Asymptotic stability of systems: Results involving the system topology

In this paper we answer the following question for a large class of (linear and nonlinear) dynamical systems. Given is a system with dissipation and given is the associated conservative system. Suppose the associated conservative system is stable. What properties of the system topology (system configuration) will ensure that the overall system with dissipation is asymptotically stable? Both linear and nonlinear (Hamiltonian) systems are treated. For the linear case, necessary and sufficient conditions for asymptotic stability are established, while for the nonlinear case, sufficient conditions and also some necessary and sufficient conditions for asymptotic stability are obtained. It is emphasized that the application of the present results to specific problems will usually not require a search for appropriate Lyapunov functions. Indeed, a stability analysis by the present method involves the following two steps: (a) given a system with dissipation, the stability of its trivial solution (equilibrium) is ascertained by determining the stability of the associated conservative system, i.e., by determining whether the potential energy is a minimum at the equilibrium; and (b) attractivity of the equilibrium of the entire system (with dissipation) is determined from the system topology (system configuration). This approach to stability analysis appears to be new. Furthermore, since the present method involves concepts from control theory (namely, the notion of observability), these results provide further insight into the mechanisms of stability (and stabilization). To provide motivation and to demonstrate the applicability of the results, some specific examples are considered.