Improving LaSalle´s Invariance Principle using Geometric Clues

In this paper, we consider the stability properties of an equilibrium located at the boundary of an open forward invariant set. This is motivated by biological systems because they are nonnegative systems in nature and it is often the case that the equilibrium of interest has some zero component (so that it is at the boundary of the nonnegative orthant). Since the equilibrium is at the boundary, a non-isolated equilibrium can be asymptotically stable with respect to the positive orthant, which is emphasized in this paper. Then, we assert that some equilibrium at the boundary that is asymptotically stable in the usual sense does not admit a Lyapunov function, which may sound like a contradiction to the Lyapunov converse theorem. In fact, this is due to the non-uniform convergence rate around the boundary of forward invariant set. Inspired by the fact that some asymptotically stable equilibrium may not admit the Lyapunov function, we consider the well-known LaSalle´s principle for proving the asymptotic stability. Since it turns out that it is not straightforward, we then provide a modified LaSalle´s theorem, which incorporates the direction of vector fields around the boundary of forward invariant set