The objective of this work was to investigate a class of Hilfer-type fractional differential equations governed by the interaction of semigroup operators, history-dependent mechanisms, and probability density functions, with particular attention to their analytical and control properties. First, the existence of mild solutions was established through semigroup theory and fixed-point arguments tailored to fractional dynamics. We then addressed the controllability problem for the corresponding $(k,\phi )$-Hilfer fractional delay differential equation, taking into account the influence of memory and delay effects induced by history-dependent operators. By combining Mönch's fixed-point theorem with the measure of noncompactness, a set of sufficient conditions for controllability was obtained. This approach not only captures the complexity of the system but also deepens the understanding of how fractional-order behavior and past-state dependence affect controllability.

Nonlinear fractional differential equations and chaotic systems can be modeled with variable-order differential operators. We propose a generalized numerical scheme to simulate variable-order fractional differential operators. Fractional calculus' fundamental theorem and Lagrange polynomial interpolation are used. Two methods, Atangana-Baleanu-Caputo and Atangana-Seda derivatives, were used to solve a chaotic Newton-Leipnik system problem with fractional operators. Our scheme examined the existence and uniqueness of the solution. We analyze the model qualitatively using its equivalent integral through an iterative convergence sequence. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.