We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...

Periodic Lyapunov differential equations can be used to formulate robust optimal periodic control problems for nonlinear systems. Typically, the added Lyapunov states are discretized in the same manner as the original states. This straightforward technique fails to guarantee conservation of positive-semidefiniteness of the Lyapunov matrix under discretization. This paper describes a discretizat...

This paper proposes a stabilizing controller for single machine power system with nonzero conductance of transmission line. The proposed controller uses a reduced order observer to estimate the power angle of the system. The stability of the reduced order observer and that of the closed loop system incorporating the observer based stabilizer are given based on Lyapunov direct method. Simulation...

This paper presents an original practical criterion of global stability analysis of nonlinear polynomial systems. This criterion derived from the application of the Lyapunov direct method with a quadratic function generalizes the famous Lyapunov stability condition for linear systems. Useful mathematical transformations have allowed the formulation of the obtained conditions as an LMI (Linear M...

The Lyapunov theory of dynamical systems is the most useful general theory for studying the stability of nonlinear systems. It includes two methods, Lyapunov's indirect method and Lyapunov's direct method. Lyapunov's indirect method states that the dynamical system ˙ x = f (x), (1) where f (0) = 0, has a locally exponentially stable equilibrium point at the origin, if and only if the real parts...

We study the stability properties of a class of time-varying nonlinear systems. We assume that nonstrict input-to-state stable (ISS) Lyapunov functions for our systems are given. We posit a mild nondegeneracy condition on our given Lyapunov functions under which strict ISS Lyapunov functions for our systems are also known to exist. However, the known existence result is based on value function ...

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