$L^p$- Partially null controllability of abstract fractional differential inclusion with nonlocal condition

TL;DR

This paper establishes $L^p$-partial null controllability for abstract semilinear fractional differential inclusions with nonlocal conditions, introducing a novel approach to handle convexity issues in uniformly convex Banach spaces.

Contribution

It presents a new method to achieve partial null controllability in fractional systems with nonlocal conditions within uniformly convex Banach spaces, overcoming convexity challenges.

Findings

Successfully reduced the control problem to finite-dimensional subspaces.

Established partial null controllability for systems in uniformly convex Banach spaces.

Provided a novel approach to handle convexity issues in control construction.

Abstract

In this work, we investigate the $L^p$- partial null controllability of the abstract semilinear fractional-order differential inclusion with nonlocal conditions. The set of admissible controls is characterized by $u\in L^p(I,U)$, $1<p<\infty$, $I=[0,\nu]$, where $U$ is a uniformly convex Banach space. Assuming partial null controllability for the fractional-order linear system with a source term, we employ an approximate solvability method to simplify the problem to reduce it to finite-dimensional subspaces. Consequently, the solutions of the original problem are obtained as limiting functions within these subspaces. The paper tackles a challenge stemming from the assumption that $U$ is a uniformly convex Banach space, which introduces convexity issues in constructing the required control. These complications do not occur if $U$ is a separable Hilbert space. This study introduces a novel approach by resolving the convexity issue, thereby enabling $L^p(I, U)$ partially null controllability of the semilinear fractional-order differential control system, with $U$ being a uniformly convex Banach space.