Kybernetika 46 no. 5, 890-906, 2010

Null controllability of a nonlinear diffusion system in reactor dynamics

Kumarasamy Sakthivel, Krishnan Balachandran, Jong-Yeoul Park and Ganeshan Devipriya

Abstract:

In this paper, we prove the exact null controllability of certain diffusion system by rewriting it as an equivalent nonlinear parabolic integrodifferential equation with variable coefficients in a bounded interval of $\mathbb R$ with a distributed control acting on a subinterval. We first prove a global null controllability result of an associated linearized integrodifferential equation by establishing a suitable observability estimate for adjoint system with appropriate assumptions on the coefficients. Then this result is successfully used with some estimates for parabolic equation in $L^k$ spaces together with classical fixed point theorem, to prove the null controllability of the nonlinear model.

Keywords:

observability, controllability, parabolic integrodifferential equation

Classification:

93B05, 93B07

paper.pdf

References:

  1. R. A. Adams and J. F. Fournier: Sobolev Spaces. Second edition. Academic Press, New York 2003.   CrossRef
  2. S. Anita and V. Barbu: Null controllability of nonlinear convective heat equation. ESAIM: Control, Optimization and Calculus of Variations 5 (2000), 157-173.   CrossRef
  3. K. Balachandran and J. P. Dauer: Controllability of nonlinear systems in Banach spaces; A survey. J. Optim. Theory Appl.   CrossRef
  4. V. Barbu: Exact controllability of superlinear heat equation. Appl. Math. Optim. 42 (2000), 73-89.   CrossRef
  5. V. Barbu: Controllability of parabolic and Navier-Stokes equations. Sci. Math. Japon. 56 (2002), 143-211.   CrossRef
  6. V. Barbu and M. Iannelli: Controllability of the heat equation with memory. Differential and Integral Equations 13 (2000), 1393-1412.   CrossRef
  7. C. Bardos, G. Lebeau and J. Rauch: Controle et stabilisation de l'equation des ondes. SIAM J. Control Optim. 30 (1992), 1024-1065.   CrossRef
  8. M. Beceanu: Local exact controllability of the diffusion equation in one dimension. Abstract Appl. Anal. 14 (2003), 793-811.   CrossRef
  9. D. Chae, O. Yu. Imanuvilov and S. M. Kim: Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dynamical and Control Systems 2 (1996), 449-483.   CrossRef
  10. H. O. Fattorini and S. S. Sritharan: Necessary and sufficient conditions for optimal controls in viscous flow problems. Proc. Royal Soc. London A {\mi 124} (1994), 211-251.   CrossRef
  11. E. Fern\'{a}ndez-Cara and S. Guerrero: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006), 1395-1446.   CrossRef
  12. E. Fern\'{a}ndez-Cara and E. Zuazua: The cost of approximate controllability for heat equations: The linear case. Adv. in Differential Equations 5 (2000), 465-514.   CrossRef
  13. E. Fern\'{a}ndez-Cara and E. Zuazua: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 17 (2000), 583-616.   CrossRef
  14. A. V. Fursikov and O. Yu. Imanuvilov: Controllability of Evolution Equations. Lecture Notes Ser., Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul 1996.   CrossRef
  15. O. Yu. Imanuvilov and M. Yamamoto: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publications of RIMS, Kyoto University {\mi 39} (2003), 227-274.   CrossRef
  16. V. Komorn\'{\i}k: Exact Controllability and Stabilization: The Multiplier Method. John Wiley, Paris 1995.   CrossRef
  17. O. A. Ladyzenskaya, V. A. Solonikov and N. Uralceva: Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs, Providence 1968.   CrossRef
  18. B. G. Pachpatte: On a nonlinear diffusion system arising in reactor dynamics. J. Math. Anal. Appl. 94 (1983), 501-508.   CrossRef
  19. C. V. Pao: Bifurcation analysis of a nonlinear diffusion system in reactor dynamics. Appl. Anal. 9 (1979), 107-119.   CrossRef
  20. K. Sakthivel, K. Balachandran and S. S. Sritharan: Controllability and observability theory of certain parabolic integrodifferential equations. Comp. and Math. Appl. 52 (2006), 1299-1316.   CrossRef
  21. K. Sakthivel, K. Balachandran and S. S. Sritharan: Exact controllability of nonlinear diffusion equations arising in reactor dynamics. Nonlinear Anal.: Real World Appl. 9 (2008) 2029-2054.   CrossRef
  22. K. Sakthivel, K. Balachandran and B. R. Nagaraj: On a class of nonlinear parabolic control systems with memory effects. Internat. J. Control 81 (2008), 764-777.   CrossRef
  23. J. Simon: Compact sets in the space $L^p(0;T;B)$. Ann. Mat. Pura Appl. 146 (1986), 65-96.   CrossRef
  24. D. Tataru: Boundary controllability for conservative partial differential equations. Appl. Math. Optim. 31 (1995), 257-295.   CrossRef
  25. E. G. Yanik and G. Fairweather: Finite element methods for parabolic and hyperbolic partial integrodifferential equation. Nonlinear Anal. 12 (1988), 785-809.   CrossRef
  26. J. Yong and X. Zhang: Exact controllability of the heat equation with hyperbolic memory kernel. Control Theory of Partial Differential Equations. Chapman & Hall/CRC, Boca Raton 2005, pp. 387-401.   CrossRef
  27. E. Zuazua: Controllability and observability of partial differential equations: some results and open problem. In: Handbook of Differential Equations: Evolutionary Equations 3 (2007), 527-621.   CrossRef