CONTROLLABILITY OF THE BURGERS EQUATION UNDER THE INFLUENCE OF IMPULSES, DELAY AND NONLOCAL CONDITIONS

Abstract

Abstract: In the case of the Burges equation, this work proves the following conjecture: impulses, delays, and nonlocal conditions, under some assumptions, do not destroy some posed system qualitative properties since they are themselves intrinsic to it. we verified that the property of controllability is robust under this type of disturbances. Specifically, we prove that the interior approximate controllability of the linear heat equation is not destroyed if we add impulses, nonlocal conditions, and a nonlinear perturbation with delay in the state. This is done by using new techniques avoiding fixed point theorems employed by A.E. Bashirov et al. In this case the delay helps us to prove the approximate controllability of this system by pulling back the control solution to a fixed curve in a short time interval, and from this position, we are able to reach a neighborhood of the final state in time τ by using the fact that the corresponding linear heat equation is approximately controllable on any interval [t0, τ ], 0 < t0 < τ .