Controllability of Semilinear Neutral Differential Equations with
Impulses and Nonlocal Conditions
Abstract
When a real-life problem is mathematically modeled by differential
equations or another type of equation, there are always intrinsic
phenomena that are not taken into account and can affect the behavior of
such a model. For example, external forces can abruptly change the
model; impulses and delay can cause a breakdown of it. Considering these
intrinsic phenomena in the mathematical model makes the difference
between a simple differential equation and a differential equation with
impulses, delay, and nonlocal conditions. So, in this work, we consider
a semilinear nonautonomous neutral differential equation under the
influence of impulses, delay, and nonlocal conditions. In this paper we
study the controllability of these semilinear neutral differential
equations with some of these intrinsic phenomena taking into
consideration. Our aim is to prove that the controllability of the
associated ordinary linear differential equation is preserved under
certain conditions imposed on these new disturbances. In order to
achieve our objective, we apply Rothe’s fixed point Theorem to prove the
exact controllability of the system. Finally, our method can be extended
to the evolution equation in Hilbert spaces with applications to control
systems governed by PDE’s equations.