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
Dev Prakash Jha, Raju K George
TL;DR
This paper studies existence/uniqueness of mild solutions and exact null controllability for a class of non-autonomous conformable fractional semi-linear systems. It defines the initial time as the intersection of two time intervals and develops an evolution-operator framework, employing Schauder's fixed-point theorem and the Banach contraction principle to derive solvability results for the abstract Cauchy problem. A key contribution is the controllability analysis via the operator $H_\alpha = (L_{\zeta})^{-1}N_{\zeta}^{t_2}$, which yields a constructive control $u(t) = -H_\alpha(x_0,F)(t)$ that steers the system to zero under a small-gain condition. The results are illustrated with an application to a fractional PDE, demonstrating practical applicability to non-autonomous conformable fractional dynamics.
Abstract
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.