$L^p$- Exact controllability of abstract differential inclusion with nonlocal condition

Abstrakt

This paper addresses the $L^p([0,\nu], U)$ exact controllability of the abstract semilinear differential inclusion with nonlocal conditions within the context of a uniformly convex Banach space ($U$). By presuming exact controllability for the linear system, we apply an approximate solvability technique to reduce the problem to finite-dimensional subspaces. Consequently, the solutions for the primary problem are the limiting functions within these finite dimensional subspaces. The paper offers a unique solution to a challenge introduced by assuming $U$ as a uniformly convex Banach space, which presents issues of convexity during the construction of the necessary control. Such issues are not present when $U$ is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for $L^p([0,\nu], U)$ controllability of the semilinear differential control system in which $U$ is a uniformly convex Banach space.