The Controllability Of P-Laplacian Equations

In this paper, we study the controllability of a control system governed by the evolutionary p-Laplacian equations with p > 1. About this, here we have a discussion about two hands of the controllability of the system, the non-controllability and the exact null controllability.In the first part of the paper, we make a statement of the results which has attained and the main deduction of this paper. We know that it has different controllability of the system under the different supposition.In the second part, we study the non-controllability to the general aim functions. The diffusion of the evolutionary p-Laplacian equations has slow diffusion and fast diffusion respectively, thus, we also distinguish this two cases in study. we obtained our results by using the well-known finite propagation property for the case p > 2(slow diffusion), and by proving a L~1 -obstruction phenomenon for the case 1 < p < 2 (fast diffusion) respectively.Finally, we consider the exact null controllability of the system. We also prove that the system considered is exact null controllability at any time T > 0 when the input control acts on a non-empty open set , which is based on the finite time extinction and localization of the solution for the equation with a strong absorption term.