An example of a gas system which can be destabilized by an integrable perturbation

Abstract

A construction of a globally asymptotically stable time-invariant system which can be destabilized by some integrable perturbation is given. Besides its intrinsic interest, this serves to provide counterexamples to an open question regarding Lyapunov functions.

Bounded-Energy-Input Convergent-State Property of Dissipative Nonlinear Systems: An iISS Approach

“…On the one hand, even in the simplest of nonlinear systems satisfying the latter hypothesis, the BEICS property may fail to hold. In [16], Sontag and Krichman construct an example of a 0-GAS system of the form _ x = f 0 (x) + u with the property that, for every " > 0, there is an integrable function, with L 1 norm kuk 1 < ", such that the system admits an unbounded solution: subsequently, in [17], Teel and Hespanha provide an example of a system of similar structure, but with the stronger property of 0-GES (that is, 0 is a globally exponentially stable equilibrium of _ x = f0(x)) for which an exponentially decaying additive input u, arbitrarily small in L p , can give rise to an unbounded solution. On the other hand, if _ x = f(x; u), with f : n 2 m !…”

On the robustness analysis of triangular nonlinear systems: iISS and practical stability

“…This transient peaking suffices to generate unbounded solutions. Similarly, as shown in [42,47], neither integrability nor even exponential decay of the solutions of the driving subsystem is sufficient to preserve global asymptotic stability in general.…”

A Separation Property of Dissipative Observers for Nonlinear Systems

“…Note that if W (x) is a Lyapunov function for f (x), then so is L (W (x)) for any K ∞ function L (·). However, this condition is too weak since, as shown in [24], [25], there are systems that are GAS and that can be destabilized by an arbitrarily small (in L 1 ) perturbation. To avoid this situation, inequality (18) is introduced.…”

“…As such, this condition is not restrictive. Moreover, if (18) is satisfied, systemẋ = f (x) + d (t) is forward complete for every bounded perturbation d (t) [26], and that x (t) → 0 as t → ∞ when d (t) ∈ L 1 [24]. The results of [24], [25] show that, in general, condition (18) cannot be relaxed.…”