In this paper Kharitonov's t h e o m for the robust stability of interval polynomials is proved using the second method of Lyapunov. The Hermite matrix is taken as the matrix of the quadratic form which is used as a Lyapunov function to prove Hurwitz stability. It is shown that if the four Hemite matrices correspondii to the four Kharitonov extreme polynomials are positive definite, the Hermite...

In the paper presents a brief survey of some new developments in Lyapunov’s approach including the generalized perturbation equation and its applications; the use of nonanalytic Lyapunov functions; an extension of the Barbashin-Krasovskii theorem related to asymptotic stability assured by a Lyapunov function with nonpositive derivative; the consistency condition for a time-space mosaic that con...

The Lyapunov stability theorem is used to derive a control law that can be used to control the spacing between vehicles in a platoon. A third order system is adopted to model the vehicle and power-train dynamics. In addition, the concept of "Expected Spacing Error" is introduced and used to form a Lyapunov function. Then a control law that always decreases the Lyapunov function is selected. A p...

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods tha...

The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...

A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent invest...

This paper investigates the nano-positioning control for a piezoelectric actuator system with states time-delay via multiple Lyapunov functions. The multiple Lyapunov functions can relax stabilization conditions derived by the traditional single Lyapunov function. First, the Bouc-Wen equation establishes the mathematical model of a piezoelectric actuator system. To reduce the nonlinear hysteres...

The first Fundamental Theorem of Asset Pricing establishes the equivalence between the absence of arbitrage in financial markets and the existence of Equivalent Martingale Measures, if appropriate conditions hold. Since the theorem may fail when dealing with infinitely many trading dates, this paper draws on the A.A. Lyapunov Theorem in order to retrieve the equivalence for complete markets suc...