Forward Invariance Lemmas for Control Problems with Convergence to Lyapunov-unstable Sets

Details Forward Invariance Lemmas for Control Problems with Convergence to Lyapunov-unstable Sets Forward Invariance Lemmas for Control Problems with Convergence to Lyapunov-unstable Sets

2009-01-22

arxiv.org/abs/0901.3577v4

Mathematics

Dynamical Systems

Scientific paper

We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. These characterizations allow one to derive families of forward invariant sets such that members of these families, on the one hand, include the invariant set as a subset, but on the other hand they are not necessarily neighborhoods of the invariant set. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable or critically stable ones. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control. In addition to providing estimates of the domains of forward invariance for this class of systems the results also allow one to derive domains of initial conditions corresponding to solutions leaving a given neighborhood of the origin at least once. In contrast to other works addressing convergence issues in unstable systems, our results require neither availability of the input-output characterizations for the stable compartment nor estimates of its convergence rates. We also do not restrict the class of systems to the ones in which almost all trajectories converge to an attractor. The results are illustrated with examples, including the analysis of phase synchronization of neural oscillators with heterogenous coupling, and the synthesis of adaptive control laws for systems with nonlinear parameterized uncertainties.

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