Controllability of Variable-Order Conformable Systems

Abstract

We study controllability of semilinear evolution systems with infinite memory governed by a generalized conformable derivative of variable order. Since the time-change reduction to semigroups fails, we construct a two-parameter evolution family and the corresponding integral representation in a Banach space. Under Lipschitz and Hale-Kato hypotheses we prove existence and uniqueness of mild solutions, introduce a variable-order controllability Gramian, and characterize exact and approximate controllability. We also obtain a quantitative observability inequality with explicit constants. Approximate controllability in the semilinear case follows from a regularized feedback and a fixed point scheme. An application and a numerical illustration are included.