Results on approximate controllability results for second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay

Abstract

In this article, we determine a set of appropriate conditions for approximate controllability of second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay. We verify the main results by applying ideas about the cosine function, sine functions, and fixed point approach. Next, we continue the discussion to the nonlocal second‐order Sobolev‐type differential system. In the end, we provide a theoretical application to assist in the effectiveness of our discussion.

A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay

“…Motivated by the Filter systems presented in previous studies, 9‐11,40,41,45,57,58 we present our filter system Figure 1. Figure 1 describes rough pattern of the block diagram, it aids in enhancing the usefulness of the solution along with minimum measure of inputs.…”

“…Motivated by the Filter systems presented in previous studies, [9][10][11]40,41,45,57,58 we present our filter system Figure 1. Finally, we move all outputs from the integrators to summer network.…”

“…The controllability problem is finding an appropriate control function to such a degree that one can steer the considered dynamical system to a final state. Accordingly, over recent years, the discussion about the controllability of numerous nonlinear frameworks has been comprehensively investigated by many researchers and one can view previous studies 13‐16,18,19,36‐42 …”

“…Accordingly, over recent years, the discussion about the controllability of numerous nonlinear frameworks has been comprehensively investigated by many researchers and one can view previous studies. [13][14][15][16]18,19,[36][37][38][39][40][41][42] Hilfer 43 initiated other kinds of derivatives of fractional order, including Riemann-Liouville and Caputo fractional derivative. Recently, in Gu and Trujillo, 44 the authors proved the existence of integral solution for evolution equation with the Hilfer fractional derivative by using noncompact measure method and fixed point theorem approach.…”

Two fractional hybrid and non‐hybrid boundary value problems

“…What is more important to researchers today than anything else is to understand some qualitative properties of solutions for these fractional dynamical systems modeled based on different complicated fractional boundary value problems (BVPs) of hybrid or non‐hybrid type. In this direction, a lot of works have been published about different types of fractional integro‐differential and differential equations, 1‐6 integro‐differential equations involving the Caputo–Fabrizio derivative, 7‐11 hybrid differential equations, 12‐19 approximate solutions of different fractional differential equations, 20‐23 fractional mathematical modelings, 24‐31 and numerical techniques in different fields of fractional calculus 32‐45 …”