Approximate Controllability of Fractional Evolution Equations with Nonlocal Conditions via Operator Theory
Dev Prakash Jha, Raju K George
TL;DR
The paper tackles approximate controllability of abstract fractional evolution equations with nonlocal initial conditions in Hilbert spaces by developing an operator-theoretic framework that uses Green's function, the Gramian controllability operator, and ${}^{C}D^{\alpha}$ with $0<\alpha<1$. It reduces controllability to a solvability problem for a semilinear operator equation, proves existence/uniqueness of mild solutions via $\alpha$-order solution/resolvent theory and Schauder-type fixed-point arguments, and provides sufficient conditions under which the nonlinear system inherits approximate controllability from the linear one. A pivotal contribution is showing how to formulate and solve $u=Kv+KNu$ in a setting with nonlocal initial data, leading to density-based controllability criteria. The approach is validated through an application to a fractional PDE with discrete nonlocal initial condition, demonstrating existence of a mild solution and approximate controllability under explicit bounds on nonlocal coefficients and data.
Abstract
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.