Passivity of Control Systems on Compact Manifolds

Abstract

This paper derives a passivity defined in the neighborhood of a non-degenerate critical point of energy functions of systems on compact manifolds from which analytical solutions cannot be always derived, e.g., systems of nonlinear partial differential equations, but the energy can be regarded as a Morse function. We first show that the change in energy of the systems defined on a closed domain in the neighborhood can be transformed into an energy flow defined on a boundary of the domain in terms of the boundary integrability of Stokes theorem. Next, this boundary power balance is detailed by using the index of the critical points; therefore, it can be considered as the extended passivity.