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On the existence of a Lyapunov function - a dual point of view


H. Peyrl

IfA Internal Seminar Series

Lyapunov's second method is one of the most important cornerstones in the study of stability of dynamical systems. The central ingredient of the approach is the search for a so-called "Lyapunov function", a function of the state that decreases monotonically along trajectories. Once such a function is found, stability of an equilibrium point is proved. The question about the existence of a Lyapunov function may be addressed using duality theory, a well-known concept in functional analysis and convex optimization. Starting with a simple finite state machine as an example, I will show how duality theory is used to derive a theorem of the alternative for the existence of a Lyapunov function. Furthermore, a very natural interpretation of the dual solutions will become apparent. Later on, the presented ideas will be used to motivate a similar result for nonlinear continuous time systems on a compact set.


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