On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valued derivative” of vector Lyapunov functions is introduced, some generalized comparison principles on discontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.

Feedback stabilization for a class of discontinuous systems driven by integrator

This paper investigates the feedback stabilization problem for a class of discontinuous systems which is characterized by Filippov differential inclusion. Lyapunov-based backstepping design method is generalized with nons- mooth Lyapunov functions to solve the control problem. A set-valued time derivative is introduced first for nonsmooth function along discontinuous vector fields, which enables us to perform Lyapunov-based design with nondifferentiable Lyapunov function. Conditions for designing a virtual control law which is shown nondifferentiable in general in the re- cursive design problem are proposed. Finally, as a special case, piecewise linear system is discussed to demonstrate the application of the presented design approach.

Uniform Invariance Principle of Discontinuous Systems with Parameter Variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.

State feedback stabilization of cascaded nonlinear systems with discontinuous connection

In practical engineering, many phenomena are described as a discontinuous function of a state variable, and the discontinuity is usually the main reason for the degradation of the control performance. For example, in the set-point control problem of mechanical systems, the static friction （described by a sgn function of velocity of the contacting faces） causes undesired positioning error. In this paper, we will investigate the stabilization problem for a class of nonlinear systems that consist of two subsystems with cascaded connection. We will show the basic idea with a special case first, and then the result will be extended to more general cases. Some interesting numerical examples will be given to demonstrate the effectiveness of the proposed design approach.

Class of numerical methods for differential-algebraic systems with discontinuous right-hand sides

Numerical methods for Differential-Algebraic systems with discontinuous right-hand sides is discussed. A class of continuous Rosenbrock methods are constructed, and numerical experiments show that the continuous Rosenbrock methods are effective. Applying the methods, a fast and high-precision numerical algorithm is given to deal with typical discontinuous parts, which occur frequently in differential-algebraic systems(DAS).

一类不连续系统Φ-有界变差解对参数的连续依赖性（英文）

The functions of bounded Φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper, by using Henstock-Kurzweil integral, we establish theorems of continuous dependence of bounded Φ-variation solutions on parameter for a class of discontinuous systems on the base of Φ-function. These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.

Investigating the effect of pyroptosis on the slow CD4+T cell depletion in HIV-1 infection,by dynamical analysis of its discontinuous mathematical model

HIV infection is one of the most serious causes of death throughout the world.CD4+T cells which play an important role in immune protection,are the primary targets for HIV infection.The hallmark of HIV infection is the progressive loss in population of CD4+T cells.However,the pathway causing this slow T cell decline is poorly understood[16].This paper studies a discontinuous mathematical model for HIV-1 infection,to investigate the effect of pyroptosis on the disease.For this purpose,we use the theory of discontinuous dynamical systems.In this way,we can better analyze the dynamical behavior of the HIV-1 system.Especially,considering the dynamics of the system on its discontinuity boundary enables us to obtain more comprehensive results rather than the previous researches.A stability region for the system,corresponding to its equilibria on the discontinuity boundary,will be determined.In such a parametric region,the trajectories of the system will be trapped on the discontinuity manifold forever.It is also shown that in the obtained stability region,the disease can lead to a steady state in which the population of uninfected T cells and viruses will preserve at a constant level of cytokines.This means that the pyroptosis will be restricted and the disease cannot progress for a long time.Some numerical simulations based on clinical and experimental data are given which are in good agreement with our theoretical results.

On the Well-posedness of the PWM Control System

One of the basic issues in the study of hybrid systems is the well-posedness(existence and uniqueness of solutions)problem of discontinuous dynamical systems.This paper addresses this problem for a class of piecewise affine discontinuous systems with affine inequalities such as systems with pulse-width modulator under the definition of Carathéodory solutions in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions.Furthermore,it is clear that when carrier signal h(t)=0,closed-loop pulse-width modulation(PWM)DC–DC converters are not well posed,and when some condition is satisfied,the closed-loop PWM DC–DC converters with a P controller are well posed.