Converse Lyapunov functions for exponentially stable periodic orbits
The authors consider an autonomous system in \(\mathbb{R}^n\), (S) \(\dot x= f(x)\), with a periodic orbit \(\eta\). First they provide a special set of local coordinates \((\rho, \theta)\) in a neighbourhood of \(\eta\), which highlight the local transversal behaviour of the flow. Then they associate to (S) a linear \(\theta\)-periodic system (P) in the \(\rho\) coordinates. They prove that \(\eta\) is exponentially stable for (S) if and only if the origin is asymptotically stable for (P). Under the hypothesis that \(\eta\) is exponentially stable for (S), they use the Lyapunov equation associated to (P) to produce a Lyapunov function \(V(x)\) that proves the exponential stability of \(\eta\). Finally, they provide an example in which this result is used to stabilize a periodic orbit of an affine control system.
scientific article; zbMATH DE number 640408