Department of Mathematics, University of Calabar, Calabar, Nigeria
E-mail: inehmichael@unical.edu.ng

Department of Mathematics and Computer Science, Ritman University, Ikot Ekpene, Nigeria
E-mail: attai.kingsley@ritmanuniversity.edu.ng

Department of Mathematics, National Open University of Nigeria, Nigeri
E-mail: cishola@noun.edu.ng

Department of Mathematics, Durban University of Technology, Durban, South Africa
E-mail: adhirm@dut.ac.za

Department of Physics, University of Calabar, Calabar, Nigeria
E-mail: beusibe@unical.edu.ng

Abstract:

This paper develops a novel framework for uniform asymptotic stability (UAS) analysis of Caputo fractional dynamic equations on time scales using vector Lyapunov functions and the newly introduced Caputo fractional delta Dini derivative. Our approach establishes rigorous UAS criteria that guarantee consistent boundedness and uniform convergence of system trajectories, extending classical stability theory to hybrid systems with memory effects. The theoretical advancements unify continuous and discrete analysis through time-scale calculus while incorporating fractional-order dynamics. We demonstrate the practical efficacy of the framework through two applications: stabilization of a fractional-order robotic arm with PD control and load balancing in distributed computing systems. The results provide both theoretical insights into fractional hybrid systems and practical tools for engineering applications where memory effects and mixed continuous-discrete dynamics coexist.

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