On a class of non-linear parabolic control systems with memory effects

Abstract

This work establishes the local exact null controllability of non-linear parabolic integrodifferential equations with non-linearities under the integral sign in a bounded domain and with homogeneous Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain ω of Ω. Using a Carleman type inequality we first prove the global exact null controllability of a linearized equation. Then, by a fixed point method, we obtain the main result for the non-linear case. This result asserts that, when the non-linearity is of class C ¹ and globally Lipschitz, the system is null controllable.

Acknowledgements

Part of this work was finished during the stay of the first author at the Tata Institute of Fundamental Research, Bangalore in April–May, 2006. The work of the first and second author is supported by the NBHM, Department of Atomic Energy, India (Grant No: 48/3/2006/R&D-II/8336).