The Null-Controllability Of Somereaction-Diffusion Systems

The reaction-diffusion systems are wildly exist in the real world and the investigation of these systems is becoming more and more important. In this paper, we investigate the null-controllability of some nonlinear reaction-diffusion systems whose nonlinearity involving gradient term.First of all, we consider the linear system associated to the nonlinear system and present the existence, the uniqueness and the regularity of the linear system. We construct a relative functional and by the minimal element of the functional, we can give the control functions such that the system to be null-controllable. In addition, the existence of the minimal element is based on the observability estimate of the adjoint system. To prove the observability inequality, we modify a Carleman inequality to fit our case with the gradient term. Finally, we prove that the system is null-controllable when there are two control forces acting on an arbitrarily small open set.Moreover, reducing the number of control forces is an important part of the control theory. Based on above argument, we point out that the reduction of the number of control forces can be realized by constructing a suitable Carleman inequality for the sytems instead of the single equation. So we prove the Carleman inequality for the sytems under some certain assumptions on the coefficients. O n the other hand, since we have proved the system is null-controllable with two internal control forces, we can apply the strategy of fictitious control functions to prove the null-controllability of the system under one control force. Thus, with some assumptions, we find the system is also null-controllable when there are only one control force.Finally, we linearize the nonlinear system and by the Schauder’s fixed point theorem, we prove that the results of the relative linear system are also suitable for the nonlinear system, thus we prove the main results of this paper.