This paper generalizes the well-known Lyapunov-type inequality for certain higher order fractional differential equations. The investigation is based on a construction of Green's functions and finding its corresponding maximum value. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
A controller based on the curve tracking algorithm is offered as a solution to the issue of the wheeled mobile robot’s trajectory not be precisely monitored. First, Utilizing a planar global coordinate system, the kinematics and dynamics models of the wheeled mobile robot is created. Then, the kinematic model of the actual and predicted trajectories is utilized to determine the trajectory tracking error of the wheeled mobile robot. By the trajectory tracking error system model, The curve tracking algorithm controller is employed in the outer loop control to remove the robot’s position and attitude deviation during trajectory tracking, while the PI controller is utilized in the inner loop control to precisely track the robot’s speed. The simulation results demonstrate that the suggested control technique can regulate the wheeled mobile robot’s pose error [xq yq θq]T within the ranges of 0.1m, 0.03m, and 0.03rad, which demonstrates the efficacy of the control approach.
In this paper, the asymptotic behavior of a multigroup SEIR model with stochastic perturbations and nonlinear incidence rate functions is studied. First, the existence and uniqueness of the solution to the model we discuss are given. Then, the global asymptotical stability in probability of the model with R0<1 is established by constructing Lyapunov functions. Next, we prove that the disease can die out exponentially under certain stochastic perturbation while it is persistent in the deterministic case when R0>1. Finally, several examples and numerical simulations are provided to illustrate the dynamic behavior of the model and verify our analytical results.
A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.
In this paper, we investigate the problem of fixed-time stabilization (FTS) for a class of second-order nonlinear time-delay systems. Using the fixed-time stability theory and Lyapunov-Krasovskii functional, a design scheme of fixed-time stability for time-delay systems is proposed. This scheme effectively solves the adverse impact of time-delay on the system, and and ensure that the corresponding closed-loop system is stable for a fixed time. The main benefit of the proposed control methods is that the influences of time-delay state and nonlinear uncertainties can be effectively attenuated with the help of feedback method. Finally, the simulation is given to show the effectiveness of the design scheme.
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