Lyapunov, direct method

Until further notice we use stable for what is technically called asymptotically stable in the sense of Lyapunov see, e.g., J. La Salle and S. Lefshetz, Stability by Liapunov s Direct Method (Academic Press, New York 1961). 

An analysis of chemical reactor stability and control-VIII The direct method of Lyapunov. Introduction and applications to simple reactions in stirred vessels (with R.B. Warden and N.R. Amundson). Chem. Eng. ScL 19, 149-172 (1963). 

Oliver, W. K. Seborg, D. E. and Fisher, D.G., "Model Reference Adaptive Control Based on Lyapunov s Direct Method," Chem. Engr. Comm., 1973, 1, 125. 

In the case of nonlinear systems that cannot be reduced to a linearized system, stability is much more difficult to assess. Lyapunov s direct method [1] requires a suitable energy function to be found. Often, only numerical time integration gives an indication of the dynamic behaviour and stability that cannot be proven otherwise. 

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

J. La Salle and S. Lefschetz, Stability by Lyapunov s Direct Method, Academic Press, New York 1961. 

S. Junco. Stability Analysis and Stabilizing Control Synthesis via Lyapunov s Second Method Directly of Bond Graphs on Nonhnear Systems. In Proceedings of IECON 93, volume 3, pages 2065-2069, IEEE, 1993. 

WOLF and SWIFT [36,37] have recently developed a method for computing the non-negative portion of the Lyapunov spectrum directly from a time series. The method does not require the construction of a map from the d ata, and in fact even when the data are well described by a one-dimensional map, the Wolf and Swift method yields exponent values that are more robust than those obtained from a map. (The difficulty with the map arises in part because ln f (X.) is very sensitive to the procedure used to determine the derivative of the map.)... 

---