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Duality Theory in the Stability Analysis of Systems with Polynomial Vector Fields


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, 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.


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