Abstract. 
The theory of non-equilibrium steady states is fairly well
understood for the case of 1-dimensional systems of masses connected
by harmonic and an-harmonic springs. In this talk I discuss the
extensions to a set of more complicated networks of connections. These
include $n$-dimensional slabs which are connected to heat baths at
their ends. Under an explicit genericity assumptions on the couplings,
arbitrary networks can be treated.
The arguments are based on controllability and conditions on the
spring potentials at infinity. I will focus my talk on the
controllability. This is work with Noe Cuneo.