Controllability of fractional backward evolution equations with order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e23"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo linebreak="goodbreak" linebreakstyle="a

This paper investigates the approximate and exact controllability of a class of fractional backward evolution equations with order 1 < α < 2 in Hilbert spaces. First, by employing the Tikhonov-type regularization method to construct an approximate control function, we establish sufficient conditions for the approximate controllability of the system via Schauder ′ s fixed point theorem. Subsequently, the exact controllability of the system is systematically studied under two distinct cases: the compact semigroup case and the noncompact semigroup case. In particular, for the noncompact case, the controllability results are derived by utilizing the Mönch ′ s fixed point theorem combined with the technique of measures of non-compactness. Finally, an example is provided to verify the validity of the proposed theoretical results.