Relative controllability of neutral semilinear$(k,\chi)$-Hilfer fractional integro-differential systemswith nonlocal initial conditions

Abstract

 This paper investigates the relative controllability of a neutral semilinear $(k,\chi)$-Hilfer fractional dynamical system in which
the derivative acts on a neutral combination of the state, the source term is driven by a $(k,\chi)$-Riemann--Liouville fractional integral
of order $p$, and the initial condition is of nonlocal type. An explicit mild solution is derived via the generalized $(k,\chi)$-Laplace transform, producing a shifted Mittag--Leffler kernel of order $(\alpha+p)/k$. For the linear system, positive definiteness of the controllability Gramian is shown to be necessary and sufficient for relativecontrollability. For the semilinear system, the Banach contraction principle yields a unique controlled trajectory, while Krasnoselskii's theorem provides an existence result under weaker conditions on the source. A numerical example confirms the theoretical findings.