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

TL;DR

This work establishes $L^p$-partial null controllability for abstract fractional-order differential inclusions with nonlocal conditions in uniformly convex Banach spaces, leveraging Caputo derivatives to account for memory effects. It develops a finite-dimensional approximate solvability scheme and a convexity-resolving fixed-point strategy to overcome nonconvexity arising from the control operator, enabling $u\in L^p(I,U)$ with $1<p<\infty$. The framework generalizes prior Hilbert-space results to Banach spaces and relaxes assumptions such as semigroup compactness, while incorporating nonlocal initial-like conditions. An applied diffusion-type fractional system illustrates how the theory yields practical controllability results for systems with memory and distributed parameters.

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 $u\in L^p(I,U)$, $1<p<\infty$, $I=[0,ν]$, 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.

Theorems & Definitions (69)

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