Convergence-to-Zero and Guaranteed-Cost Synchronization of Caputo–Hadamard Fractional-Order Systems with a Time-Varying Delay

In this paper, the convergence-to-zero and finite-horizon guaranteed-cost synchronization criteria are developed for linear Caputo–Hadamard fractional-order systems with an admissible time-varying delay. The delay assumption is expressed in such a way that is consistent with the Caputo–Hadamard Halanay inequality and the memory structure of the model, which is logarithmic in time. This analysis includes a quadratic Caputo–Hadamard Lyapunov method, Schur-complement bounds for the delayed channel and a supremum argument in logarithmic time. An important novelty in the proposed approach is that the current state, the delayed state and the Caputo–Hadamard derivative are not considered as independent augmented variables; this prevents the structural feasibility obstacle from occurring when using full-space residual LMI formulations. The convergence-to-zero condition is first established for the drive system. Next, a fixed-gain guaranteed-cost synchronization theorem is established and, by using a standard change of variables, a convex controller-synthesis condition is arrived at. An explicit logarithmic-time form of the finite-horizon cost estimate is derived. The criteria are further extended to systems with several admissible delays and to systems with norm-bounded parametric uncertainty. Four numerical examples are reported, in which feasible matrices, the controller gain, strict eigenvalue margins and a comparison of the simulated cost and the theoretical upper bound are given, together with a quantitative comparison against augmented-state linear matrix inequality formulations, a scalability study up to a dimension of 30 and a sensitivity study. The simulations show the dynamics that the theory predicts; the convergence to the asymptotics is valid for the LMI certificates checked in the simulations and for the Caputo–Hadamard Halanay inequality. In conclusion, the paper delivers a complete and numerically verifiable design chain for Caputo–Hadamard synchronization: admissibility of a possibly unbounded time-varying delay is checked directly, a stabilizing gain is obtained from a convex program whose largest block has size 2n instead of 3n, and an a priori cost certificate JT* is produced from the same feasible variables; on the reported benchmark, the method retains 95.7% of the admissible delay-channel gain of an augmented-state formulation while solving up to 21 times faster at dimension 30.