Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay

Abstract

LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. The system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? This is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? This is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impulses, delays, memory, nonlocal conditions, and noises, which are intrinsic phenomena no taken into account by the mathematical model that is representing by a differential equation. For many control systems in real life, delays, impulses, and noises are natural properties that do not change their behavior. So, we conjecture that, under certain conditions, the abrupt changes, delays, and noises as perturbations of a system do not modify certain properties such as controllability. In this regard, we prove the interior $S^{\ast }$ -controllability of the semilinear stochastic heat equation with impulses and delay on the state variable, and this is done by using new techniques avoiding fixed point theorems employed by Bashirov et al.