Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria
The paper deals with stability and asymptotic stability of time-varying systems of ordinary differential equations \[ \dot x= f(x,t). \] It is well known that the origin is uniformly stable if there is a Lyapunov function \(V\) which is positive definite, decrescent, and whose derivative \(\dot V\) is negative semidefinite. The authors are interested in conditions which guarantee asymptotic stability, but which are weaker than the usual negative definiteness assumption about \(\dot V\). It is shown that this problem can be reinterpreted as an observability problem, the observation map \(h\) being related to \(V\) by \(| h(x,t)|^2= -\dot V(x, t)\). The observability achievements require positiveness of the integrals \(\int^{t+ T}_t| h|^2\) for some interval length \(T\). For certain classes of systems, by this way it is possible to reduce the stability analysis to systems of simpler form. Some illustrative examples are discussed in the last section.
scientific article; zbMATH DE number 1161158
scientific article