L^p- Exact controllability of abstract differential inclusion with nonlocal condition - Kumbhakar, Bholanath; Pandey, Dwijendra Narain (Topological Methods in Nonlinear Analysis, 2025)
This paper addresses the L^p([0, u], 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, u], U) controllability of the semilinear differential control system in which U is a uniformly convex Banach space.