An iterative method for the low-rank approximate solution of a class of generalized Lyapunov equations is studied. At each iteration, a standard Lyapunov is solved using Galerkin projection with an extended Krylov subspace method. This Lyapunov equation is solved inexactly, thus producing a nonstationary iteration. Several theoretical and computational issues are discussed so as to make the ite...

The discrete-time periodic Lyapunov equation has several important applications in the analysis and design of linear periodic control systems. Specific applications considered in the paper are the solution of stateand output-feedback optimal periodic control problems, the stabilization by periodic state feedback and the square-root balancing of discrete-time periodic systems. Efficient numerica...

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...

Using the method of Lyapunov functionals construction, it is shown that investigation of stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation. Difference equations usually appear by investigation of systems with discrete time or by ...

Lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic expression) of the function may be difficult. Herein we propose an approach to avoid ...

The modified generalized Van der Pol-Mathieu equation is generalization of the equation that is investigated by authors Momeni et al. (2007), Veerman and Verhulst (2009) and Kadeřábek (2012). In this article the Bautin bifurcation of the autonomous system associated with the modified generalized Van der Pol-Mathieu equation has been proved. The existence of limit cycles is studied and the Lyapu...

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...

We generalized a direct method for generalized Lyapunov matrix equation, using proper orthogonal decomposition (POD). Such equations arise in model reduction of descriptor systems.