E. Achhab, B. Aylaj, and M. Laabissi, “Global existence of state trajectoties for a class of tubular reactor nonlinear models,” in Proceedings CD-ROM of the 16th International Symposium on the Mathematical Theory of Networks and Systems (MTNS '04), Leuven, Belgium, July 2004.
  • B. Aylaj, M. E. Achhab, and M. Laabissi, “Asymptotic behaviour of state trajectories for a class of tubular reactor nonlinear models,” IMA Journal of Mathematical Control and Information, vol. 24, no. 2, pp. 163–175, 2007.
  • R. H. Martin Jr., “Mathematical models in gas-liquid reactions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 4, no. 3, pp. 509–527, 1980.
  • N. D. Alikakos, “$L^p$ Bounds of solutions of reaction-diffusion equations,” Communications in Partial Differential Equations, vol. 4, no. 8, pp. 827–868, 1979.
  • K. Masuda, “On the global existence and asymptotic behavior of solutions of reaction-diffusion equations,” Hokkaido Mathematical Journal, vol. 12, no. 3, pp. 360–370, 1983.
  • E. P. Van Elk, Gas-liquid reactions: influence of liquid bulk and mass transfer on process performance, Ph.D. thesis, University of Twente, Enschede, The Netherlands, 2001.
  • P. V. Danckwerts, Gas-Liquid Reactions, McGraw-Hill, New York, NY, USA, 1970.
  • R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995.
  • R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1976.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
  • T. Iwamiya, “Global existence of mild solutions to semilinear differential equations in Banach spaces,” Hiroshima Mathematical Journal, vol. 16, no. 3, pp. 499–530, 1986.
  • B. Aulbach and N. Van Minh, “Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations,” Abstract and Applied Analysis, vol. 1, no. 4, pp. 351–380, 1996.
  • V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest, 1976.
  • D. Bothe, “Flow invariance for perturbed nonlinear evolution equations,” Abstract and Applied Analysis, vol. 1, no. 4, pp. 417–433, 1996.
  • Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series, Springer, London, UK, 1999.
  • C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
  • N. Pavel, “Invariant sets for a class of semi-linear equations of evolution,” Nonlinear Analysis. Theory, Methods & Applications, vol. 1, no. 2, pp. 187–196, 1977.
  • T. Iwamiya, “Global existence of solutions to nonautonomous differential equations in Banach spaces,” Hiroshima Mathematical Journal, vol. 13, no. 1, pp. 65–81, 1983.
  • N. Kenmochi and T. Takahashi, “Nonautonomous differential equations in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 4, no. 6, pp. 1109–1121, 1980.
  • W. Arendt, A. Grabosch, G. Greiner, et al., One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.
  • M. Laabissi, M. E. Achhab, J. J. Winkin, and D. Dochain, “Multiple equilibrium profiles for nonisothermal tubular reactor nonlinear models,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications & Algorithms, vol. 11, no. 3, pp. 339–352, 2004.
  • M. E. Achhab, B. Aylaj, and M. Laabissi, “Equilibrium profiles for a class of tubular reactor nonlinear models,” in Proceedings of the 13th Mediteranean Conference on Control and Automation (MED '05), Limassol, Cyprus, June 2005.