Controllability of infinite‐dimensional conformable <i>Θ</i> ‐ <i>H</i> ‐systems

Abstract This paper extends the concept of a generator to ‐ ‐semigroups and investigates solutions to the ‐ ‐systems in infinite‐dimensional spaces. Additionally, the study delves into the homogeneous problem and analyzes conformable fractional calculus and ‐ ‐semigroups. The research investigates the uniqueness and existence of a solution to the ‐ ‐systems. Furthermore, the current paper discusses the controllability of ‐ ‐systems governed by infinite‐dimensional ‐ ‐differential equations. By stating the controllability map, we provide necessary and sufficient conditions of the exact controllability for the ‐ ‐systems. Finally, some examples are given to illustrate our theoretical results.