arXiv 2505.034762025-05-06math.OCmath.CA

具有非局部条件的抽象分数阶微分包含的 $L^p$-部分零能控性

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

Bholanath Kumbhakar, Deeksha, Dwijendra Narain Pandey

AI总结：

本文研究一致凸Banach空间中具有非局部条件的半线性分数阶微分包含的$L^p$-部分零能控性，通过近似可解性方法约化到有限维子空间，并解决凸性问题以实现控制构造。

摘要：

本文研究具有非局部条件的抽象半线性分数阶微分包含的 $L^p$-部分零能控性。容许控制集由 $u\in L^p(I,U)$ 刻画，其中 $1<p<\infty$，$I=[0,\nu]$，$U$ 是一致凸 Banach 空间。在假设带源项的分数阶线性系统具有部分零能控性的前提下，我们采用近似可解性方法将问题简化，将其约化到有限维子空间。由此，原问题的解作为这些子空间中的极限函数获得。本文解决了一个由 $U$ 为一致凸 Banach 空间这一假设所带来的困难，该假设在构造所需控制时引入了凸性问题。若 $U$ 是可分 Hilbert 空间，则不会出现这些复杂情况。本研究通过解决凸性问题引入了一种新方法，从而在 $U$ 为一致凸 Banach 空间的条件下实现了半线性分数阶微分控制系统的 $L^p(I,U)$ 部分零能控性。

英文摘要：

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.

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