Proving Asymptotic Stability with LaSalle’s Invariance Principle: On the Automatic Computation of Invariant Sets Using Quantifier Elimination
Daniel Gerbet, Klaus Röbenack · 2020
Lyapunov’s second method is the most commonly used stability test for nonlinear systems. While Lyapunov’s approach requires that the time-derivative of an energy-like function is strictly positive or negative, the generalisation by LaSalle allows the sign to become zero. With the latter theorem convergence to an invariant set contained in the zero-set of the time-derivative can be shown. In this contribution we propose a method to compute such an invariant set for polynomial systems using algebraic geometry. This is done by extending the concept of Lie derivatives on polynomial ideals. The proposed method is illustrated on some simple examples.