Controllability of the Burgers equation under the influence of impulses, delay and nonlocal conditions

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 them-selves intrinsic to it. we verified that the property of controllability is robust under this type of disturbances. Specifically, we prove that the interior ap-proximate 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 em-ployed 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 cor-responding linear heat equation is approximately controllable on any interval [t₀, τ], 0 < t₀ < τ.