Analytic resolvent approach to $\Psi$-fractional control systems with linear closed operators

Abstract

UDC 517.9; 517.97; 517.98

We investigate the existence of mild solutions and the approximate controllability of a class of $\Psi$-fractional evolution control systems governed by a closed linear operator generating a compact analytic resolvent family. Combining the $\Psi$-Laplace transform technique with the theory of analytic resolvent operators and leveraging the key continuity properties in the uniform operator topology, we derive explicit variation-of-constants formulas and establish new sufficient conditions guaranteeing the existence of mild solutions via Schauder's fixed-point theorem. The approximate controllability is then analyzed through a regularized cost functional minimization approach, leading to a precise characterization in terms of  strong convergence of the resolvent operator $\lambda R(\lambda, \Lambda_b) \to 0$ as $\lambda \to 0^+.$ A concrete application to a $\Psi$-fractional partial differential equation with  Laplacian under Dirichlet boundary conditions in $L^2[0,\pi]$ is provided, whereas the resolvent is expressed explicitly through the Mittag–Leffler functions and the approximate controllability condition is verified analytically.