Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks

This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results

Neural Networks in Engineering Design: Robust Practical Stability Analysis

In recent years, we are witnessing artificial intelligence being deployed on embedded platforms in our everyday life, including engineering design practice problems starting from early stage design ideas to the final decision. One of the most challenging problems is related to the design and implementation of neural networks in engineering design tasks. The successful design and practical applications of neural network models depend on their qualitative properties. Elaborating efficient stability is known to be of a high importance. Also, different stability notions are applied for differently behaving models. In addition, uncertainties are ubiquitous in neural network systems, and may result in performance degradation, hazards or system damage. Driven by practical needs and theoretical challenges, the rigorous handling of uncertainties in the neural network design stage is an essential research topic. In this research, the concept of robust practical stability is introduced for generalized discrete neural network models under uncertainties applied in engineering design. A robust practical stability analysis is offered using the Lyapunov function method. Since practical stability concept is more appropriate for engineering applications, the obtained results can be of a practical significance to numerous engineering design problems of diverse interest.

Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability

Due to the worldwide outbreak of COVID-19, many strategies and models have been put forward by researchers who intend to control the current situation with the given means. In particular, compartmental models are being used to model and analyze the COVID-19 dynamics of different considered populations as Susceptible, Exposed, Infected and Recovered compartments (SEIR). This study derives control-oriented compartmental models of the pandemic, together with constructive control laws based on the Lyapunov theory. The paper presents the derivation of new vaccination and quarantining strategies, found using compartmental models and design methods from the field of Lyapunov theory. The Lyapunov theory offers the possibility to track desired trajectories, guaranteeing the stability of the controlled system. Computer simulations aid to demonstrate the efficacy of the results. Stabilizing control laws are obtained and analyzed for multiple variants of the model. The stability, constructivity, and feasibility are proven for each Lyapunov-like function. Obtaining the proof of practical stability for the controlled system, several interesting system properties such as herd immunity are shown. On the basis of a generalized SEIR model and an extended variant with additional Protected and Quarantined compartments, control strategies are conceived by using two fundamental system inputs, vaccination and quarantine, whose influence on the system is a crucial part of the model. Simulation results prove that Lyapunov-based approaches yield effective control of the disease transmission.

Calculation of the optimal parameters of corrective elements in induction acceleration systems

The formulations of a number of optimization problems for a linear induction acceleration system with respect to the adjustment parameters are considered. The dynamics of the transverse motion of electrons in the horizontal plane is investigated in the presence of given energy values for each resonator period: the particles at the initial moment of time are somewhat displaced relative to the accelerator axis (we neglect the displacements of the ends of the solenoids and the centers of the accelerating gaps relative to the accelerator axis). A connection is established between the initial and final coordinates and the components of the momentum. The presence of parasitic electric and magnetic fields arising as a result of the displacement of particles relative to the axis of the accelerator, which change the transverse components of the pulses, is taken into account. For the mathematical formulation of problems, in order to apply algorithms of practical stability, the original difference model of the induction system was converted to a linear form. By introducing into consideration the vector of parameters, the scatter of phase coordinates, and tolerances on the parameters, the problem of calculating the tolerances for given linear constraints on the scatter of phase coordinates for the corresponding inhomogeneous system is formulated. For the case of nonlinear dynamic constraints on the spread of the vector of phase coordinates, it is proposed to approximate a convex closed set by tangent hyperplanes. Numerical estimation of the range of tolerances for the parameters of correcting elements is reduced to the problems of practical stability of discrete parametric systems. In this case, the region of the initial conditions on the state vector, the tolerances on the parameters, are given structurally in the form of an ellipsoid, which makes it possible to numerically solve the original problem as an extremal one. From the standpoint of practical stability in the corresponding space of functions, the problem of assessing the range of tolerances for the parameters of correcting elements in the presence of specified restrictions on the spread of the quality criterion is considered. Attention is focused on an important class of problems of numerical modelling of a linear induction acceleration system − problems of practical stability. Numerical estimation of the region of initial displacements of the transverse coordinates of the linear induction acceleration system in the given structures in the presence of linear constraints on the vector of phase coordinates in dynamics is carried out. Key words: modeling, induction system of acceleration, solenoid, parameters, elements of correction, optimization, stability

Practical stability of nonlinear time-varying impulsive cascade systems

Practical stability of impulsive time-varying cascade systems is investigated. In this way, we extend some existing results under more generalized assumptions. Examples are given to show the feasibility of our results.

Practical stability for Riemann–Liouville delay fractional differential equations

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.