Symmetry Properties and Their Application to Hilfer Fractional Systems

The paper investigates semilinear Hilfer fractional systems. A symmetric fractional derivative and its properties are discussed. A symmetrized model for these systems is proposed and examined. A bounded nonlinear function f is applied, depending on the time as well as on the state. The Laplace transformation is used to derive the solution formula for the systems under consideration. The primary contribution of the paper is the formulation and proof of controllability criteria for symmetrized Hilfer systems. To deepen the understanding of the dynamics of such systems, the concept of reflection symmetries is introduced with a detailed analysis of their essential features, including projection functions and a reflection operator. Furthermore, a decomposition of the symmetric Hilfer fractional derivative is presented, utilizing the projection function and reflection operator. This decomposition not only provides a controllability condition for symmetrized Hilfer systems but also clarifies the relationship between the system’s trajectory across subintervals. Two illustrative examples are presented to demonstrate the computational and practical significance of the theoretical results.