Lyapunov equations arise in several areas of control, including stability theory and the study of the rms behavior of systems. Vx xtpx xn i1 xn j1 x ix jp ij where p pt is a real matrix. Lyapunov functionals and stability of stochastic difference equations is primarily addressed to experts in stability theory but will also be of use in the work of pure and computational mathematicians and researchers using the ideas of optimal control to study economic, mechanical and biological systems. Nov, 2018 defining stability using lyapunov functions energy functions duration. Lyapunov s direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. We denote by the stable set and by the unstable set of the theorem states that is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of at.
Rothblum, a proof of the convexity of the range of a nonatomic vector measure using linear inequalities, linear algebra and its applications, volume 199, supplement 1, 1 march 1994, pages 373379. Lyapunov theorem for beginners mathematics stack exchange. Choose a web site to get translated content where available and see local events and offers. The general problem of stability of motion en ingles primera edicion. Files are available under licenses specified on their description page. Based on your location, we recommend that you select. Lyapunov stability is named after aleksandr mikhailovich lyapunov, a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university in 1892. The method is a generalization of the idea that if there is some measure of energy in a system, then. Lyapunov criteria for uniform convergence of conditional. On characterizations of the inputtostate stability pro perty.
Unidad 3 estabilidad 1 by pedro daniel lara maldonado. Outline autonomous systemsinvariance principlelinear system and linearization lyapunov and lasalle appl. All structured data from the file and property namespaces is available under the creative commons cc0 license. Pdf lyapunov based stability analysis for metro lines. Most proofs of lyapunovs theorem i know are highly nonconstructive. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by. Distributed under a creative commons attribution 4.