Control theory of a differential system - García Pinargote, Katherine Jazmín (2021)
For many years, one of the goals of control theory has been to prove the following conjecture: under conditions such as impulses, delays, and non-local conditions, which are intrinsic phenomena, the controllability of a system does not change. That is, if we consider these three characteristics as disturbances of the system, which is very natural in real-life problems, the controllability of the system turns out to be robust. Taking into account this phenomena, this work is devoted to study the existence, uniqueness of solutions, and the controllability of an impulsive semilinear system with infinite delay and non-local conditions. To achieve this goal, we first select the phase space adequately in such a way that it satisfies the axiomatic theory formulated by Hale and Kato to study differential equations with infinite delay. After defining the space we will be working on, we develop the three main proofs of our study. The existence of solution, and the exact controllability are reduced to the problem of finding the fixed points of an operator, for doing so, we apply Karakosta's Fixed Point Theorem (an Extension of Krasnosel'skii's Fixed Point Theorem) and Rothe's Fixed Point Theorem, respectively. The third proof use a technique developed by A. Bashirov et. al, which evades the use of fixed point theorems and will be applied to prove the approximate controllability of the semi-linear system. At the end of the existence proof, we show an example that involves impulses, infinite delay, and non-local conditions.