Internal Controllability of Parabolic Equations with Inputs in Coefficients

  • Viorel Barbu
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 90)


Very often, the input control arises in the coefficients of a parabolic equation and the exact controllability of initial data to origin or to a given stationary state is a delicate problem which cannot be treated by the linearization method developed in the previous chapter. However, in some situations, one can construct explicit feedback controllers which steer initial data to a given stationary state. In general, such a controller is nonlinear, eventually multivalued mapping, and its controllability effect is based on the property of solutions to certain nonlinear partial differential equations to have extinction in finite time. Here we shall study a few examples of this type.


  1. 4.
    Andreu, F., Vaselles, V., Díaz, J.I., Mazón, J.M.: Some qualitative properties for the total variation flow. J. Funct. Anal. 188(2), 516–547 (2002)MathSciNetCrossRefGoogle Scholar
  2. 14.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–394 (1987)CrossRefGoogle Scholar
  3. 26.
    Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, New York (2010)CrossRefGoogle Scholar
  4. 27.
    Barbu, V.: Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems. Annu. Rev. Control. 340, 52–61 (2010)Google Scholar
  5. 33.
    Barbu, V.: Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into a critical one. Math. Methods Appl. Sci. 1–8 (2013)Google Scholar
  6. 40.
    Barbu, V., Röckner, M.: Stochastic porous media equation and self-organized criticality: convergence to the critical state in all dimensions. Commun. Math. Phys. 311, 539–555 (2012)MathSciNetCrossRefGoogle Scholar
  7. 41.
    Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209(3), 797–834 (2013)MathSciNetCrossRefGoogle Scholar
  8. 46.
    Barbu, V., Da Prato, G., Röckner, M.: Stochastic porous media equation and self-organized criticality. Commun. Math. Phys. 285, 901–923 (2009)MathSciNetCrossRefGoogle Scholar
  9. 48.
    Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Springer, Berlin (2016)Google Scholar
  10. 55.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)Google Scholar
  11. 56.
    Cafiero, R., Loreto, V., Pietronero, A., Zapperi, Z.: Local rigidity and self-organized criticality for avalanches. Europhys. Lett. 29, 111–116 (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Viorel Barbu
    • 1

Personalised recommendations