This paper aims to design Observer-Based (OB) controllers that ensure the exponential stability of a class of nonlinear time-delay systems. The nonlinear part of the system satisfies a weak Quasi-One-Sided Lipschitz (QOSL) condition characterized by the matrices $({\mathcal{L}}_1,{\mathcal{M}}_1,{\mathcal{N}}_1)$, as well as a QOSL condition characterized by the matrices $({\mathcal{L}}_2,{\mathcal{M}}_2,{\mathcal{N}}_2)$. First, we derive a sufficient condition formulated as a Linear Matrix Inequality (LMI) via a Lyapunov–Krasovskii (LK) functional. The main advantage of this design is that the controller and observer gains are computed in a single step. However, its main drawback is that the matrices ${\mathcal{L}}_1$, ${\mathcal{M}}_1$, and ${\mathcal{N}}_1$ are fixed rather than treated as decision variables. To overcome this limitation, we propose an improved design in which the matrices ${\mathcal{L}}_1,{\mathcal{M}}_1$, and ${\mathcal{N}}_1$ are treated as decision-variable matrices with a fixed structure. By using an appropriate decoupling technique, this approach provides greater flexibility in the selection of matrices and reduces conservatism. The efficacy of the developed OB controllers is demonstrated via a suitable numerical example.

This study investigates how to control chaos in discrete-time systems that have variable fractional orders. We examine a general type of discrete fractional-order equations where the order changes with time, and we show that these systems can become chaotic when certain parameters are chosen. To address this, we develop and apply tailored control techniques to suppress chaos and achieve system stabilization. Using detailed numerical simulations, we confirm that the suggested control method works effectively in two example cases. Our findings underscore that chaos control in variable fractional-order systems provides significant flexibility in modulating dynamic behavior, offering valuable insights into the broader applicability of these methods in discrete fractional-order systems.

In this research, we present a novel discrete fractional-order model designed to simulate computer virus propagation. We performed a thorough dynamical analysis, encompassing phase portrait visualization, bifurcation diagram construction, maximal Lyapunov exponent computation, and equilibrium point stability assessment using the basic reproduction number ( $R_0$). To characterize system complexity and validate chaotic dynamics, we employed Approximate Entropy, $C_0$ Complexity, and Permutation Entropy. Furthermore, control and synchronization methodologies were developed to mitigate chaotic behavior and achieve coordinated system dynamics. The findings proved the efficacy of the proposed fractional model in accurately simulating viral spread and illustrated the considerable implications of fractional-order parameters on system dynamics. In order to validate the results, MATLAB simulations were run.