Dissipation Obstacle hampers Control—by—Interconnection Methodology

It is well—known that the presence of dissipation hampers our ability to shape the energy of port—Hamiltonian systems using control.by.interconnection methods—a phenomenon called the dissipation obstacle. In particular, the Casimir functions that are used to shift the energy function cannot depend on the coordinates where dissipation is present if we use, as is usually the case, passive controllers. Recently, it was proposed to relax the latter condition, namely, to use controllers that inject energy into the system to be able to create the required Casimir functions. In this note we prove that, alas, even if the Casimirs can be created with active controllers the dissipation obstacle stymies the possibility to assign an energy function with the minimum at an equilibrium point. As a corollary we prove that the subtle, deleterious effect of pervasive dissipation does not only stem from the inability of the controller to inject the (infinite) energy required for stabilization—as it was conjectured in earlier publications