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A theorem of the alternative for the existence of a Lyapunov function

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, global stability of an equilibrium point is proved. In practice one only considers functions of a certain class (e.g., polynomials) and parameterizes the candidate function accordingly. The problem is then posed as a feasibility problem: if it is feasible, stability has been proved. However, if the problem is infeasible, no firm conclusion about stability can be drawn. 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. Elements in the dual space have a natural interpretation in terms of occupation measures of system trajectories and can be used to provide a certificate of infeasibilty of the Lyapunov stability problem. As a special case the existence of a sums of squares (SOS) Lyapunov function for systems described by a polynomial vector fields will be considered.

Type of Seminar:
IfA Seminar
Helfried Peyrl
Jul 05, 2007   11am

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