Raju K George | Searcharxiv

arXiv2025

Existence and uniqueness of mild solutions and evolution operators for a class of non-autonomous conformable fractional semi-linear systems and Their Exact Null Controllability

This paper investigates the controllability of systems governed by conformable fractional order derivatives. It first establishes the existence and uniqueness of evolution operators for non-autonomous fractional-order homogeneous systems, using a suitable initial time defined as the intersection of two specific time intervals. Using the theory of linear evolution operators, Schauder's fixed-point theorem, and the Banach contraction principle, the study derives a new set of sufficient conditions for the existence and uniqueness of a mild solution to non-autonomous conformable fractional semi-linear systems. Additionally, the paper examines the exact null controllability of abstract systems based on the mild solution. We provide a comprehensive example to demonstrate the applicability of the established theoretical results.

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arXiv2025

Approximate Controllability of Fractional Evolution Equations with Nonlocal Conditions via Operator Theory

This paper investigates the existence and uniqueness of mild solutions, as well as the approximate controllability, of a class of fractional evolution equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for approximate controllability are established through a novel approach to the approximate solvability of semilinear operator equations. The methodology utilizes Green's function and constructs a control function based on the Gramian controllability operator. The analysis is based on Schauder's fixed point theorem and the theory of fractional order solution operators and resolvent operators. To demonstrate the feasibility of the proposed theoretical results, an illustrative example is provided.

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