Arc-length-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria

This paper focuses on the stability analysis of systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit point that are Lyapunov stable. In this paper, we relate convergence and stability to arc length of the orbits. More specifically, we show that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next we derive arc-length-base Lyapunov results for convergence and stability. These results do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the righthand side of the differential equation and the Lyapunov function derivative. The inequality makes it possible to deduce properties of the arc length function and this leads to sufficient conditions for convergence and stability. Finally, we give additional assumptions under which the converses of all the main results hold.