Exact Controllability for Nonlinear Implicit Caputo Fractional Models with Hybrid Structure
Abstract
Existing research on fractional hybrid differential equations (abbrev. FHDEs) has largely focused on results concerning the existence, uniqueness or stability of solutions, often under restrictive assumptions and relying on contraction mappings with constants less than one or auxiliary conditions. However, the controllability of nonlinear hybrid implicit fractional systems with control inputs has only been partially addressed. In this paper, we advance the theory by analysing a generalised class of Caputo FHDEs that include explicit control functions. Our contributions are threefold. Firstly, we establish an equivalence theorem that clarifies the precise notion of a solution, forming the basis for all subsequent arguments. Secondly, we prove the existence of solutions under fixed control using Schauder's fixed-point theorem, with auxiliary constructions supported by the Banach contraction principle. Third, we demonstrate exact controllability under assumptions that are strictly weaker than those required in earlier works. These results broaden the class of admissible systems and highlight the flexibility of fixed-point techniques beyond conventional approaches. A physical example illustrates the applicability of the theoretical findings.
Keywords
Controllability, Existence of solution, Fractional hybrid differential equation
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